https://quanta-study.de/en/formula-sheet 2026-10-16 monthly 0.76 https://quanta-study.de/en/formula-sheet/newton-zweites-gesetz 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/newton-zweites-gesetz-en-light.svg?v=7497d987bb5fc401 A block on a horizontal surface. Normal force N points upward and weight mg downward; they cancel. The resultant external force F_res and acceleration a point to the right. Schematic free-body diagram on a horizontal contact surface: normal force and weight balance. The horizontal net force summarizes the horizontal forces; it is not an additional force. The dashed arrow shows acceleration. https://quanta-study.de/tafelwerk/diagramme/newton-zweites-gesetz-en-dark.svg?v=7497d987bb5fc401 A block on a horizontal surface. Normal force N points upward and weight mg downward; they cancel. The resultant external force F_res and acceleration a point to the right. Schematic free-body diagram on a horizontal contact surface: normal force and weight balance. The horizontal net force summarizes the horizontal forces; it is not an additional force. The dashed arrow shows acceleration. https://quanta-study.de/en/formula-sheet/gravitationsgesetz 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/gravitationsgesetz-en-light.svg?v=44b8ce60cc306376 Diagram: a large sphere m₁ on the left, a small sphere m₂ on the right, from each an equally long force arrow F_G points toward the other, below them a dimension arrow r from centre to centre. Two masses m₁ and m₂ attract each other with equal forces F_G, r is the distance between their centres. https://quanta-study.de/tafelwerk/diagramme/gravitationsgesetz-en-dark.svg?v=44b8ce60cc306376 Diagram: a large sphere m₁ on the left, a small sphere m₂ on the right, from each an equally long force arrow F_G points toward the other, below them a dimension arrow r from centre to centre. Two masses m₁ and m₂ attract each other with equal forces F_G, r is the distance between their centres. https://quanta-study.de/en/formula-sheet/energie-kinetisch 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/energie-kinetisch-en-light.svg?v=c6c1cee9165d49e4 Original normalised parabola Ekin/E₀ = (v/v₀)². Marked points: speeds 1, 2 and 3 give energies 1, 4 and 9. Mass remains constant. At fixed mass, doubling speed gives four times the kinetic energy. v₀ > 0 is a reference speed and E₀ = ½mv₀². https://quanta-study.de/tafelwerk/diagramme/energie-kinetisch-en-dark.svg?v=c6c1cee9165d49e4 Original normalised parabola Ekin/E₀ = (v/v₀)². Marked points: speeds 1, 2 and 3 give energies 1, 4 and 9. Mass remains constant. At fixed mass, doubling speed gives four times the kinetic energy. v₀ > 0 is a reference speed and E₀ = ½mv₀². https://quanta-study.de/en/formula-sheet/energie-potentiell 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/energie-potentiell-en-light.svg?v=44e60e7daaf6dd48 Diagram: a body of mass m sits at height h above the ground, a dimension arrow marks its centre-of-mass height h, an arrow g points straight down. In a uniform gravitational field, Epot = mgh. h measures centre-of-mass height above the chosen zero level; g points downward. https://quanta-study.de/tafelwerk/diagramme/energie-potentiell-en-dark.svg?v=44e60e7daaf6dd48 Diagram: a body of mass m sits at height h above the ground, a dimension arrow marks its centre-of-mass height h, an arrow g points straight down. In a uniform gravitational field, Epot = mgh. h measures centre-of-mass height above the chosen zero level; g points downward. https://quanta-study.de/en/formula-sheet/coulomb-gesetz 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/coulomb-gesetz-en-light.svg?v=7cb7df0c3b56553d Diagram: two positive charges q₁ and q₂ sit at distance r, from each charge a force arrow F points outward, away from the other charge. Two like charges q₁ and q₂ at distance r repel each other with the equal force F. https://quanta-study.de/tafelwerk/diagramme/coulomb-gesetz-en-dark.svg?v=7cb7df0c3b56553d Diagram: two positive charges q₁ and q₂ sit at distance r, from each charge a force arrow F points outward, away from the other charge. Two like charges q₁ and q₂ at distance r repel each other with the equal force F. https://quanta-study.de/en/formula-sheet/lorentzkraft 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/lorentzkraft-en-light.svg?v=6268642c9ef6564a Two charges +e and −e moving right in a field out of the plane. Force on +e points down, on −e up, with equal magnitudes. Transverse speed 300 km/s rightward and B=0.200 T out of the plane give force magnitude 9.61·10⁻¹⁵ N for ±e. Negative charge reverses direction on the right. Purely magnetic power is zero; arrow lengths are schematic. https://quanta-study.de/tafelwerk/diagramme/lorentzkraft-en-dark.svg?v=6268642c9ef6564a Two charges +e and −e moving right in a field out of the plane. Force on +e points down, on −e up, with equal magnitudes. Transverse speed 300 km/s rightward and B=0.200 T out of the plane give force magnitude 9.61·10⁻¹⁵ N for ±e. Negative charge reverses direction on the right. Purely magnetic power is zero; arrow lengths are schematic. https://quanta-study.de/en/formula-sheet/wirkungsgrad-carnot 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/wirkungsgrad-carnot-en-light.svg?v=cd69de0052830fc6 Carnot bound for a 300-kelvin cold reservoir: zero at hot-reservoir temperature 300 kelvin, 0.52 at 625 kelvin and 0.8 at 1500 kelvin; the curve remains below 1. Reversible bound at fixed cold-reservoir temperature TC = 300 K. ηC = 1 − TC/TH rises with TH in kelvin. At TH = 625 K, ηC = 0.52; a different cold reservoir gives a different curve. https://quanta-study.de/tafelwerk/diagramme/wirkungsgrad-carnot-en-dark.svg?v=cd69de0052830fc6 Carnot bound for a 300-kelvin cold reservoir: zero at hot-reservoir temperature 300 kelvin, 0.52 at 625 kelvin and 0.8 at 1500 kelvin; the curve remains below 1. Reversible bound at fixed cold-reservoir temperature TC = 300 K. ηC = 1 − TC/TH rises with TH in kelvin. At TH = 625 K, ηC = 0.52; a different cold reservoir gives a different curve. https://quanta-study.de/en/formula-sheet/de-broglie-wellenlaenge 2026-07-09 monthly 0.82 https://quanta-study.de/en/formula-sheet/photoeffekt-einstein 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/photoeffekt-einstein-en-light.svg?v=336b86c7be9c126f Diagram: E_kin over f as a rising line from the zero W_A/h, a dashed extension down to −W_A on the E_kin axis, and a slope triangle with slope h. The Einstein line: E_kin grows linearly with f with slope h; electrons leave only above the threshold frequency W_A/h, and the extension meets the axis at −W_A. https://quanta-study.de/tafelwerk/diagramme/photoeffekt-einstein-en-dark.svg?v=336b86c7be9c126f Diagram: E_kin over f as a rising line from the zero W_A/h, a dashed extension down to −W_A on the E_kin axis, and a slope triangle with slope h. The Einstein line: E_kin grows linearly with f with slope h; electrons leave only above the threshold frequency W_A/h, and the extension meets the axis at −W_A. https://quanta-study.de/en/formula-sheet/induktion-faraday 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/induktion-faraday-en-light.svg?v=9dbb25dc17bf4d76 Flux per turn rises from zero to 0.9 milliwebers in thirty milliseconds and returns to zero by ninety milliseconds. Emf for eighty turns is first minus 2.4 volts and then plus 1.2 volts. Original pulse example: N = 80 equal turns. Steeper flux rise produces the larger negative emf magnitude. The two volt-second areas cancel; derivative is not unique at ideal corners. OpenStax §§13.1–13.2. https://quanta-study.de/tafelwerk/diagramme/induktion-faraday-en-dark.svg?v=9dbb25dc17bf4d76 Flux per turn rises from zero to 0.9 milliwebers in thirty milliseconds and returns to zero by ninety milliseconds. Emf for eighty turns is first minus 2.4 volts and then plus 1.2 volts. Original pulse example: N = 80 equal turns. Steeper flux rise produces the larger negative emf magnitude. The two volt-second areas cancel; derivative is not unique at ideal corners. OpenStax §§13.1–13.2. https://quanta-study.de/en/formula-sheet/hookes-gesetz 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/hookes-gesetz-en-light.svg?v=abb702f2c4d9828e Force–displacement plot for positive x: line F hold = kx above and line Fx = −kx below the horizontal x axis. The triangular area under holding force is labelled spring energy U. Positive displacement x: holding force rises with slope k; directed spring force Fx = −kx opposes displacement. The triangular area under holding force is U = ½kx². https://quanta-study.de/tafelwerk/diagramme/hookes-gesetz-en-dark.svg?v=abb702f2c4d9828e Force–displacement plot for positive x: line F hold = kx above and line Fx = −kx below the horizontal x axis. The triangular area under holding force is labelled spring energy U. Positive displacement x: holding force rises with slope k; directed spring force Fx = −kx opposes displacement. The triangular area under holding force is U = ½kx². https://quanta-study.de/en/formula-sheet/waermekapazitaet 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/waermekapazitaet-en-light.svg?v=6ce6541994fe48ef Heat versus temperature change: two lines with slopes 3.1365 and 3.3165 kilojoules per kelvin. Both start at zero; at 34 kelvin total heat exceeds water heat by 6.120 kilojoules. Original heating balance with constant appropriate heat capacities: water and vessel need 112.761 kJ over 34 K; water alone needs 106.641 kJ. The vessel contributes 6.120 kJ. https://quanta-study.de/tafelwerk/diagramme/waermekapazitaet-en-dark.svg?v=6ce6541994fe48ef Heat versus temperature change: two lines with slopes 3.1365 and 3.3165 kilojoules per kelvin. Both start at zero; at 34 kelvin total heat exceeds water heat by 6.120 kilojoules. Original heating balance with constant appropriate heat capacities: water and vessel need 112.761 kJ over 34 K; water alone needs 106.641 kJ. The vessel contributes 6.120 kJ. https://quanta-study.de/en/formula-sheet/brechungsgesetz-snellius 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/brechungsgesetz-snellius-en-light.svg?v=71235ea07a941d30 Diagram: a light ray hits an interface at an angle, the dashed normal is perpendicular to it, angle of incidence α₁ above and smaller angle of refraction α₂ in the denser medium below. Refraction at the interface: entering the optically denser medium (n₂ > n₁) the ray bends toward the normal, α₂ < α₁. https://quanta-study.de/tafelwerk/diagramme/brechungsgesetz-snellius-en-dark.svg?v=71235ea07a941d30 Diagram: a light ray hits an interface at an angle, the dashed normal is perpendicular to it, angle of incidence α₁ above and smaller angle of refraction α₂ in the denser medium below. Refraction at the interface: entering the optically denser medium (n₂ > n₁) the ray bends toward the normal, α₂ < α₁. https://quanta-study.de/en/formula-sheet/heisenberg-unschaerfe 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/heisenberg-unschaerfe-en-light.svg?v=283f5754525ad880 Diagram: Δp over Δx, a falling hyperbola Δx·Δp = ℏ/2 separates the shaded forbidden region near the origin from the allowed region above it. Every pair of position uncertainty Δx and momentum uncertainty Δp lies on or above the curve Δx·Δp = ℏ/2; both cannot be made arbitrarily small at once. https://quanta-study.de/tafelwerk/diagramme/heisenberg-unschaerfe-en-dark.svg?v=283f5754525ad880 Diagram: Δp over Δx, a falling hyperbola Δx·Δp = ℏ/2 separates the shaded forbidden region near the origin from the allowed region above it. Every pair of position uncertainty Δx and momentum uncertainty Δp lies on or above the curve Δx·Δp = ℏ/2; both cannot be made arbitrarily small at once. https://quanta-study.de/en/formula-sheet/gauss-gesetz-elektrisches-feld 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/gauss-gesetz-elektrisches-feld-en-light.svg?v=9ff8a17a23c7f1c7 Imaginary Gaussian cube in a rightward uniform field. Flux enters left and leaves right; enclosed charge and net flux are zero. A closed imaginary cube surface in a uniform external field encloses no charge. Fluxes through opposing faces cancel; the field itself is not zero. https://quanta-study.de/tafelwerk/diagramme/gauss-gesetz-elektrisches-feld-en-dark.svg?v=9ff8a17a23c7f1c7 Imaginary Gaussian cube in a rightward uniform field. Flux enters left and leaves right; enclosed charge and net flux are zero. A closed imaginary cube surface in a uniform external field encloses no charge. Fluxes through opposing faces cancel; the field itself is not zero. https://quanta-study.de/en/formula-sheet/erster-hauptsatz-thermodynamik 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/erster-hauptsatz-thermodynamik-en-light.svg?v=a8ba30996a5b9d97 Closed gas releases 180 joules of heat and receives 640 joules of work. A heat arrow points outward and a work arrow inward. Internal energy rises by 460 joules. With unchanged bulk kinetic and external potential energy, heat leaves the closed gas and work enters. In ΔU = Q − W, W counts work by the gas; it is negative in this compression. Arrow lengths are schematic. https://quanta-study.de/tafelwerk/diagramme/erster-hauptsatz-thermodynamik-en-dark.svg?v=a8ba30996a5b9d97 Closed gas releases 180 joules of heat and receives 640 joules of work. A heat arrow points outward and a work arrow inward. Internal energy rises by 460 joules. With unchanged bulk kinetic and external potential energy, heat leaves the closed gas and work enters. In ΔU = Q − W, W counts work by the gas; it is negative in this compression. Arrow lengths are schematic. https://quanta-study.de/en/formula-sheet/ohmsches-gesetz 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/ohmsches-gesetz-en-light.svg?v=f920f13ab0766ee8 Diagram: a line through the origin in the U-I diagram; a slope triangle shows ΔU over ΔI as the resistance R. The voltage is proportional to the current; the slope of the line is the resistance R = U/I. https://quanta-study.de/tafelwerk/diagramme/ohmsches-gesetz-en-dark.svg?v=f920f13ab0766ee8 Diagram: a line through the origin in the U-I diagram; a slope triangle shows ΔU over ΔI as the resistance R. The voltage is proportional to the current; the slope of the line is the resistance R = U/I. https://quanta-study.de/en/formula-sheet/elektrische-leistung 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/elektrische-leistung-en-light.svg?v=3d88d9454c1d1b56 Closed circuit with 18 and 36 ohm parallel branches; signed power bars alongside show minus 12, plus 8 and plus 4 watts. Two parallel loads across 12 V absorb 8 W and 4 W. The source delivers 12 W; its absorbed power is negative. https://quanta-study.de/tafelwerk/diagramme/elektrische-leistung-en-dark.svg?v=3d88d9454c1d1b56 Closed circuit with 18 and 36 ohm parallel branches; signed power bars alongside show minus 12, plus 8 and plus 4 watts. Two parallel loads across 12 V absorb 8 W and 4 W. The source delivers 12 W; its absorbed power is negative. https://quanta-study.de/en/formula-sheet/widerstand-reihen-parallel 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/widerstand-reihen-parallel-en-light.svg?v=9e128715872a1a3a A 3 kilohm upper resistor in series with parallel 6 and 10 kilohm branches; two bars show 8 V unloaded and 6.67 V loaded on the same scale. A 12 V divider with 3 kilohms above and 6 kilohms below is loaded by 10 kilohms. Output falls from 8 V to 20/3 V. https://quanta-study.de/tafelwerk/diagramme/widerstand-reihen-parallel-en-dark.svg?v=9e128715872a1a3a A 3 kilohm upper resistor in series with parallel 6 and 10 kilohm branches; two bars show 8 V unloaded and 6.67 V loaded on the same scale. A 12 V divider with 3 kilohms above and 6 kilohms below is loaded by 10 kilohms. Output falls from 8 V to 20/3 V. https://quanta-study.de/en/formula-sheet/gleichmaessig-beschleunigte-bewegung 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/gleichmaessig-beschleunigte-bewegung-en-light.svg?v=2c497595be54cde8 Original v-t line with positive v₀ and positive constant a. The marked trapezoid gives displacement s=Δx and, in this case, distance travelled. At constant a, v(t) is a straight line. Signed area gives s=Δx. In the illustrated case v is positive, so displacement equals distance travelled. https://quanta-study.de/tafelwerk/diagramme/gleichmaessig-beschleunigte-bewegung-en-dark.svg?v=2c497595be54cde8 Original v-t line with positive v₀ and positive constant a. The marked trapezoid gives displacement s=Δx and, in this case, distance travelled. At constant a, v(t) is a straight line. Signed area gives s=Δx. In the illustrated case v is positive, so displacement equals distance travelled. https://quanta-study.de/en/formula-sheet/geschwindigkeit-beschleunigung 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/geschwindigkeit-beschleunigung-en-light.svg?v=30246699bd0793cd Diagram: a v-t line meets the axis at v₀; a slope triangle shows Δv over Δt as the acceleration a. At constant a, directed velocity changes linearly. The illustrated line has a > 0; its slope is a=Δv/Δt and its intercept is v₀. https://quanta-study.de/tafelwerk/diagramme/geschwindigkeit-beschleunigung-en-dark.svg?v=30246699bd0793cd Diagram: a v-t line meets the axis at v₀; a slope triangle shows Δv over Δt as the acceleration a. At constant a, directed velocity changes linearly. The illustrated line has a > 0; its slope is a=Δv/Δt and its intercept is v₀. https://quanta-study.de/en/formula-sheet/freier-fall 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/freier-fall-en-light.svg?v=60effc0b19fc3b02 Original parabola h=½gt² for a fall from rest at constant g without air resistance. Downward displacement h increases with time. For a fall from rest without air resistance and with constant g, the fallen height h grows quadratically with t. h is downward displacement, not height above the ground. https://quanta-study.de/tafelwerk/diagramme/freier-fall-en-dark.svg?v=60effc0b19fc3b02 Original parabola h=½gt² for a fall from rest at constant g without air resistance. Downward displacement h increases with time. For a fall from rest without air resistance and with constant g, the fallen height h grows quadratically with t. h is downward displacement, not height above the ground. https://quanta-study.de/en/formula-sheet/impuls 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/impuls-en-light.svg?v=298423b026971c62 Original plot of directed momentum components: horizontal axis vx/v₀, vertical axis px/(m·v₀), with positive v₀. A line through the origin runs from the negative to the positive quadrant. For the same mass in a fixed reference frame, px = m·vx. Both components change sign together. Normalised points (−1,−1), (0,0), (1,1) lie on a straight line. https://quanta-study.de/tafelwerk/diagramme/impuls-en-dark.svg?v=298423b026971c62 Original plot of directed momentum components: horizontal axis vx/v₀, vertical axis px/(m·v₀), with positive v₀. A line through the origin runs from the negative to the positive quadrant. For the same mass in a fixed reference frame, px = m·vx. Both components change sign together. Normalised points (−1,−1), (0,0), (1,1) lie on a straight line. https://quanta-study.de/en/formula-sheet/impulserhaltung 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/impulserhaltung-en-light.svg?v=f5fe5b1d838fdd7f Diagram in two rows: at the top the cart m₁ = 2·m₂, drawn twice as wide, moves with v₁ toward the resting cart m₂, at the bottom m₁ continues with v₁' = v₁/3 and m₂ moves off with v₂' = 4/3·v₁. Elastic collision with m₁ = 2·m₂, the velocity arrows are to scale: the total momentum is the same before and after, m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'. https://quanta-study.de/tafelwerk/diagramme/impulserhaltung-en-dark.svg?v=f5fe5b1d838fdd7f Diagram in two rows: at the top the cart m₁ = 2·m₂, drawn twice as wide, moves with v₁ toward the resting cart m₂, at the bottom m₁ continues with v₁' = v₁/3 and m₂ moves off with v₂' = 4/3·v₁. Elastic collision with m₁ = 2·m₂, the velocity arrows are to scale: the total momentum is the same before and after, m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'. https://quanta-study.de/en/formula-sheet/arbeit-mechanisch 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/arbeit-mechanisch-en-light.svg?v=b010606f55c26c98 Diagram: a box is pulled by a force F directed obliquely upward at the angle α, dashed the horizontal component F·cos α, below it the path s. Only the part F·cos α of the force along the path does work: W = F·s·cos α. https://quanta-study.de/tafelwerk/diagramme/arbeit-mechanisch-en-dark.svg?v=b010606f55c26c98 Diagram: a box is pulled by a force F directed obliquely upward at the angle α, dashed the horizontal component F·cos α, below it the path s. Only the part F·cos α of the force along the path does work: W = F·s·cos α. https://quanta-study.de/en/formula-sheet/leistung-mechanisch 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/leistung-mechanisch-en-light.svg?v=5ad461ca78f8ec88 Work–time curve W=40t² with joules and seconds. A secant connects (1 s,40 J) and (3 s,360 J); its slope is 160 W. The steeper tangent at 3 s has slope 240 W. Original example W(t) = 40 J·(t/s)²: average power from 1 s to 3 s is 160 W. The tangent at 3 s gives instantaneous power of 240 W. https://quanta-study.de/tafelwerk/diagramme/leistung-mechanisch-en-dark.svg?v=5ad461ca78f8ec88 Work–time curve W=40t² with joules and seconds. A secant connects (1 s,40 J) and (3 s,360 J); its slope is 160 W. The steeper tangent at 3 s has slope 240 W. Original example W(t) = 40 J·(t/s)²: average power from 1 s to 3 s is 160 W. The tangent at 3 s gives instantaneous power of 240 W. https://quanta-study.de/en/formula-sheet/zentripetalkraft-kreisbewegung 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/zentripetalkraft-kreisbewegung-en-light.svg?v=b02a3f81ad235c3a Diagram: a body of mass m moves on a circular path of radius r; the velocity v is tangential, the force F_z points along the radius to the centre. The centripetal force F_z always points to the centre, perpendicular to the orbital velocity v. https://quanta-study.de/tafelwerk/diagramme/zentripetalkraft-kreisbewegung-en-dark.svg?v=b02a3f81ad235c3a Diagram: a body of mass m moves on a circular path of radius r; the velocity v is tangential, the force F_z points along the radius to the centre. The centripetal force F_z always points to the centre, perpendicular to the orbital velocity v. https://quanta-study.de/en/formula-sheet/harmonische-schwingung-federpendel 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/harmonische-schwingung-federpendel-en-light.svg?v=c95896c9a6b05e84 Diagram: a cosine curve of displacement x over time t; the amplitude A and the period T between two crests are marked. Ideal undamped linear mass-spring model: x is measured from stable equilibrium. The illustrated cosine phase starts at the positive maximum; T = 2π√(m/k). https://quanta-study.de/tafelwerk/diagramme/harmonische-schwingung-federpendel-en-dark.svg?v=c95896c9a6b05e84 Diagram: a cosine curve of displacement x over time t; the amplitude A and the period T between two crests are marked. Ideal undamped linear mass-spring model: x is measured from stable equilibrium. The illustrated cosine phase starts at the positive maximum; T = 2π√(m/k). https://quanta-study.de/en/formula-sheet/wellengleichung 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/wellengleichung-en-light.svg?v=5ee7fc1adde1b73b Spatial sinusoidal profile with wavelength λ between adjacent crests and phase-speed arrow to the right. It does not depict an individual string particle’s motion. Snapshot of a travelling harmonic component: λ is the separation of adjacent crests. The arrow marks phase speed vₚ = λf, not transverse particle motion. https://quanta-study.de/tafelwerk/diagramme/wellengleichung-en-dark.svg?v=5ee7fc1adde1b73b Spatial sinusoidal profile with wavelength λ between adjacent crests and phase-speed arrow to the right. It does not depict an individual string particle’s motion. Snapshot of a travelling harmonic component: λ is the separation of adjacent crests. The arrow marks phase speed vₚ = λf, not transverse particle motion. https://quanta-study.de/en/formula-sheet/dichte 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/dichte-en-light.svg?v=e958bd2eecd71bbe Original schematic cube with side a=4 cm, labelled mass 172.8 g and volume 64 cm³. Below it, density ρ=m/V and 1 cm³=10⁻⁶ m³. Original homogeneous model cube: a = 4 cm gives V = a³ = 64 cm³. With m = 172.8 g, ρ = 2.7 g/cm³ = 2700 kg/m³. The volume conversion factor is cubed. https://quanta-study.de/tafelwerk/diagramme/dichte-en-dark.svg?v=e958bd2eecd71bbe Original schematic cube with side a=4 cm, labelled mass 172.8 g and volume 64 cm³. Below it, density ρ=m/V and 1 cm³=10⁻⁶ m³. Original homogeneous model cube: a = 4 cm gives V = a³ = 64 cm³. With m = 172.8 g, ρ = 2.7 g/cm³ = 2700 kg/m³. The volume conversion factor is cubed. https://quanta-study.de/en/formula-sheet/druck 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/druck-en-light.svg?v=c48013f0c2b6213d A 150 N force acts on a horizontal surface at 60° from the dashed normal. Components show 75 N normal and about 129.9 N tangential. The correct quotient uses 75 N and 3 cm², not total force. 150 N at 60° from the normal gives 75 N normal force. Over 3 cm² this produces 250 kPa average normal contact loading. The tangential component does not enter that pressure quotient. https://quanta-study.de/tafelwerk/diagramme/druck-en-dark.svg?v=c48013f0c2b6213d A 150 N force acts on a horizontal surface at 60° from the dashed normal. Components show 75 N normal and about 129.9 N tangential. The correct quotient uses 75 N and 3 cm², not total force. 150 N at 60° from the normal gives 75 N normal force. Over 3 cm² this produces 250 kPa average normal contact loading. The tangential component does not enter that pressure quotient. https://quanta-study.de/en/formula-sheet/zerfallsgesetz 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/zerfallsgesetz-en-light.svg?v=25bad7be6d48b36c Diagram: a falling exponential curve N over t; dashed lines mark N₀/2 at the half-life T½. The number of undecayed nuclei falls exponentially; after the half-life T½ half of them remain. https://quanta-study.de/tafelwerk/diagramme/zerfallsgesetz-en-dark.svg?v=25bad7be6d48b36c Diagram: a falling exponential curve N over t; dashed lines mark N₀/2 at the half-life T½. The number of undecayed nuclei falls exponentially; after the half-life T½ half of them remain. https://quanta-study.de/en/formula-sheet/masse-energie-aequivalenz 2026-07-09 monthly 0.82 https://quanta-study.de/en/formula-sheet/kondensator-kapazitaet 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/kondensator-kapazitaet-en-light.svg?v=efdcba61e079ab6d Charge-voltage plot with two straight lines through the origin. The steeper line represents 100 µF, the shallower 50 µF. Endpoints at 12 V mark 1,200 and 600 µC. At constant C, Q is proportional to U. At 12 V, one plate carries 600 µC for 50 µF or 1,200 µC for 100 µF. The other plate carries the opposite charge. https://quanta-study.de/tafelwerk/diagramme/kondensator-kapazitaet-en-dark.svg?v=efdcba61e079ab6d Charge-voltage plot with two straight lines through the origin. The steeper line represents 100 µF, the shallower 50 µF. Endpoints at 12 V mark 1,200 and 600 µC. At constant C, Q is proportional to U. At 12 V, one plate carries 600 µC for 50 µF or 1,200 µC for 100 µF. The other plate carries the opposite charge. https://quanta-study.de/en/formula-sheet/kondensator-energie 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/kondensator-energie-en-light.svg?v=3e064affe5e5bd4a Linear voltage-charge curve. Only the area between initial charge 28.2 mC and final charge 84.6 mC is shaded as additional energy; it is 6.768 J. An already charged 470 µF capacitor is charged from 60 V to 180 V. Additional stored energy is the trapezoidal area between 28.2 and 84.6 mC: 6.768 J. https://quanta-study.de/tafelwerk/diagramme/kondensator-energie-en-dark.svg?v=3e064affe5e5bd4a Linear voltage-charge curve. Only the area between initial charge 28.2 mC and final charge 84.6 mC is shaded as additional energy; it is 6.768 J. An already charged 470 µF capacitor is charged from 60 V to 180 V. Additional stored energy is the trapezoidal area between 28.2 and 84.6 mC: 6.768 J. https://quanta-study.de/en/formula-sheet/linsengleichung 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/linsengleichung-en-light.svg?v=fa62d1b13ced511a Ray diagram: an upright object on the left at distance g, a converging lens, focal points F; the parallel ray and the central ray form an inverted real image on the right at distance b. Construction at a converging lens: the parallel ray and the central ray meet at the real, inverted image. 1/f = 1/g + 1/b. https://quanta-study.de/tafelwerk/diagramme/linsengleichung-en-dark.svg?v=fa62d1b13ced511a Ray diagram: an upright object on the left at distance g, a converging lens, focal points F; the parallel ray and the central ray form an inverted real image on the right at distance b. Construction at a converging lens: the parallel ray and the central ray meet at the real, inverted image. 1/f = 1/g + 1/b. https://quanta-study.de/en/formula-sheet/schiefer-wurf 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/schiefer-wurf-en-light.svg?v=ce3558088421a5fd Diagram: a projectile parabola from the origin, the launch arrow v₀ at the angle α with its components v₀·cos α and v₀·sin α, at the apex an arrow g pointing down. Ideal projectile parabola at constant g with no other forces. The illustrated special case has equal launch and landing heights; horizontal and vertical motion share the same time. https://quanta-study.de/tafelwerk/diagramme/schiefer-wurf-en-dark.svg?v=ce3558088421a5fd Diagram: a projectile parabola from the origin, the launch arrow v₀ at the angle α with its components v₀·cos α and v₀·sin α, at the apex an arrow g pointing down. Ideal projectile parabola at constant g with no other forces. The illustrated special case has equal launch and landing heights; horizontal and vertical motion share the same time. https://quanta-study.de/en/formula-sheet/auftrieb-archimedes 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/auftrieb-archimedes-en-light.svg?v=c75d43ab945dd8aa Diagram: a block floats partly submerged in a liquid of density ρ_Fl, the submerged part V is shaded, from there the buoyant force F_A points upward, an equally large weight F_G points downward from the center of mass; g is shown separately. The buoyant force F_A equals the weight of the displaced fluid: In the illustrated stationary homogeneous fluid, the block floats without other forces: F_A = F_G; V is displaced liquid volume. https://quanta-study.de/tafelwerk/diagramme/auftrieb-archimedes-en-dark.svg?v=c75d43ab945dd8aa Diagram: a block floats partly submerged in a liquid of density ρ_Fl, the submerged part V is shaded, from there the buoyant force F_A points upward, an equally large weight F_G points downward from the center of mass; g is shown separately. The buoyant force F_A equals the weight of the displaced fluid: In the illustrated stationary homogeneous fluid, the block floats without other forces: F_A = F_G; V is displaced liquid volume. https://quanta-study.de/en/formula-sheet/dopplereffekt 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/dopplereffekt-en-light.svg?v=6e8d537d1e1b06f3 Diagram: three circular waves of a source moving to the right, their centres shifted to the left; on the right the fronts are close together, on the left far apart; on the right stands an observer f_B, below an arrow v_Q. A moving source: ahead of it the wave fronts are closer together, so the observer there hears the higher frequency f_B > f_Q. https://quanta-study.de/tafelwerk/diagramme/dopplereffekt-en-dark.svg?v=6e8d537d1e1b06f3 Diagram: three circular waves of a source moving to the right, their centres shifted to the left; on the right the fronts are close together, on the left far apart; on the right stands an observer f_B, below an arrow v_Q. A moving source: ahead of it the wave fronts are closer together, so the observer there hears the higher frequency f_B > f_Q. https://quanta-study.de/en/formula-sheet/fadenpendel 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/fadenpendel-en-light.svg?v=f497f745869f90aa Diagram: a ball on a string of length l is deflected from the dashed rest position, a dashed arc shows the swing to the opposite end position, next to it an arrow g pointing down. Small-angle schematic at ±8° about vertical equilibrium. l runs from pivot to bob center of mass; T ≈ 2π√(l/g) assumes an ideal undamped simple pendulum and small angles. https://quanta-study.de/tafelwerk/diagramme/fadenpendel-en-dark.svg?v=f497f745869f90aa Diagram: a ball on a string of length l is deflected from the dashed rest position, a dashed arc shows the swing to the opposite end position, next to it an arrow g pointing down. Small-angle schematic at ±8° about vertical equilibrium. l runs from pivot to bob center of mass; T ≈ 2π√(l/g) assumes an ideal undamped simple pendulum and small angles. https://quanta-study.de/en/formula-sheet/reibungskraft 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/reibungskraft-en-light.svg?v=12609d239496acd1 Diagram: a block slides to the right across the floor, the friction force F_R points to the left at the contact surface, the normal force F_N points straight up, μ is marked at the contact surface. The illustration shows rightward sliding relative to stationary ground. Kinetic friction points leftward; its magnitude in the dry model is μkF_N. https://quanta-study.de/tafelwerk/diagramme/reibungskraft-en-dark.svg?v=12609d239496acd1 Diagram: a block slides to the right across the floor, the friction force F_R points to the left at the contact surface, the normal force F_N points straight up, μ is marked at the contact surface. The illustration shows rightward sliding relative to stationary ground. Kinetic friction points leftward; its magnitude in the dry model is μkF_N. https://quanta-study.de/en/formula-sheet/transformator 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/transformator-en-light.svg?v=83274fed43987655 Original transformer sketch with shared core and dotted terminals. Values are RMS magnitudes. The secondary circuit closes through a 12 Ω load; voltages scale 20:1 and currents 1:20. The few drawn loops do not represent 800 or 40 actual turns. Ideal loaded 20:1 model: primary 240 V and 50 mA, secondary 12 V and 1 A. Both sides transfer 12 W to the 12 Ω load. Positive current enters the primary and leaves the matching dotted secondary terminal; windings are schematic. https://quanta-study.de/tafelwerk/diagramme/transformator-en-dark.svg?v=83274fed43987655 Original transformer sketch with shared core and dotted terminals. Values are RMS magnitudes. The secondary circuit closes through a 12 Ω load; voltages scale 20:1 and currents 1:20. The few drawn loops do not represent 800 or 40 actual turns. Ideal loaded 20:1 model: primary 240 V and 50 mA, secondary 12 V and 1 A. Both sides transfer 12 W to the 12 Ω load. Positive current enters the primary and leaves the matching dotted secondary terminal; windings are schematic. https://quanta-study.de/en/formula-sheet/schwingkreis-thomson 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/schwingkreis-thomson-en-light.svg?v=4b37f50c5959b110 Original quantitative curves over t/T₀ from 0 to 1: upper q/Q=cos(2πt/T₀), i/Imax=−sin(2πt/T₀); lower EC/E=cos² and EL/E=sin². At ¼ period q=0 and i=−1; at ½ period q=−1 and i=0. Example 8 mH, 50 nF, 12 V: T₀=125.664 μs. Free ideal LC circuit with initially positive charge and zero current. Positive current enters the selected plate. Charge and current are separated by a quarter-period; the two energy fractions repeat after half a charge period and always sum to one. https://quanta-study.de/tafelwerk/diagramme/schwingkreis-thomson-en-dark.svg?v=4b37f50c5959b110 Original quantitative curves over t/T₀ from 0 to 1: upper q/Q=cos(2πt/T₀), i/Imax=−sin(2πt/T₀); lower EC/E=cos² and EL/E=sin². At ¼ period q=0 and i=−1; at ½ period q=−1 and i=0. Example 8 mH, 50 nF, 12 V: T₀=125.664 μs. Free ideal LC circuit with initially positive charge and zero current. Positive current enters the selected plate. Charge and current are separated by a quarter-period; the two energy fractions repeat after half a charge period and always sum to one. https://quanta-study.de/en/formula-sheet/kepler-drittes-gesetz 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/kepler-drittes-gesetz-en-light.svg?v=c4edc44cda729b23 Diagram: an (exaggerated) elliptical orbit with a planet, the central body M at one focus, the semi-major axis a is marked from the centre to the edge. Ellipse of a light companion around M in the unperturbed model. a is the semimajor axis; for m ≪ M, T²/a³ approximates 4π²/(GM). https://quanta-study.de/tafelwerk/diagramme/kepler-drittes-gesetz-en-dark.svg?v=c4edc44cda729b23 Diagram: an (exaggerated) elliptical orbit with a planet, the central body M at one focus, the semi-major axis a is marked from the centre to the edge. Ellipse of a light companion around M in the unperturbed model. a is the semimajor axis; for m ≪ M, T²/a³ approximates 4π²/(GM). https://quanta-study.de/en/formula-sheet/drehmoment 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/drehmoment-en-light.svg?v=f83241c0b52dfd97 Diagram: a lever of length r about a pivot axis, at its end the force F acts at the angle θ, dashed the perpendicular component F·sin θ, at the axis a turning arrow M. r is distance to the application point; perpendicular component F sinθ produces torque magnitude rFsinθ. The perpendicular lever arm is d = r sinθ. https://quanta-study.de/tafelwerk/diagramme/drehmoment-en-dark.svg?v=f83241c0b52dfd97 Diagram: a lever of length r about a pivot axis, at its end the force F acts at the angle θ, dashed the perpendicular component F·sin θ, at the axis a turning arrow M. r is distance to the application point; perpendicular component F sinθ produces torque magnitude rFsinθ. The perpendicular lever arm is d = r sinθ. https://quanta-study.de/en/formula-sheet/bernoulli-gleichung 2026-10-01 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/bernoulli-gleichung-en-light.svg?v=4d5f28c4d179f4d5 Diagram: a horizontal pipe with a constriction, a short arrow v₁ in the wide part and an arrow v₂ four times as long in the constriction; the standpipe above the wide part (H₁) is filled higher than the one above the constriction (H₂). Steady, incompressible, ideally lossless circular-pipe flow at a common axis elevation. Halving diameter quadruples speed; the lower standpipe level indicates a smaller piezometric head and lower axis pressure. https://quanta-study.de/tafelwerk/diagramme/bernoulli-gleichung-en-dark.svg?v=4d5f28c4d179f4d5 Diagram: a horizontal pipe with a constriction, a short arrow v₁ in the wide part and an arrow v₂ four times as long in the constriction; the standpipe above the wide part (H₁) is filled higher than the one above the constriction (H₂). Steady, incompressible, ideally lossless circular-pipe flow at a common axis elevation. Halving diameter quadruples speed; the lower standpipe level indicates a smaller piezometric head and lower axis pressure. https://quanta-study.de/en/formula-sheet/doppelspalt-interferenz 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/doppelspalt-interferenz-en-light.svg?v=4b5c3767f0c709b7 Diagram: a barrier with two slits at distance d, parallel rays leave both slits at the angle α_k, a dashed perpendicular cuts off the segment k·λ from the lower ray. The rays from the two slits differ in path by d·sin α_k; a maximum appears when this equals a whole multiple k·λ. https://quanta-study.de/tafelwerk/diagramme/doppelspalt-interferenz-en-dark.svg?v=4b5c3767f0c709b7 Diagram: a barrier with two slits at distance d, parallel rays leave both slits at the angle α_k, a dashed perpendicular cuts off the segment k·λ from the lower ray. The rays from the two slits differ in path by d·sin α_k; a maximum appears when this equals a whole multiple k·λ. https://quanta-study.de/en/formula-sheet/elektrisches-feld 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/elektrisches-feld-en-light.svg?v=845e2444d4a31597 Positive left and negative right plates, field arrows pointing right and electron force left. Potential drops by 360 V toward the right. In the idealized interior plate region, 360 V across 6 mm gives a field of 60 kV/m to the right. Force on an electron acts leftward. Plate geometry is schematic. https://quanta-study.de/tafelwerk/diagramme/elektrisches-feld-en-dark.svg?v=845e2444d4a31597 Positive left and negative right plates, field arrows pointing right and electron force left. Potential drops by 360 V toward the right. In the idealized interior plate region, 360 V across 6 mm gives a field of 60 kV/m to the right. Force on an electron acts leftward. Plate geometry is schematic. https://quanta-study.de/en/formula-sheet/spezifischer-widerstand 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/spezifischer-widerstand-en-light.svg?v=a3833ccdcae6f44c Closed loop with 18 m outgoing and return conductors, load and source. The graph below shows total loop resistance against prescribed temperature. In the supplied linear model, resistance of both wires together rises from 0.864 ohm at 20 °C to 1.03248 ohm at 70 °C. Loop length is 36 m. https://quanta-study.de/tafelwerk/diagramme/spezifischer-widerstand-en-dark.svg?v=a3833ccdcae6f44c Closed loop with 18 m outgoing and return conductors, load and source. The graph below shows total loop resistance against prescribed temperature. In the supplied linear model, resistance of both wires together rises from 0.864 ohm at 20 °C to 1.03248 ohm at 70 °C. Loop length is 36 m. https://quanta-study.de/en/formula-sheet/winkelgeschwindigkeit-kreisfrequenz 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/winkelgeschwindigkeit-kreisfrequenz-en-light.svg?v=e7b1415b71dc2354 Diagram: a rotating disc with a rotation arrow ω, below it a dimension arrow r from the centre to the rim. At the rim a velocity arrow v, at half the radius an arrow v/2 half as long, both tips on a dashed straight line through the centre. All points on the rigid disk share angular velocity. Tangential speed is v = |ωz|r and grows in proportion to distance from the fixed axis. https://quanta-study.de/tafelwerk/diagramme/winkelgeschwindigkeit-kreisfrequenz-en-dark.svg?v=e7b1415b71dc2354 Diagram: a rotating disc with a rotation arrow ω, below it a dimension arrow r from the centre to the rim. At the rim a velocity arrow v, at half the radius an arrow v/2 half as long, both tips on a dashed straight line through the centre. All points on the rigid disk share angular velocity. Tangential speed is v = |ωz|r and grows in proportion to distance from the fixed axis. https://quanta-study.de/en/formula-sheet/stefan-boltzmann-strahlung 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/stefan-boltzmann-strahlung-en-light.svg?v=c4d1ed696717b0de Diagram: P over T as a very steeply rising curve, dashed guide lines show that P rises from P₁ at T₁ to 2·P₁ at 1.19·T₁. The radiated power P grows with T⁴: just 19 % more temperature doubles P. https://quanta-study.de/tafelwerk/diagramme/stefan-boltzmann-strahlung-en-dark.svg?v=c4d1ed696717b0de Diagram: P over T as a very steeply rising curve, dashed guide lines show that P rises from P₁ at T₁ to 2·P₁ at 1.19·T₁. The radiated power P grows with T⁴: just 19 % more temperature doubles P. https://quanta-study.de/en/formula-sheet/magnetfeld-spule 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/magnetfeld-spule-en-light.svg?v=40834a287e1acdb9 Finite axial field of a short air-core solenoid falls from 2.355 mT at centre to 1.400 mT at an end. Long approximation 3.016 mT is drawn as a horizontal comparison. Thin uniformly wound air-core solenoid: N=300, length 0.100 m, radius 0.0400 m, current 0.800 A. Finite axial expression gives 2.355 mT at centre and 1.400 mT at z=±0.050 m. Long comparison value 3.016 mT is not the finite interior-field curve. https://quanta-study.de/tafelwerk/diagramme/magnetfeld-spule-en-dark.svg?v=40834a287e1acdb9 Finite axial field of a short air-core solenoid falls from 2.355 mT at centre to 1.400 mT at an end. Long approximation 3.016 mT is drawn as a horizontal comparison. Thin uniformly wound air-core solenoid: N=300, length 0.100 m, radius 0.0400 m, current 0.800 A. Finite axial expression gives 2.355 mT at centre and 1.400 mT at z=±0.050 m. Long comparison value 3.016 mT is not the finite interior-field curve. https://quanta-study.de/en/formula-sheet/spannenergie-feder 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/spannenergie-feder-en-light.svg?v=053cae0978b26892 Diagram: a straight line through the origin shows the spring force D·s over the displacement s, the shaded triangle beneath it is the elastic energy E_Spann = ½·D·s². Quasistatic external stretching force Ds grows linearly from force-free natural length. Triangular area gives stored energy U = ½Ds² with U(0) = 0. https://quanta-study.de/tafelwerk/diagramme/spannenergie-feder-en-dark.svg?v=053cae0978b26892 Diagram: a straight line through the origin shows the spring force D·s over the displacement s, the shaded triangle beneath it is the elastic energy E_Spann = ½·D·s². Quasistatic external stretching force Ds grows linearly from force-free natural length. Triangular area gives stored energy U = ½Ds² with U(0) = 0. https://quanta-study.de/en/formula-sheet/selbstinduktion-spule 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/selbstinduktion-spule-en-light.svg?v=d3cfe7fb092a07dc Upper current trace rises 0→1 A, stays at 1 A and falls to 0.5 A. Lower passive voltage is +50,0 and−50 V over the open intervals. Ideal constant L=0.100 H: current rises linearly during 0–2 ms, is constant during 2–5 ms and falls linearly during 5–6 ms. Passive voltage is +50 V,0 V,−50 V; emf has opposite sign. Derivative is not uniquely defined at corners. Final current remains 0.500 A. https://quanta-study.de/tafelwerk/diagramme/selbstinduktion-spule-en-dark.svg?v=d3cfe7fb092a07dc Upper current trace rises 0→1 A, stays at 1 A and falls to 0.5 A. Lower passive voltage is +50,0 and−50 V over the open intervals. Ideal constant L=0.100 H: current rises linearly during 0–2 ms, is constant during 2–5 ms and falls linearly during 5–6 ms. Passive voltage is +50 V,0 V,−50 V; emf has opposite sign. Derivative is not uniquely defined at corners. Final current remains 0.500 A. https://quanta-study.de/en/formula-sheet/rc-zeitkonstante 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/rc-zeitkonstante-en-light.svg?v=5b4f230277c3c07f Exponential rise from 1 V toward a horizontal 5 V asymptote. Markers at one and two time constants show 3.5285 and 4.4587 V; after five time constants, 5 V is still not reached exactly. The capacitor starts at 1 V and approaches 5 V with τ = 103.4 ms. After τ, 63.2% of the initial 4 V deviation has decayed. https://quanta-study.de/tafelwerk/diagramme/rc-zeitkonstante-en-dark.svg?v=5b4f230277c3c07f Exponential rise from 1 V toward a horizontal 5 V asymptote. Markers at one and two time constants show 3.5285 and 4.4587 V; after five time constants, 5 V is still not reached exactly. The capacitor starts at 1 V and approaches 5 V with τ = 103.4 ms. After τ, 63.2% of the initial 4 V deviation has decayed. https://quanta-study.de/en/formula-sheet/bewegungsinduktion-motional-emk 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/bewegungsinduktion-motional-emk-en-light.svg?v=6f038cf3dbe2909b For a rotating radial rod, the motional integrand Bωr rises linearly from zero to 1.296 volts per metre at 0.24 metres. Triangle area gives emf 0.15552 volts; the tip value does not apply along the entire rod. Original radial-rod example: B = 0.300 T, ω = 18.0 rad/s, L = 0.240 m. B and ω point along +z; integration runs from axis to tip. Triangle area: +0.15552 V. Tip value times full length would incorrectly double it. OpenStax §13.3, MIT 8.02 Ch. 10. https://quanta-study.de/tafelwerk/diagramme/bewegungsinduktion-motional-emk-en-dark.svg?v=6f038cf3dbe2909b For a rotating radial rod, the motional integrand Bωr rises linearly from zero to 1.296 volts per metre at 0.24 metres. Triangle area gives emf 0.15552 volts; the tip value does not apply along the entire rod. Original radial-rod example: B = 0.300 T, ω = 18.0 rad/s, L = 0.240 m. B and ω point along +z; integration runs from axis to tip. Triangle area: +0.15552 V. Tip value times full length would incorrectly double it. OpenStax §13.3, MIT 8.02 Ch. 10. https://quanta-study.de/en/formula-sheet/stehende-welle-saite 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/stehende-welle-saite-en-light.svg?v=8f7bfa5feca3b821 A standing string mode with three half-waves and nodes at both fixed ends and between them; the length L and node spacing λ₃/2 are marked. Snapshot of the third ideal mode with two fixed ends: L = 3λ₃/2. Nodes stay fixed while displacements change in time. The dashed profile shows the opposite phase half a period later. https://quanta-study.de/tafelwerk/diagramme/stehende-welle-saite-en-dark.svg?v=8f7bfa5feca3b821 A standing string mode with three half-waves and nodes at both fixed ends and between them; the length L and node spacing λ₃/2 are marked. Snapshot of the third ideal mode with two fixed ends: L = 3λ₃/2. Nodes stay fixed while displacements change in time. The dashed profile shows the opposite phase half a period later. https://quanta-study.de/en/formula-sheet/photoeffekt-gegen-spannung 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/photoeffekt-gegen-spannung-en-light.svg?v=49b9226a8b7018b5 Graph of stopping-potential magnitude U_g versus light frequency f: below f₀ there is no photoelectric effect in the single-photon model; from f₀ a line with slope h/e begins. Above the threshold frequency f₀, the stopping-potential magnitude rises linearly with slope h/e. https://quanta-study.de/tafelwerk/diagramme/photoeffekt-gegen-spannung-en-dark.svg?v=49b9226a8b7018b5 Graph of stopping-potential magnitude U_g versus light frequency f: below f₀ there is no photoelectric effect in the single-photon model; from f₀ a line with slope h/e begins. Above the threshold frequency f₀, the stopping-potential magnitude rises linearly with slope h/e. https://quanta-study.de/en/formula-sheet/gedaempfte-harmonische-schwingung 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/gedaempfte-harmonische-schwingung-en-light.svg?v=f6194b8ce3fa84f3 Damped cosine between two exponential envelopes. A₀ denotes initial envelope magnitude; displacement and its actual extrema depend on phase. Underdamped linear model with chosen zero phase. Envelope ±A₀e⁻ᵝᵗ decays exponentially; the depicted motion does not start from rest. Instantaneous total energy is not generally E₀e⁻²ᵝᵗ. https://quanta-study.de/tafelwerk/diagramme/gedaempfte-harmonische-schwingung-en-dark.svg?v=f6194b8ce3fa84f3 Damped cosine between two exponential envelopes. A₀ denotes initial envelope magnitude; displacement and its actual extrema depend on phase. Underdamped linear model with chosen zero phase. Envelope ±A₀e⁻ᵝᵗ decays exponentially; the depicted motion does not start from rest. Instantaneous total energy is not generally E₀e⁻²ᵝᵗ. https://quanta-study.de/en/formula-sheet/generator-induktionsspannung 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/generator-induktionsspannung-en-light.svg?v=2467947fb1e72035 Normalised cosine and sine curves over one period. Dashed flux starts at plus one, blue emf starts at zero and rises. Time marks 0, T/4, T/2, 3T/4 and T. Uniform rotation with θ₀ = 0: normalised flux follows cos(ωt), while normalised emf follows sin(ωt). At maximum flux the emf is zero; a quarter period later it reaches its positive maximum. https://quanta-study.de/tafelwerk/diagramme/generator-induktionsspannung-en-dark.svg?v=2467947fb1e72035 Normalised cosine and sine curves over one period. Dashed flux starts at plus one, blue emf starts at zero and rises. Time marks 0, T/4, T/2, 3T/4 and T. Uniform rotation with θ₀ = 0: normalised flux follows cos(ωt), while normalised emf follows sin(ωt). At maximum flux the emf is zero; a quarter period later it reaches its positive maximum. https://quanta-study.de/en/formula-sheet/hydrostatischer-druck 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/hydrostatischer-druck-en-light.svg?v=1656c18b74456c80 Liquid column with a free surface and depth marker h; sideways pressure arrows grow longer with depth. The hydrostatic pressure difference grows with depth h: Δp = ρgh; p₀ is absolute pressure at the free surface. The pressure arrows show the depth-proportional increment Δp. https://quanta-study.de/tafelwerk/diagramme/hydrostatischer-druck-en-dark.svg?v=1656c18b74456c80 Liquid column with a free surface and depth marker h; sideways pressure arrows grow longer with depth. The hydrostatic pressure difference grows with depth h: Δp = ρgh; p₀ is absolute pressure at the free surface. The pressure arrows show the depth-proportional increment Δp. https://quanta-study.de/en/formula-sheet/kontinuitaetsgleichung-stroemung 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/kontinuitaetsgleichung-stroemung-en-light.svg?v=3820b78b9ae68bb1 Circular pipe with halved diameter: left drawn height80 and right40 units. Mean speed on the right is four times larger, with four-times-longer arrows. Diameters and constant volume flows are labelled. Equal density, steady flow and circular sections: D₁ = 2D₂ gives A₁ = 4A₂ and v̄₂ = 4v̄₁. Velocity arrows are drawn in ratio 1:4; Q₁ = Q₂. https://quanta-study.de/tafelwerk/diagramme/kontinuitaetsgleichung-stroemung-en-dark.svg?v=3820b78b9ae68bb1 Circular pipe with halved diameter: left drawn height80 and right40 units. Mean speed on the right is four times larger, with four-times-longer arrows. Diameters and constant volume flows are labelled. Equal density, steady flow and circular sections: D₁ = 2D₂ gives A₁ = 4A₂ and v̄₂ = 4v̄₁. Velocity arrows are drawn in ratio 1:4; Q₁ = Q₂. https://quanta-study.de/en/formula-sheet/traegheitsmoment-punktmassen 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/traegheitsmoment-punktmassen-en-light.svg?v=20bdd5b5c6907c32 Three particles on a massless frame: 1.20 kg on the axis, 0.60 kg at perpendicular distance 0.20 m and 0.40 kg at 0.35 m. Distances are measured from the axis line. Particles about an axis normal to the drawing: 0.60·0.20² + 0.40·0.35² = 0.073 kg·m². The on-axis particle contributes zero in this model. https://quanta-study.de/tafelwerk/diagramme/traegheitsmoment-punktmassen-en-dark.svg?v=20bdd5b5c6907c32 Three particles on a massless frame: 1.20 kg on the axis, 0.60 kg at perpendicular distance 0.20 m and 0.40 kg at 0.35 m. Distances are measured from the axis line. Particles about an axis normal to the drawing: 0.60·0.20² + 0.40·0.35² = 0.073 kg·m². The on-axis particle contributes zero in this model. https://quanta-study.de/en/formula-sheet/drehimpuls-starrer-koerper 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/drehimpuls-starrer-koerper-en-light.svg?v=451a054a93f057b2 xz-plane section: equal masses on a tilted rod through O. Angular velocity along positive z; angular momentum points up-left, perpendicular to the rod. L and ω are not parallel. Non-principal-axis rotation: ω points along z, but L also has a negative x component. This two-particle body gives L = (−1.44, 0, 1.08) kg·m²/s. L and ω use separate arrow scales. https://quanta-study.de/tafelwerk/diagramme/drehimpuls-starrer-koerper-en-dark.svg?v=451a054a93f057b2 xz-plane section: equal masses on a tilted rod through O. Angular velocity along positive z; angular momentum points up-left, perpendicular to the rod. L and ω are not parallel. Non-principal-axis rotation: ω points along z, but L also has a negative x component. This two-particle body gives L = (−1.44, 0, 1.08) kg·m²/s. L and ω use separate arrow scales. https://quanta-study.de/en/formula-sheet/lineare-waermeausdehnung 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/lineare-waermeausdehnung-en-light.svg?v=7d32f99ef9927b41 Diametral clearance g in millimetres versus temperature change: initially minus 0.040 millimetre, zero at 76.994 kelvin and finally plus 0.011952 millimetre at 100 kelvin. Original free-fit calculation: both separate parts undergo the same temperature change. Diametral clearance reaches zero near 77 K and is about 11.95 µm at 100 K. The small difference is shown magnified. https://quanta-study.de/tafelwerk/diagramme/lineare-waermeausdehnung-en-dark.svg?v=7d32f99ef9927b41 Diametral clearance g in millimetres versus temperature change: initially minus 0.040 millimetre, zero at 76.994 kelvin and finally plus 0.011952 millimetre at 100 kelvin. Original free-fit calculation: both separate parts undergo the same temperature change. Diametral clearance reaches zero near 77 K and is about 11.95 µm at 100 K. The small difference is shown magnified. https://quanta-study.de/en/formula-sheet/latente-waerme-phasenwechsel 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/latente-waerme-phasenwechsel-en-light.svg?v=83d0b59265cae862 Temperature versus supplied heat: ice starts at minus 12 degrees Celsius, reaches zero after 3.762 kilojoules and melts at constant temperature until 53.712 kilojoules. Water then warms to 25 degrees at total heat 69.3945 kilojoules. Original balance for 0.150 kg ice near normal atmospheric pressure: 3.762 kJ warms ice, 49.950 kJ melts it at 0 °C and 15.6825 kJ warms water to 25 °C. The plateau assumes a pure substance at equilibrium and fixed pressure. https://quanta-study.de/tafelwerk/diagramme/latente-waerme-phasenwechsel-en-dark.svg?v=83d0b59265cae862 Temperature versus supplied heat: ice starts at minus 12 degrees Celsius, reaches zero after 3.762 kilojoules and melts at constant temperature until 53.712 kilojoules. Water then warms to 25 degrees at total heat 69.3945 kilojoules. Original balance for 0.150 kg ice near normal atmospheric pressure: 3.762 kJ warms ice, 49.950 kJ melts it at 0 °C and 15.6825 kJ warms water to 25 °C. The plateau assumes a pure substance at equilibrium and fixed pressure. https://quanta-study.de/en/formula-sheet/magnetischer-fluss 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/magnetischer-fluss-en-light.svg?v=21caa03f6a5b3cb9 Oriented flux Φ divided by BA is the cosine of angle to the positive surface normal: plus one at zero degrees, zero at ninety and minus one at one hundred eighty. Uniform field, plane area: measure angle to the normal. Oriented flux is +0.5 BA at 60° and −0.5 BA at 120°. Original diagram of the flux definition in OpenStax §13.1. https://quanta-study.de/tafelwerk/diagramme/magnetischer-fluss-en-dark.svg?v=21caa03f6a5b3cb9 Oriented flux Φ divided by BA is the cosine of angle to the positive surface normal: plus one at zero degrees, zero at ninety and minus one at one hundred eighty. Uniform field, plane area: measure angle to the normal. Oriented flux is +0.5 BA at 60° and −0.5 BA at 120°. Original diagram of the flux definition in OpenStax §13.1. https://quanta-study.de/en/formula-sheet/wechselstrom-impedanz-rlc 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/wechselstrom-impedanz-rlc-en-light.svg?v=66f794396c81c593 Two impedance triangles with resistance 24 ohms: at 500 radians per second reactance is plus 24 ohms and magnitude 33.94 ohms; at 250 radians per second reactance is minus 12 ohms and magnitude 26.83 ohms. Original series circuit: R = 24 Ω, L = 80 mH, C = 125 μF. Both axes use 5 px/Ω. Blue: ω = 500 rad/s, voltage leads current by 45°. Dashed: ω = 250 rad/s, voltage lags current by 26.565°. Signed net reactance determines direction, rather than adding positive reactances. Based on OpenStax §15.3. https://quanta-study.de/tafelwerk/diagramme/wechselstrom-impedanz-rlc-en-dark.svg?v=66f794396c81c593 Two impedance triangles with resistance 24 ohms: at 500 radians per second reactance is plus 24 ohms and magnitude 33.94 ohms; at 250 radians per second reactance is minus 12 ohms and magnitude 26.83 ohms. Original series circuit: R = 24 Ω, L = 80 mH, C = 125 μF. Both axes use 5 px/Ω. Blue: ω = 500 rad/s, voltage leads current by 45°. Dashed: ω = 250 rad/s, voltage lags current by 26.565°. Signed net reactance determines direction, rather than adding positive reactances. Based on OpenStax §15.3. https://quanta-study.de/en/formula-sheet/rotationsenergie-starrer-koerper 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/rotationsenergie-starrer-koerper-en-light.svg?v=335334668de755b4 Rolling-disk energy balance: an upper bar shows 13.2435 joules available gravitational energy. The equally long lower bar splits two to one into translation and rotation about the centre of mass. Uniform disk from the original rolling example: M = 3.00 kg, R = 0.200 m, h = 0.450 m. Without slipping or losses, 8.829 J translation plus 4.4145 J rotation equals Mgh = 13.2435 J. Bars use the same energy scale. https://quanta-study.de/tafelwerk/diagramme/rotationsenergie-starrer-koerper-en-dark.svg?v=335334668de755b4 Rolling-disk energy balance: an upper bar shows 13.2435 joules available gravitational energy. The equally long lower bar splits two to one into translation and rotation about the centre of mass. Uniform disk from the original rolling example: M = 3.00 kg, R = 0.200 m, h = 0.450 m. Without slipping or losses, 8.829 J translation plus 4.4145 J rotation equals Mgh = 13.2435 J. Bars use the same energy scale. https://quanta-study.de/en/formula-sheet/entropie-reversibel-isotherm 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/entropie-reversibel-isotherm-en-light.svg?v=f9b5d8e964653b24 Two paths from 2 to 5 litres at equal final temperature. Top: free expansion, initially gas occupying two fifths with vacuum beside it. Bottom: reversible comparison. Entropy changes are equal but transferred heats differ. The same ideal-gas endpoints at 340 K: free expansion in an isolated rigid vessel has Q = W = 0. The reversible isothermal comparison gives Qrev = Wrev ≈ 1942.6 J. Both paths have ΔS ≈ 5.7135 J/K. https://quanta-study.de/tafelwerk/diagramme/entropie-reversibel-isotherm-en-dark.svg?v=f9b5d8e964653b24 Two paths from 2 to 5 litres at equal final temperature. Top: free expansion, initially gas occupying two fifths with vacuum beside it. Bottom: reversible comparison. Entropy changes are equal but transferred heats differ. The same ideal-gas endpoints at 340 K: free expansion in an isolated rigid vessel has Q = W = 0. The reversible isothermal comparison gives Qrev = Wrev ≈ 1942.6 J. Both paths have ΔS ≈ 5.7135 J/K. https://quanta-study.de/en/formula-sheet/magnetfeld-langer-gerader-leiter 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/magnetfeld-langer-gerader-leiter-en-light.svg?v=09a6fe8f13b57530 Long round conductor with uniform current density: normalized field rises linearly inside to one at radius a, then falls inversely outside. At half radius, field is half the surface value; on axis it is zero. Original interior/exterior-model diagram following OpenStax §12.5. B(a) is surface field magnitude. For I = 10.0 A and a = 2.00 mm, B(a) ≈ 1.00 mT. External inverse-distance form does not apply inside; determine circulation separately with the right hand. https://quanta-study.de/tafelwerk/diagramme/magnetfeld-langer-gerader-leiter-en-dark.svg?v=09a6fe8f13b57530 Long round conductor with uniform current density: normalized field rises linearly inside to one at radius a, then falls inversely outside. At half radius, field is half the surface value; on axis it is zero. Original interior/exterior-model diagram following OpenStax §12.5. B(a) is surface field magnitude. For I = 10.0 A and a = 2.00 mm, B(a) ≈ 1.00 mT. External inverse-distance form does not apply inside; determine circulation separately with the right hand. https://quanta-study.de/en/formula-sheet/wechselstrom-effektivwert 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/wechselstrom-effektivwert-en-light.svg?v=2ff1ccfacdfbe0eb Pulse 12 volts for 2 out of 8 milliseconds. Its square is 144 volts squared during the pulse and otherwise zero; mean square 36 volts squared and RMS 6 volts. Signed mean voltage is 3 volts. Original quarter-duty pulse, period 8 ms. Voltage above, its square below; shaded squared area is 288 V²·ms. Dividing by 8 ms gives 36 V², whose square root is 6 V. Dashed lines show mean voltage 3 V and mean square 36 V². Values at individual ideal jump times do not affect the integral. Definition based on MIT 8.02 Ch. 12. https://quanta-study.de/tafelwerk/diagramme/wechselstrom-effektivwert-en-dark.svg?v=2ff1ccfacdfbe0eb Pulse 12 volts for 2 out of 8 milliseconds. Its square is 144 volts squared during the pulse and otherwise zero; mean square 36 volts squared and RMS 6 volts. Signed mean voltage is 3 volts. Original quarter-duty pulse, period 8 ms. Voltage above, its square below; shaded squared area is 288 V²·ms. Dividing by 8 ms gives 36 V², whose square root is 6 V. Dashed lines show mean voltage 3 V and mean square 36 V². Values at individual ideal jump times do not affect the integral. Definition based on MIT 8.02 Ch. 12. https://quanta-study.de/en/formula-sheet/elektrische-potentialenergie-spannung 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/elektrische-potentialenergie-spannung-en-light.svg?v=c4a4f63cd06c61eb Two lines through the origin: positive charge rises with potential difference, negative charge falls. Energy change is normalized to charge magnitude times 200 V. For equally large positive and negative test charges, ΔU = qΔV. Normalized energy changes have opposite slopes; the graph is not a time trajectory. https://quanta-study.de/tafelwerk/diagramme/elektrische-potentialenergie-spannung-en-dark.svg?v=c4a4f63cd06c61eb Two lines through the origin: positive charge rises with potential difference, negative charge falls. Energy change is normalized to charge magnitude times 200 V. For equally large positive and negative test charges, ΔU = qΔV. Normalized energy changes have opposite slopes; the graph is not a time trajectory. https://quanta-study.de/en/formula-sheet/schallintensitaet 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/schallintensitaet-en-light.svg?v=93dd3d6edf22c42b Four equally spaced points on a logarithmic intensity axis: I over I zero equals 1, 10, 100 and 1000 correspond to 0, 10, 20 and 30 decibels. The level rises by 10 dB per intensity decade: β=10 log₁₀(I/I₀), with I₀=10⁻¹² W/m². https://quanta-study.de/tafelwerk/diagramme/schallintensitaet-en-dark.svg?v=93dd3d6edf22c42b Four equally spaced points on a logarithmic intensity axis: I over I zero equals 1, 10, 100 and 1000 correspond to 0, 10, 20 and 30 decibels. The level rises by 10 dB per intensity decade: β=10 log₁₀(I/I₀), with I₀=10⁻¹² W/m². https://quanta-study.de/en/formula-sheet/beugungsgitter-maxima 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/malus-gesetz-polarisation 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/malus-gesetz-polarisation-en-light.svg?v=1d09ed6574a771a7 Four intensity bars on one linear scale: 120, 60, 45 and 11.25. Three filter-axis glyphs show 0°, 30° and 90°; the first halves unpolarized input, then adjacent axis differences apply. 120 W/m² of unpolarized input becomes 60, 45 and 11.25 W/m² after ideal filters at 0°, 30° and 90°. Adjacent axis differences are 30° and 60°. Beam direction remains unchanged. https://quanta-study.de/tafelwerk/diagramme/malus-gesetz-polarisation-en-dark.svg?v=1d09ed6574a771a7 Four intensity bars on one linear scale: 120, 60, 45 and 11.25. Three filter-axis glyphs show 0°, 30° and 90°; the first halves unpolarized input, then adjacent axis differences apply. 120 W/m² of unpolarized input becomes 60, 45 and 11.25 W/m² after ideal filters at 0°, 30° and 90°. Adjacent axis differences are 30° and 60°. Beam direction remains unchanged. https://quanta-study.de/en/formula-sheet/radioaktive-aktivitaet 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kernbindungsenergie-massendefekt 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/zeitdilatation-spezielle-relativitaet 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/relativistische-energie-impuls-beziehung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/winkelgeschwindigkeit-konstante-winkelbeschleunigung 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/winkelgeschwindigkeit-konstante-winkelbeschleunigung-en-light.svg?v=de03e0daa9979d2b Angular velocity against time: line from 12 rad/s at 0 s through zero at 4 s to −3 rad/s at 5 s. Positive triangular area 24 rad, negative area −1.5 rad, net angle 22.5 rad. Constant α = −3 rad/s² passes through rest at t = 4 s. Over 0 to 5 s, signed area is 22.5 rad; total angular travel is 25.5 rad. https://quanta-study.de/tafelwerk/diagramme/winkelgeschwindigkeit-konstante-winkelbeschleunigung-en-dark.svg?v=de03e0daa9979d2b Angular velocity against time: line from 12 rad/s at 0 s through zero at 4 s to −3 rad/s at 5 s. Positive triangular area 24 rad, negative area −1.5 rad, net angle 22.5 rad. Constant α = −3 rad/s² passes through rest at t = 4 s. Over 0 to 5 s, signed area is 22.5 rad; total angular travel is 25.5 rad. https://quanta-study.de/en/formula-sheet/newton-rotation-starrer-koerper 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/newton-rotation-starrer-koerper-en-light.svg?v=b7e090754f3fa8cb Signed net-torque versus angular-acceleration line through zero: from −6 rad/s², −1.920 N·m to +6 rad/s², +1.920 N·m. Rotor example is marked upper right; zero torque means no angular acceleration, not necessarily rest. For the same rotor about a fixed axis with constant I = 0.320 kg·m², τnet = Iα. Sum motor and load torques first: +2.40 − 0.480 = +1.920 N·m gives α = +6 rad/s². https://quanta-study.de/tafelwerk/diagramme/newton-rotation-starrer-koerper-en-dark.svg?v=b7e090754f3fa8cb Signed net-torque versus angular-acceleration line through zero: from −6 rad/s², −1.920 N·m to +6 rad/s², +1.920 N·m. Rotor example is marked upper right; zero torque means no angular acceleration, not necessarily rest. For the same rotor about a fixed axis with constant I = 0.320 kg·m², τnet = Iα. Sum motor and load torques first: +2.40 − 0.480 = +1.920 N·m gives α = +6 rad/s². https://quanta-study.de/en/formula-sheet/rollbedingung-ohne-rutschen 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/rollbedingung-ohne-rutschen-en-light.svg?v=6feb7b166dd666cc Wheel of radius 0.280 m rotating clockwise, with positive z out of the page. In the ground frame, centre velocity is 1.680 m/s rightward; the top arrow is twice as long. Bottom contact velocity is zero. Below, equal translation and rotation arrows cancel. On stationary ground, rotation cancels translation at contact. With R = 0.280 m, ωz = −6 rad/s and vCM = 1.680 m/s, the top point moves at 3.360 m/s; contact velocity is vC = 0. This does not imply aC = 0. https://quanta-study.de/tafelwerk/diagramme/rollbedingung-ohne-rutschen-en-dark.svg?v=6feb7b166dd666cc Wheel of radius 0.280 m rotating clockwise, with positive z out of the page. In the ground frame, centre velocity is 1.680 m/s rightward; the top arrow is twice as long. Bottom contact velocity is zero. Below, equal translation and rotation arrows cancel. On stationary ground, rotation cancels translation at contact. With R = 0.280 m, ωz = −6 rad/s and vCM = 1.680 m/s, the top point moves at 3.360 m/s; contact velocity is vC = 0. This does not imply aC = 0. https://quanta-study.de/en/formula-sheet/massentraegheitsmoment-parallelachsensatz 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/massentraegheitsmoment-parallelachsensatz-en-light.svg?v=6f5c776f85b04d35 Horizontal uniform rod of length ℓ with center of mass at its midpoint. Two parallel axes normal to the drawing pass through its left end and center; their separation is d = ℓ/2. Uniform thin rod: parallel axes through its center of mass and end. With d = ℓ/2, Iend = Mℓ²/12 + Mℓ²/4 = Mℓ²/3. https://quanta-study.de/tafelwerk/diagramme/massentraegheitsmoment-parallelachsensatz-en-dark.svg?v=6f5c776f85b04d35 Horizontal uniform rod of length ℓ with center of mass at its midpoint. Two parallel axes normal to the drawing pass through its left end and center; their separation is d = ℓ/2. Uniform thin rod: parallel axes through its center of mass and end. With d = ℓ/2, Iend = Mℓ²/12 + Mℓ²/4 = Mℓ²/3. https://quanta-study.de/en/formula-sheet/gravitations-potentielle-energie 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/gravitations-potentielle-energie-en-light.svg?v=5cff0e57fa6595f0 Normalised energy increase versus h/R from zero to 0.6. Exact curve x/(1+x) lies below the local approximation line x. At x = 0.25, values 0.20 and 0.25 are marked. Own normalised energy curve: at h/R = 0.25, the exact lift is ΔU/(μm/R) = 0.20. Constant-g₀ approximation gives 0.25, overestimating the energy increase by 25%. The models agree only near h = 0. https://quanta-study.de/tafelwerk/diagramme/gravitations-potentielle-energie-en-dark.svg?v=5cff0e57fa6595f0 Normalised energy increase versus h/R from zero to 0.6. Exact curve x/(1+x) lies below the local approximation line x. At x = 0.25, values 0.20 and 0.25 are marked. Own normalised energy curve: at h/R = 0.25, the exact lift is ΔU/(μm/R) = 0.20. Constant-g₀ approximation gives 0.25, overestimating the energy increase by 25%. The models agree only near h = 0. https://quanta-study.de/en/formula-sheet/kreisbahn-geschwindigkeit 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/kreisbahn-geschwindigkeit-en-light.svg?v=9c183a437e8b918e Two bodies on concentric circular orbits about common center O: M left at radius rM, m right at three times the radius. Tangential velocities are opposite, with magnitude ratio 1:3; separation r is the sum of radii. Two-body circular orbit around common center of mass O, here M = 3m. Radii and speed magnitudes have ratio 1:3; vrel = √(G(M+m)/r) is relative speed, not each body’s speed. https://quanta-study.de/tafelwerk/diagramme/kreisbahn-geschwindigkeit-en-dark.svg?v=9c183a437e8b918e Two bodies on concentric circular orbits about common center O: M left at radius rM, m right at three times the radius. Tangential velocities are opposite, with magnitude ratio 1:3; separation r is the sum of radii. Two-body circular orbit around common center of mass O, here M = 3m. Radii and speed magnitudes have ratio 1:3; vrel = √(G(M+m)/r) is relative speed, not each body’s speed. https://quanta-study.de/en/formula-sheet/fluchtgeschwindigkeit 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/mittlere-kinetische-energie-molekuel 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/mittlere-kinetische-energie-molekuel-en-light.svg?v=5afa3c7a9f29fdd6 Normalised mean translational energy versus normalised temperature. Values 0.5, 1, 1.5 and 2 lie on the same line; T₀ is 300 kelvin. No state at zero kelvin is plotted. At equal temperature, mean translational energy in classical equilibrium is the same for light and heavy particles. Normalisation: T₀ = 300 K and E₀ = 3kBT₀/2; the plotted line covers 150–600 K. https://quanta-study.de/tafelwerk/diagramme/mittlere-kinetische-energie-molekuel-en-dark.svg?v=5afa3c7a9f29fdd6 Normalised mean translational energy versus normalised temperature. Values 0.5, 1, 1.5 and 2 lie on the same line; T₀ is 300 kelvin. No state at zero kelvin is plotted. At equal temperature, mean translational energy in classical equilibrium is the same for light and heavy particles. Normalisation: T₀ = 300 K and E₀ = 3kBT₀/2; the plotted line covers 150–600 K. https://quanta-study.de/en/formula-sheet/innere-energie-einatomiges-idealgas 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/innere-energie-einatomiges-idealgas-en-light.svg?v=1016e6f05bc1de80 Energy–temperature line for 0.80 mole. The 290 and 410 kelvin states are marked, with a difference triangle showing energy change 1.1973 kilojoules. 0.80 mol of a classical monatomic ideal gas: between 290 and 410 K, U rises from 2.8934 to 4.0907 kJ. In a rigid closed vessel without other work or external energy changes, Q = ΔU ≈ 1.1973 kJ. The vertical scale begins at 2 kJ. https://quanta-study.de/tafelwerk/diagramme/innere-energie-einatomiges-idealgas-en-dark.svg?v=1016e6f05bc1de80 Energy–temperature line for 0.80 mole. The 290 and 410 kelvin states are marked, with a difference triangle showing energy change 1.1973 kilojoules. 0.80 mol of a classical monatomic ideal gas: between 290 and 410 K, U rises from 2.8934 to 4.0907 kJ. In a rigid closed vessel without other work or external energy changes, Q = ΔU ≈ 1.1973 kJ. The vertical scale begins at 2 kJ. https://quanta-study.de/en/formula-sheet/rms-geschwindigkeit-gasmolekuel 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/rms-geschwindigkeit-gasmolekuel-en-light.svg?v=df668559ab3b0a45 Normalised Maxwell density. Maximum at x equal to 1, mean speed at 1.1284 and RMS at 1.2247. Separate label leaders identify their closely spaced actual positions. Three distinct characteristics of one Maxwell distribution at equilibrium at rest: cmp is most probable speed, ⟨c⟩ arithmetic mean and crms the square root of its second moment. Normalised density g(x) = 4x²exp(−x²)/√π with x = c/cmp. https://quanta-study.de/tafelwerk/diagramme/rms-geschwindigkeit-gasmolekuel-en-dark.svg?v=df668559ab3b0a45 Normalised Maxwell density. Maximum at x equal to 1, mean speed at 1.1284 and RMS at 1.2247. Separate label leaders identify their closely spaced actual positions. Three distinct characteristics of one Maxwell distribution at equilibrium at rest: cmp is most probable speed, ⟨c⟩ arithmetic mean and crms the square root of its second moment. Normalised density g(x) = 4x²exp(−x²)/√π with x = c/cmp. https://quanta-study.de/en/formula-sheet/kinetische-gastheorie-druck 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/kinetische-gastheorie-druck-en-light.svg?v=30044af0d913769d A particle hits a stationary right wall. Normal component cx reverses sign while tangential component cy stays unchanged. The wall receives positive x momentum 2mcx. Elastic wall collisions transfer normal momentum. The factor 1/3 arises only in the isotropic statistical average, not from one collision’s geometry. Velocities refer to the local gas rest frame. https://quanta-study.de/tafelwerk/diagramme/kinetische-gastheorie-druck-en-dark.svg?v=30044af0d913769d A particle hits a stationary right wall. Normal component cx reverses sign while tangential component cy stays unchanged. The wall receives positive x momentum 2mcx. Elastic wall collisions transfer normal momentum. The factor 1/3 arises only in the isotropic statistical average, not from one collision’s geometry. Velocities refer to the local gas rest frame. https://quanta-study.de/en/formula-sheet/wellengeschwindigkeit-saite 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/wellengeschwindigkeit-saite-en-light.svg?v=d9d4673084dcff33 Slightly curved string element μΔx: equal tension magnitudes FT point outward along left and right tangents. Small transverse components add to downward ΔFy. A separate horizontal arrow marks phase speed, not transverse string motion. Ideal string element with small local slopes: tangential tensions have opposing longitudinal components and a resultant transverse force. Phase speed along the string is v = √(FT/μ); the resultant arrow is schematic. https://quanta-study.de/tafelwerk/diagramme/wellengeschwindigkeit-saite-en-dark.svg?v=d9d4673084dcff33 Slightly curved string element μΔx: equal tension magnitudes FT point outward along left and right tangents. Small transverse components add to downward ΔFy. A separate horizontal arrow marks phase speed, not transverse string motion. Ideal string element with small local slopes: tangential tensions have opposing longitudinal components and a resultant transverse force. Phase speed along the string is v = √(FT/μ); the resultant arrow is schematic. https://quanta-study.de/en/formula-sheet/grenzwinkel-totalreflexion 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/grenzwinkel-totalreflexion-en-light.svg?v=4858baffb8824805 Three separate ray diagrams with angles from the normal: 35° refracts to 59.36°; at 41.81° the limiting direction is tangent; at 50° reflection and a decaying evanescent field remain. The same model interface from n₁ = 1.50 to n₂ = 1.00: below, at and above the critical angle 41.81°. The decay profile is schematic. Above it an evanescent field remains, but no propagating refracted ray. https://quanta-study.de/tafelwerk/diagramme/grenzwinkel-totalreflexion-en-dark.svg?v=4858baffb8824805 Three separate ray diagrams with angles from the normal: 35° refracts to 59.36°; at 41.81° the limiting direction is tangent; at 50° reflection and a decaying evanescent field remain. The same model interface from n₁ = 1.50 to n₂ = 1.00: below, at and above the critical angle 41.81°. The decay profile is schematic. Above it an evanescent field remains, but no propagating refracted ray. https://quanta-study.de/en/formula-sheet/brewster-winkel 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/brewster-winkel-en-light.svg?v=dc51eb44353ae638 Interface with normal, incident ray at 56.66°, s-reflected ray and transmitted ray at 33.34°. The right angle is between reflection and transmission, not incidence and reflection. For the lossless model n₁ = 1.000 → n₂ = 1.520, θB ≈ 56.66°. Reflected and transmitted rays are perpendicular. Rp = 0 applies to p polarization; unpolarized input leaves s reflection. https://quanta-study.de/tafelwerk/diagramme/brewster-winkel-en-dark.svg?v=dc51eb44353ae638 Interface with normal, incident ray at 56.66°, s-reflected ray and transmitted ray at 33.34°. The right angle is between reflection and transmission, not incidence and reflection. For the lossless model n₁ = 1.000 → n₂ = 1.520, θB ≈ 56.66°. Reflected and transmitted rays are perpendicular. Rp = 0 applies to p polarization; unpolarized input leaves s reflection. https://quanta-study.de/en/formula-sheet/rayleigh-kriterium-kreisblende 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/rayleigh-kriterium-kreisblende-en-light.svg?v=6f36004191276515 Two true Bessel-Airy sections centered at q = 0 and 1. Their linear intensity sum is about 0.735 at the midpoint; both centers have total 1. Each first zero reaches the other center. Two equal incoherent Airy point images at Rayleigh separation: each maximum falls on the first minimum of the other. Curves show intensities and their sum, not coherent field amplitudes. For 550 nm and D = 0.200 m, approximately θmin ≈ 3.355 µrad. https://quanta-study.de/tafelwerk/diagramme/rayleigh-kriterium-kreisblende-en-dark.svg?v=6f36004191276515 Two true Bessel-Airy sections centered at q = 0 and 1. Their linear intensity sum is about 0.735 at the midpoint; both centers have total 1. Each first zero reaches the other center. Two equal incoherent Airy point images at Rayleigh separation: each maximum falls on the first minimum of the other. Curves show intensities and their sum, not coherent field amplitudes. For 550 nm and D = 0.200 m, approximately θmin ≈ 3.355 µrad. https://quanta-study.de/en/formula-sheet/plattenkondensator-kapazitaet-geometrie 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/plattenkondensator-kapazitaet-geometrie-en-light.svg?v=c0d3252714cc613d Inverse curve with normalized points: half gap double capacitance, unit gap unit capacitance, double gap half capacitance. At fixed plate area and permittivity, capacitance is inversely proportional to gap. Half the gap doubles capacitance; twice the gap halves it. https://quanta-study.de/tafelwerk/diagramme/plattenkondensator-kapazitaet-geometrie-en-dark.svg?v=c0d3252714cc613d Inverse curve with normalized points: half gap double capacitance, unit gap unit capacitance, double gap half capacitance. At fixed plate area and permittivity, capacitance is inversely proportional to gap. Half the gap doubles capacitance; twice the gap halves it. https://quanta-study.de/en/formula-sheet/magnetische-kraft-leiter 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/magnetische-kraft-leiter-en-light.svg?v=367455e5061ad3a6 Bent conductor: current first travels 0.120 metres right, then 0.160 metres up. Field points out of plane. First force is 0.0900 newtons down, second 0.120 newtons right, resultant 0.150 newtons diagonally down-right. Original example with I = 3.00 A and B = 0.250 T along +z. Wire geometry: 750 px/m; force arrows: 500 px/N. Forces sum to (+0.120, −0.0900) N. Adding wire lengths would incorrectly give 0.210 N. OpenStax §11.4. https://quanta-study.de/tafelwerk/diagramme/magnetische-kraft-leiter-en-dark.svg?v=367455e5061ad3a6 Bent conductor: current first travels 0.120 metres right, then 0.160 metres up. Field points out of plane. First force is 0.0900 newtons down, second 0.120 newtons right, resultant 0.150 newtons diagonally down-right. Original example with I = 3.00 A and B = 0.250 T along +z. Wire geometry: 750 px/m; force arrows: 500 px/N. Forces sum to (+0.120, −0.0900) N. Adding wire lengths would incorrectly give 0.210 N. OpenStax §11.4. https://quanta-study.de/en/formula-sheet/drehmoment-stromschleife 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/drehmoment-stromschleife-en-light.svg?v=a608e0fe0e7d2894 Fixed dipole: normalized orientation potential minus cosine rises from minus one at parallel to plus one at antiparallel alignment. Signed angular-coordinate torque is minus sine and acts towards decreasing angle between zero and one hundred eighty degrees. Original diagram for fixed dipole magnitude in stationary uniform field, following OpenStax §11.5. θ is angle to current-oriented normal. τθ is signed; its magnitude is NIAB sinθ. Parallel alignment has minimum potential and is stable; antiparallel is maximum and unstable, although both torques vanish. https://quanta-study.de/tafelwerk/diagramme/drehmoment-stromschleife-en-dark.svg?v=a608e0fe0e7d2894 Fixed dipole: normalized orientation potential minus cosine rises from minus one at parallel to plus one at antiparallel alignment. Signed angular-coordinate torque is minus sine and acts towards decreasing angle between zero and one hundred eighty degrees. Original diagram for fixed dipole magnitude in stationary uniform field, following OpenStax §11.5. θ is angle to current-oriented normal. τθ is signed; its magnitude is NIAB sinθ. Parallel alignment has minimum potential and is stable; antiparallel is maximum and unstable, although both torques vanish. https://quanta-study.de/en/formula-sheet/elektrische-feldenergiedichte 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/elektrische-feldenergiedichte-en-light.svg?v=c198f65bc10b6cfd Quadratic normalized energy density versus normalized position. Shaded area equals one third; horizontal mean line lies at one third. For the prescribed vacuum profile E(x)=E₀x/L, energy density grows quadratically. Its spatial mean is one third of the endpoint density, not one quarter. https://quanta-study.de/tafelwerk/diagramme/elektrische-feldenergiedichte-en-dark.svg?v=c198f65bc10b6cfd Quadratic normalized energy density versus normalized position. Shaded area equals one third; horizontal mean line lies at one third. For the prescribed vacuum profile E(x)=E₀x/L, energy density grows quadratically. Its spatial mean is one third of the endpoint density, not one quarter. https://quanta-study.de/en/formula-sheet/elektrisches-feld-unendliche-linienladung 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/elektrisches-feld-unendliche-linienladung-en-light.svg?v=2062cea98ee303b6 Positive line in cross-section with outward radial arrows and radius r₀. Below, the normalised curve decreases from one to a quarter as distance ratio increases from one to four. Points at one and two show halving. Original cross-section for λ>0; the line is normal to the plane. With r₀=0.10 m and λ=20 nC/m, E₀≈3595 V/m; at 2r₀ it is E₀/2. The graph shows E/E₀=1/(r/r₀); no field value is assigned on the singular axis. https://quanta-study.de/tafelwerk/diagramme/elektrisches-feld-unendliche-linienladung-en-dark.svg?v=2062cea98ee303b6 Positive line in cross-section with outward radial arrows and radius r₀. Below, the normalised curve decreases from one to a quarter as distance ratio increases from one to four. Points at one and two show halving. Original cross-section for λ>0; the line is normal to the plane. With r₀=0.10 m and λ=20 nC/m, E₀≈3595 V/m; at 2r₀ it is E₀/2. The graph shows E/E₀=1/(r/r₀); no field value is assigned on the singular axis. https://quanta-study.de/en/formula-sheet/maxwell-integralgleichungen 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/maxwell-integralgleichungen-en-light.svg?v=c168698ed67ace3f Idealized magnetic field of a slowly charging rotationally symmetric thin vacuum capacitor: at 3 milliamperes and plate radius 30 millimetres, linear growth to 20 nanoteslas, then inverse-radius decrease. At 10 millimetres 6.667 nanoteslas; at 50 millimetres 12 nanoteslas. Original approximate gap-model curve: R = 30 mm, instantaneous charging current I = 3 mA. Thin gap, rotationally symmetric geometry and excitation, slowly changing fields; fringing and additional wiring contributions neglected. B is a magnitude. Inside, the enclosed displacement-current area fraction matters; outside, the full simplified field term. Based on MIT 6.013 §3.3.1; not an exact general solution for a finite device. https://quanta-study.de/tafelwerk/diagramme/maxwell-integralgleichungen-en-dark.svg?v=c168698ed67ace3f Idealized magnetic field of a slowly charging rotationally symmetric thin vacuum capacitor: at 3 milliamperes and plate radius 30 millimetres, linear growth to 20 nanoteslas, then inverse-radius decrease. At 10 millimetres 6.667 nanoteslas; at 50 millimetres 12 nanoteslas. Original approximate gap-model curve: R = 30 mm, instantaneous charging current I = 3 mA. Thin gap, rotationally symmetric geometry and excitation, slowly changing fields; fringing and additional wiring contributions neglected. B is a magnitude. Inside, the enclosed displacement-current area fraction matters; outside, the full simplified field term. Based on MIT 6.013 §3.3.1; not an exact general solution for a finite device. https://quanta-study.de/en/formula-sheet/poynting-vektor 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/poynting-vektor-en-light.svg?v=d52bc6ec508cd8ef At a fixed point in a wave travelling along +z, normalized x-E and y-B fields share the identical cosine curve. Below, S along +z follows cosine squared, stays nonnegative and has cycle average one half. Axes use separate normalized scales. One travelling sinusoidal plane vacuum wave, shown at the same fixed point against time phase θ = ωt: E along x, B along y, energy along +z. Above, identical normalized cosine fields; below, normalized energy flux as cosine squared, mean one half. Field and energy-flux axes are normalized independently and do not share physical units. Both fields reverse together; directed energy flow remains along +z. Based on OpenStax §16.3. https://quanta-study.de/tafelwerk/diagramme/poynting-vektor-en-dark.svg?v=d52bc6ec508cd8ef At a fixed point in a wave travelling along +z, normalized x-E and y-B fields share the identical cosine curve. Below, S along +z follows cosine squared, stays nonnegative and has cycle average one half. Axes use separate normalized scales. One travelling sinusoidal plane vacuum wave, shown at the same fixed point against time phase θ = ωt: E along x, B along y, energy along +z. Above, identical normalized cosine fields; below, normalized energy flux as cosine squared, mean one half. Field and energy-flux axes are normalized independently and do not share physical units. Both fields reverse together; directed energy flow remains along +z. Based on OpenStax §16.3. https://quanta-study.de/en/formula-sheet/schroedinger-gleichung-zeitabhaengig 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/quantenteilchen-im-kasten 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/quantenteilchen-im-kasten-en-light.svg?v=4f2f5edad871f5d9 Two wavefunctions between walls at x=0 and x=L: ψ1 has one lobe, ψ2 has two opposite lobes and an internal node. Energy levels E1 and E2=4E1 are shown beside them. In an infinite well, ψ(0)=ψ(L)=0 enforces standing states; E₂=4E₁ follows from Eₙ∝n². https://quanta-study.de/tafelwerk/diagramme/quantenteilchen-im-kasten-en-dark.svg?v=4f2f5edad871f5d9 Two wavefunctions between walls at x=0 and x=L: ψ1 has one lobe, ψ2 has two opposite lobes and an internal node. Energy levels E1 and E2=4E1 are shown beside them. In an infinite well, ψ(0)=ψ(L)=0 enforces standing states; E₂=4E₁ follows from Eₙ∝n². https://quanta-study.de/en/formula-sheet/compton-verschiebung 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/compton-verschiebung-en-light.svg?v=d26a4bdac9091fc1 An incident photon of wavelength λ travels from the left toward an electron initially at rest. The scattered photon of wavelength λ′ travels up and right; θ is between the original and new photon directions. The electron recoils. Scattering from an initially resting electron deflects the photon by θ; λ′−λ = h/(mₑc)(1−cos θ). https://quanta-study.de/tafelwerk/diagramme/compton-verschiebung-en-dark.svg?v=d26a4bdac9091fc1 An incident photon of wavelength λ travels from the left toward an electron initially at rest. The scattered photon of wavelength λ′ travels up and right; θ is between the original and new photon directions. The electron recoils. Scattering from an initially resting electron deflects the photon by θ; λ′−λ = h/(mₑc)(1−cos θ). https://quanta-study.de/en/formula-sheet/gewichtskraft 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/gewichtskraft-en-light.svg?v=89b764c46b13235a A body of mass m sits in a gravitational field. A downward arrow marks weight F_G; beside it, g points in the same direction. The relation F_G = m·g appears above. Near Earth’s surface, weight F_G = m·g points downward; its magnitude increases in proportion to mass. https://quanta-study.de/tafelwerk/diagramme/gewichtskraft-en-dark.svg?v=89b764c46b13235a A body of mass m sits in a gravitational field. A downward arrow marks weight F_G; beside it, g points in the same direction. The relation F_G = m·g appears above. Near Earth’s surface, weight F_G = m·g points downward; its magnitude increases in proportion to mass. https://quanta-study.de/en/formula-sheet/wirkungsgrad 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/wirkungsgrad-en-light.svg?v=cabcef7772337711 An energy converter receives E_in. Useful energy E_use leaves to the right, while loss energy E_loss exits downward. Below is η = E_use/E_in. Efficiency η compares useful output energy with input energy; losses remain part of the balance. https://quanta-study.de/tafelwerk/diagramme/wirkungsgrad-en-dark.svg?v=cabcef7772337711 An energy converter receives E_in. Useful energy E_use leaves to the right, while loss energy E_loss exits downward. Below is η = E_use/E_in. Efficiency η compares useful output energy with input energy; losses remain part of the balance. https://quanta-study.de/en/formula-sheet/elektrische-energie 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/elektrische-energie-en-light.svg?v=9b045f823dadb3ca Power-time plot: the first rectangle represents 5184 J, the second 2016 J. Widths have ratio 3 to 7, heights 6 to 1. The 12 W mean line has the same total area. Three minutes at 28.8 W and seven minutes at 4.8 W give 7200 J. Duration-weighted mean power is 12 W. https://quanta-study.de/tafelwerk/diagramme/elektrische-energie-en-dark.svg?v=9b045f823dadb3ca Power-time plot: the first rectangle represents 5184 J, the second 2016 J. Widths have ratio 3 to 7, heights 6 to 1. The 12 W mean line has the same total area. Three minutes at 28.8 W and seven minutes at 4.8 W give 7200 J. Duration-weighted mean power is 12 W. https://quanta-study.de/en/formula-sheet/bohr-energieniveaus-wasserstoff 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/bohr-energieniveaus-wasserstoff-en-light.svg?v=17912753362a4f64 Three horizontal energy levels E1, E2 and E3 approach the zero ionization limit with smaller spacing. An accent arrow points from E3 to E2; beside it is hν=|E3−E2|. Discrete Bohr energy levels and a photon for the n=3 to n=2 transition; the arrow marks the positive emitted photon energy. https://quanta-study.de/tafelwerk/diagramme/bohr-energieniveaus-wasserstoff-en-dark.svg?v=17912753362a4f64 Three horizontal energy levels E1, E2 and E3 approach the zero ionization limit with smaller spacing. An accent arrow points from E3 to E2; beside it is hν=|E3−E2|. Discrete Bohr energy levels and a photon for the n=3 to n=2 transition; the arrow marks the positive emitted photon energy. https://quanta-study.de/en/formula-sheet/maxwell-boltzmann-geschwindigkeitsverteilung 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/maxwell-boltzmann-geschwindigkeitsverteilung-en-light.svg?v=5c435f45f3b7cbbe Maxwell density 4x²exp(−x²)/√π from x = 0 to 3.5, peak at 1 with the right tail retained in the displayed range. Area 0.8 to 1.2 is shaded. Correctly normalized dimensionless speed density with x=v/vmp. Area from 0.8 to 1.2 is 32.3388 % of particles; density height is not a probability at one exact speed. https://quanta-study.de/tafelwerk/diagramme/maxwell-boltzmann-geschwindigkeitsverteilung-en-dark.svg?v=5c435f45f3b7cbbe Maxwell density 4x²exp(−x²)/√π from x = 0 to 3.5, peak at 1 with the right tail retained in the displayed range. Area 0.8 to 1.2 is shaded. Correctly normalized dimensionless speed density with x=v/vmp. Area from 0.8 to 1.2 is 32.3388 % of particles; density height is not a probability at one exact speed. https://quanta-study.de/en/formula-sheet/schwerpunkt-massensystem 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/schwerpunkt-massensystem-en-light.svg?v=a1673e606b5afd51 A 2 kg mass is at zero and a 1 kg mass at four metres on one axis. The marked centre of mass is at 1.33 metres. The centre of mass lies closer to the larger mass because positions are weighted by mass. https://quanta-study.de/tafelwerk/diagramme/schwerpunkt-massensystem-en-dark.svg?v=a1673e606b5afd51 A 2 kg mass is at zero and a 1 kg mass at four metres on one axis. The marked centre of mass is at 1.33 metres. The centre of mass lies closer to the larger mass because positions are weighted by mass. https://quanta-study.de/en/formula-sheet/planck-spektralgesetz 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/planck-spektralgesetz-en-light.svg?v=ca70e5bd403ebf07 A qualitative Planck curve marks its peak at λmax; the normalized wavelength is λ divided by λmax. Spectral radiance per wavelength interval has a peak; increasing temperature shifts it to shorter wavelengths. https://quanta-study.de/tafelwerk/diagramme/planck-spektralgesetz-en-dark.svg?v=ca70e5bd403ebf07 A qualitative Planck curve marks its peak at λmax; the normalized wavelength is λ divided by λmax. Spectral radiance per wavelength interval has a peak; increasing temperature shifts it to shorter wavelengths. https://quanta-study.de/en/formula-sheet/bragg-bedingung-kristallbeugung 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/bragg-bedingung-kristallbeugung-en-light.svg?v=15d16507215b81d1 Two parallel crystal planes separated by d reflect parallel X-ray rays. The Bragg angle θ is measured from the plane. Rays reflected from adjacent crystal planes interfere constructively when their path difference is 2d sin θ = nλ. https://quanta-study.de/tafelwerk/diagramme/bragg-bedingung-kristallbeugung-en-dark.svg?v=15d16507215b81d1 Two parallel crystal planes separated by d reflect parallel X-ray rays. The Bragg angle θ is measured from the plane. Rays reflected from adjacent crystal planes interfere constructively when their path difference is 2d sin θ = nλ. https://quanta-study.de/en/formula-sheet/hall-effekt-spannung 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/hall-effekt-spannung-en-light.svg?v=5252e88508434bfb Hall plate with +y upward, +x rightward and field dots out of the plane. Upper edge positive and lower edge negative. Electron drift points left, magnetic force down, electric force up. Positive probe terminal below gives negative voltage. Original electron convention: conventional I along +x, B along +z out of the plane. Electron drift along −x; negative charge below. UH=V(lower)−V(upper)≈−3.121 mV for I=5 mA, B=0.40 T, n=2·10²² m⁻³, t=0.20 mm. Thickness t lies along z, width w along y; not drawn to scale. https://quanta-study.de/tafelwerk/diagramme/hall-effekt-spannung-en-dark.svg?v=5252e88508434bfb Hall plate with +y upward, +x rightward and field dots out of the plane. Upper edge positive and lower edge negative. Electron drift points left, magnetic force down, electric force up. Positive probe terminal below gives negative voltage. Original electron convention: conventional I along +x, B along +z out of the plane. Electron drift along −x; negative charge below. UH=V(lower)−V(upper)≈−3.121 mV for I=5 mA, B=0.40 T, n=2·10²² m⁻³, t=0.20 mm. Thickness t lies along z, width w along y; not drawn to scale. https://quanta-study.de/en/formula-sheet/stokes-quadratischer-widerstand 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/stokes-quadratischer-widerstand-en-light.svg?v=7b5a059ccc811ae3 A log-log plot of relative speed and normalized drag. A slope-one line shows linear Stokes scaling, and a steeper slope-two curve shows quadratic scaling; they cross at one. Schematic scaling comparison: in the Stokes regime FD∝vrel, while the quadratic model gives FD∝vrel². The crossing is set only by normalization. https://quanta-study.de/tafelwerk/diagramme/stokes-quadratischer-widerstand-en-dark.svg?v=7b5a059ccc811ae3 A log-log plot of relative speed and normalized drag. A slope-one line shows linear Stokes scaling, and a steeper slope-two curve shows quadratic scaling; they cross at one. Schematic scaling comparison: in the Stokes regime FD∝vrel, while the quadratic model gives FD∝vrel². The crossing is set only by normalization. https://quanta-study.de/en/formula-sheet/einzelspaltbeugung-minima 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/biot-savart-gesetz 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/biot-savart-gesetz-en-light.svg?v=6f77dced7b732c3e Finite horizontal wire at y=0 with current rightward. P is above its midpoint. An element left of centre points right; its displacement points diagonally right-up to P and the cross product points out of the plane. R is slanted distance, d perpendicular distance. Original wire geometry: I=3 A along +x, segment from −0.15 to +0.15 m, P at y=0.04 m. R points from the sample element at x=−0.08 m to P. All segment contributions point along +z; Bz≈14.4935 µT. The element arrow is schematic, not a finite replacement for integration. https://quanta-study.de/tafelwerk/diagramme/biot-savart-gesetz-en-dark.svg?v=6f77dced7b732c3e Finite horizontal wire at y=0 with current rightward. P is above its midpoint. An element left of centre points right; its displacement points diagonally right-up to P and the cross product points out of the plane. R is slanted distance, d perpendicular distance. Original wire geometry: I=3 A along +x, segment from −0.15 to +0.15 m, P at y=0.04 m. R points from the sample element at x=−0.08 m to P. All segment contributions point along +z; Bz≈14.4935 µT. The element arrow is schematic, not a finite replacement for integration. https://quanta-study.de/en/formula-sheet/elektrischer-dipol-moment 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/gegenseitige-induktivitaet 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/driftmodell-stromdichte 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/equipartition-energie 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/equipartition-energie-en-light.svg?v=067cb4d6814d7a98 Three stacked bars total 1.5, 2.5 and 3.5 RT on one scale. The final mode has separate kinetic and potential half-contributions. Classical active model limits per mole: three translations give 1.5 RT; two rotations add RT; one harmonic mode adds two half RT terms. No universal temperature threshold. https://quanta-study.de/tafelwerk/diagramme/equipartition-energie-en-dark.svg?v=067cb4d6814d7a98 Three stacked bars total 1.5, 2.5 and 3.5 RT on one scale. The final mode has separate kinetic and potential half-contributions. Classical active model limits per mole: three translations give 1.5 RT; two rotations add RT; one harmonic mode adds two half RT terms. No universal temperature threshold. https://quanta-study.de/en/formula-sheet/isotherme-arbeit-ideales-gas 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/isotherme-arbeit-ideales-gas-en-light.svg?v=6b9948a6e14e72d4 Normalized reversible pressure 1/u between volume factors 1 and 3, above constant external load 1/3 with a smaller work area. For n=0.300mol,T = 360 K and volume ratio 3: reversible 986.51 J; constant external load 598.64 J. The horizontal line is external load, not an equilibrium gas path. https://quanta-study.de/tafelwerk/diagramme/isotherme-arbeit-ideales-gas-en-dark.svg?v=6b9948a6e14e72d4 Normalized reversible pressure 1/u between volume factors 1 and 3, above constant external load 1/3 with a smaller work area. For n=0.300mol,T = 360 K and volume ratio 3: reversible 986.51 J; constant external load 598.64 J. The horizontal line is external load, not an equilibrium gas path. https://quanta-study.de/en/formula-sheet/boltzmann-entropie-mikrozustaende 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/boltzmann-entropie-mikrozustaende-en-light.svg?v=88685b0c47714bfb Discrete binomial multiplicity bars peak at 70 for four upper occupations. Small-model entropy S/kB=ln70≈4.24850. Eight labeled binary sites: multiplicities are 1, 8, 28, 56, 70, 56, 28, 8, 1. Equal microstate probabilities do not make macrostates equally probable. https://quanta-study.de/tafelwerk/diagramme/boltzmann-entropie-mikrozustaende-en-dark.svg?v=88685b0c47714bfb Discrete binomial multiplicity bars peak at 70 for four upper occupations. Small-model entropy S/kB=ln70≈4.24850. Eight labeled binary sites: multiplicities are 1, 8, 28, 56, 70, 56, 28, 8, 1. Equal microstate probabilities do not make macrostates equally probable. https://quanta-study.de/en/formula-sheet/zyklotronradius-periode 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kraft-parallele-stroeme 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/quantenharmonischer-oszillator-energieniveaus 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/tunnelwahrscheinlichkeit-potentialbarriere 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/phasen-und-gruppengeschwindigkeit 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/stosszahl-rueckprall 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/strahlungsdruck-absorption-reflexion 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/ideales-gasgesetz 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/ideales-gasgesetz-en-light.svg?v=841e48d0c3203597 Diagram: two hyperbolas in the p-V diagram, the highlighted isotherm T and further out the isotherm 2T; a shaded rectangle from the origin to a point on the isotherm T stands for p·V = n·R·T. At a fixed amount n, all states of equal temperature T lie on a hyperbola, the isotherm: the rectangle p·V always equals n·R·T, and at twice the temperature the isotherm lies further out. https://quanta-study.de/tafelwerk/diagramme/ideales-gasgesetz-en-dark.svg?v=841e48d0c3203597 Diagram: two hyperbolas in the p-V diagram, the highlighted isotherm T and further out the isotherm 2T; a shaded rectangle from the origin to a point on the isotherm T stands for p·V = n·R·T. At a fixed amount n, all states of equal temperature T lie on a hyperbola, the isotherm: the rectangle p·V always equals n·R·T, and at twice the temperature the isotherm lies further out. https://quanta-study.de/en/formula-sheet/arrhenius-gleichung 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/arrhenius-gleichung-en-light.svg?v=fff265cd5b685b98 Arrhenius plot: a falling straight line of ln k over 1/T with four data points, extended as a dashed line to the ln k axis where ln A can be read; a slope triangle shows the slope −E_A/R. Plotting ln k against 1/T puts the measured values on a straight line with slope −E_A/R; its intercept at 1/T = 0 is ln A. https://quanta-study.de/tafelwerk/diagramme/arrhenius-gleichung-en-dark.svg?v=fff265cd5b685b98 Arrhenius plot: a falling straight line of ln k over 1/T with four data points, extended as a dashed line to the ln k axis where ln A can be read; a slope triangle shows the slope −E_A/R. Plotting ln k against 1/T puts the measured values on a straight line with slope −E_A/R; its intercept at 1/T = 0 is ln A. https://quanta-study.de/en/formula-sheet/henderson-hasselbalch 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/henderson-hasselbalch-en-light.svg?v=b917245ee3e3e5c4 Diagram: the pH over the ratio [A⁻] : [HA] on a logarithmic axis is a rising straight line; at 1 : 10 it lies at pKa − 1, at 1 : 1 exactly at pKa and at 10 : 1 at pKa + 1, a shaded field marks this buffer range. With the ratio [A⁻] : [HA] on a logarithmic axis, the pH rises along a straight line: every tenfold increase of the ratio raises it by one pH unit. At 1 : 1 the pH equals pKa, and between 1 : 10 and 10 : 1 lies the buffer range pKa ± 1. https://quanta-study.de/tafelwerk/diagramme/henderson-hasselbalch-en-dark.svg?v=b917245ee3e3e5c4 Diagram: the pH over the ratio [A⁻] : [HA] on a logarithmic axis is a rising straight line; at 1 : 10 it lies at pKa − 1, at 1 : 1 exactly at pKa and at 10 : 1 at pKa + 1, a shaded field marks this buffer range. With the ratio [A⁻] : [HA] on a logarithmic axis, the pH rises along a straight line: every tenfold increase of the ratio raises it by one pH unit. At 1 : 1 the pH equals pKa, and between 1 : 10 and 10 : 1 lies the buffer range pKa ± 1. https://quanta-study.de/en/formula-sheet/nernst-gleichung 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/nernst-gleichung-en-light.svg?v=407be60f7a243521 Diagram: a falling straight line E over ln Q crosses the E axis at Q = 1 at the value E°; to the left Q < 1 and E is larger, to the right Q > 1 and E is smaller, a slope triangle shows −RT/(zF). The electrode potential E falls linearly with ln Q; at Q = 1, E equals E°, and the slope is −RT/(zF). https://quanta-study.de/tafelwerk/diagramme/nernst-gleichung-en-dark.svg?v=407be60f7a243521 Diagram: a falling straight line E over ln Q crosses the E axis at Q = 1 at the value E°; to the left Q < 1 and E is larger, to the right Q > 1 and E is smaller, a slope triangle shows −RT/(zF). The electrode potential E falls linearly with ln Q; at Q = 1, E equals E°, and the slope is −RT/(zF). https://quanta-study.de/en/formula-sheet/gibbs-energie 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/gibbs-energie-en-light.svg?v=8fe58d48493cbec0 Diagram: a falling straight line ΔG over T starts at the value ΔH at T = 0; a dimension arrow shows the gap T·ΔS to the dashed ΔH line, and at the zero crossing T = ΔH/ΔS, ΔG turns negative. For ΔH > 0 and ΔS > 0, ΔG falls linearly with T by T·ΔS; above T = ΔH/ΔS, ΔG becomes negative and the reaction proceeds spontaneously. https://quanta-study.de/tafelwerk/diagramme/gibbs-energie-en-dark.svg?v=8fe58d48493cbec0 Diagram: a falling straight line ΔG over T starts at the value ΔH at T = 0; a dimension arrow shows the gap T·ΔS to the dashed ΔH line, and at the zero crossing T = ΔH/ΔS, ΔG turns negative. For ΔH > 0 and ΔS > 0, ΔG falls linearly with T by T·ΔS; above T = ΔH/ΔS, ΔG becomes negative and the reaction proceeds spontaneously. https://quanta-study.de/en/formula-sheet/massenwirkungsgesetz 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/massenwirkungsgesetz-en-light.svg?v=d2461c3e8995b683 Diagram: concentrations over time t; the curve [A] falls, the curve [C] rises, both level off into flat plateaus, a tinted region marks equilibrium with Kc = [C]/[A]. In the reaction A ⇌ C, [A] falls and [C] rises until both stay constant; at equilibrium the ratio Kc = [C]/[A] is fixed, here 2, even though the concentrations differ. https://quanta-study.de/tafelwerk/diagramme/massenwirkungsgesetz-en-dark.svg?v=d2461c3e8995b683 Diagram: concentrations over time t; the curve [A] falls, the curve [C] rises, both level off into flat plateaus, a tinted region marks equilibrium with Kc = [C]/[A]. In the reaction A ⇌ C, [A] falls and [C] rises until both stay constant; at equilibrium the ratio Kc = [C]/[A] is fixed, here 2, even though the concentrations differ. https://quanta-study.de/en/formula-sheet/ph-definition 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/ph-definition-en-light.svg?v=77b86b4d18da9697 Diagram: a pH scale from 0 to 14 with a mark at 7, acidic on the left, basic on the right, below it the directions of increasing H3O+ and OH− concentration. The pH scale from 0 to 14: acidic below 7, neutral at 7, basic above 7. https://quanta-study.de/tafelwerk/diagramme/ph-definition-en-dark.svg?v=77b86b4d18da9697 Diagram: a pH scale from 0 to 14 with a mark at 7, acidic on the left, basic on the right, below it the directions of increasing H3O+ and OH− concentration. The pH scale from 0 to 14: acidic below 7, neutral at 7, basic above 7. https://quanta-study.de/en/formula-sheet/lambert-beer-gesetz 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/lambert-beer-gesetz-en-light.svg?v=3290ae8cfd1ecac7 Diagram: a light beam I₀ enters a cuvette of width d containing dissolved particles from the left and leaves it on the right weakened as I; next to it reads A = lg(I₀/I). Light of intensity I₀ is weakened by the dissolved particles of concentration c in the cuvette of path length d and leaves with intensity I; the absorbance A = lg(I₀/I) grows in proportion to c and d. https://quanta-study.de/tafelwerk/diagramme/lambert-beer-gesetz-en-dark.svg?v=3290ae8cfd1ecac7 Diagram: a light beam I₀ enters a cuvette of width d containing dissolved particles from the left and leaves it on the right weakened as I; next to it reads A = lg(I₀/I). Light of intensity I₀ is weakened by the dissolved particles of concentration c in the cuvette of path length d and leaves with intensity I; the absorbance A = lg(I₀/I) grows in proportion to c and d. https://quanta-study.de/en/formula-sheet/stoffmenge-mol 2026-07-09 monthly 0.82 https://quanta-study.de/en/formula-sheet/molare-masse 2026-07-09 monthly 0.82 https://quanta-study.de/en/formula-sheet/stoffmengenkonzentration 2026-07-09 monthly 0.82 https://quanta-study.de/en/formula-sheet/avogadro-teilchenzahl 2026-07-09 monthly 0.82 https://quanta-study.de/en/formula-sheet/pks-wert-saeurekonstante 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/pks-wert-saeurekonstante-en-light.svg?v=897117eb0bf002db Diagram: a pKa scale from −6 to 14 with the ranges very strong below 0, moderate from 0 to 4 and weak above 4, marked with HCl, CH₃COOH and NH₄⁺; an arrow to the left shows increasing acid strength. On the pKa scale strong acids sit on the left: the smaller the pKa, the stronger the acid, from HCl (about −6) through acetic acid (4.75) to the ammonium ion (9.25). https://quanta-study.de/tafelwerk/diagramme/pks-wert-saeurekonstante-en-dark.svg?v=897117eb0bf002db Diagram: a pKa scale from −6 to 14 with the ranges very strong below 0, moderate from 0 to 4 and weak above 4, marked with HCl, CH₃COOH and NH₄⁺; an arrow to the left shows increasing acid strength. On the pKa scale strong acids sit on the left: the smaller the pKa, the stronger the acid, from HCl (about −6) through acetic acid (4.75) to the ammonium ion (9.25). https://quanta-study.de/en/formula-sheet/ionenprodukt-wasser 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/ionenprodukt-wasser-en-light.svg?v=979a9fd87c380bfe Diagram with a logarithmic concentration axis: the line [H₃O⁺] falls from 10⁰ at pH 0 to 10⁻¹⁴ at pH 14, the line [OH⁻] rises as its mirror image; they cross, highlighted, at pH 7 and 10⁻⁷ mol/L. As the pH rises, [H₃O⁺] falls while [OH⁻] rises by the same factor, because their product stays Kw = 10⁻¹⁴ mol²/L²; at pH 7 both equal 10⁻⁷ mol/L and the solution is neutral (25 °C). https://quanta-study.de/tafelwerk/diagramme/ionenprodukt-wasser-en-dark.svg?v=979a9fd87c380bfe Diagram with a logarithmic concentration axis: the line [H₃O⁺] falls from 10⁰ at pH 0 to 10⁻¹⁴ at pH 14, the line [OH⁻] rises as its mirror image; they cross, highlighted, at pH 7 and 10⁻⁷ mol/L. As the pH rises, [H₃O⁺] falls while [OH⁻] rises by the same factor, because their product stays Kw = 10⁻¹⁴ mol²/L²; at pH 7 both equal 10⁻⁷ mol/L and the solution is neutral (25 °C). https://quanta-study.de/en/formula-sheet/reaktionsenthalpie-hess 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/reaktionsenthalpie-hess-en-light.svg?v=fbcda6b12dff07aa Enthalpy diagram: at the top the level of the elements, below it the reactants and at the bottom the products; the arrows ΣΔH°f(E) and ΣΔH°f(P) lead from the elements to reactants and products, the highlighted arrow ΔH°R leads directly from reactants to products. Because enthalpy is a state function, the direct path ΔH°R from reactants to products reaches the same level as the detour via the elements, so ΔH°R = ΣΔH°f(P) − ΣΔH°f(E). https://quanta-study.de/tafelwerk/diagramme/reaktionsenthalpie-hess-en-dark.svg?v=fbcda6b12dff07aa Enthalpy diagram: at the top the level of the elements, below it the reactants and at the bottom the products; the arrows ΣΔH°f(E) and ΣΔH°f(P) lead from the elements to reactants and products, the highlighted arrow ΔH°R leads directly from reactants to products. Because enthalpy is a state function, the direct path ΔH°R from reactants to products reaches the same level as the detour via the elements, so ΔH°R = ΣΔH°f(P) − ΣΔH°f(E). https://quanta-study.de/en/formula-sheet/faraday-elektrolyse 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/faraday-elektrolyse-en-light.svg?v=cd2ac19f3daa710e Diagram of an electrolysis: two electrodes dip into a solution and are connected to a battery, the current I flows from the positive pole to the anode. A Cu²⁺ ion migrates to the cathode on the left, where a copper layer of mass m is deposited; below it reads Cu²⁺ + 2 e⁻ → Cu. When the current I flows for the time t, the mass m is deposited at the cathode. For Cu²⁺ each particle needs z = 2 electrons. https://quanta-study.de/tafelwerk/diagramme/faraday-elektrolyse-en-dark.svg?v=cd2ac19f3daa710e Diagram of an electrolysis: two electrodes dip into a solution and are connected to a battery, the current I flows from the positive pole to the anode. A Cu²⁺ ion migrates to the cathode on the left, where a copper layer of mass m is deposited; below it reads Cu²⁺ + 2 e⁻ → Cu. When the current I flows for the time t, the mass m is deposited at the cathode. For Cu²⁺ each particle needs z = 2 electrons. https://quanta-study.de/en/formula-sheet/loeslichkeitsprodukt 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/loeslichkeitsprodukt-en-light.svg?v=101446afd79a6581 Diagram: a hyperbola [Bⁿ⁻] over [Aᵐ⁺] marks K_L; the region below it is labelled unsaturated, the tinted region above it precipitate. On the curve the solution is saturated and the ion product equals K_L; below it the solution stays unsaturated, above it a precipitate forms (drawn for a = b = 1). https://quanta-study.de/tafelwerk/diagramme/loeslichkeitsprodukt-en-dark.svg?v=101446afd79a6581 Diagram: a hyperbola [Bⁿ⁻] over [Aᵐ⁺] marks K_L; the region below it is labelled unsaturated, the tinted region above it precipitate. On the curve the solution is saturated and the ion product equals K_L; below it the solution stays unsaturated, above it a precipitate forms (drawn for a = b = 1). https://quanta-study.de/en/formula-sheet/verduennungsgleichung 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/verduennungsgleichung-en-light.svg?v=70d7f9077a5bcbd5 Diagram: on the left a vessel with a small volume V₁ and closely packed particles (c₁), on the right, after adding water, three times the volume V₂ with the same twelve particles, now more spread out (c₂). Diluting only adds solvent: the number of dissolved particles, and so n = c·V, stays the same, hence c₁·V₁ = c₂·V₂; here the volume triples and the concentration drops to one third. https://quanta-study.de/tafelwerk/diagramme/verduennungsgleichung-en-dark.svg?v=70d7f9077a5bcbd5 Diagram: on the left a vessel with a small volume V₁ and closely packed particles (c₁), on the right, after adding water, three times the volume V₂ with the same twelve particles, now more spread out (c₂). Diluting only adds solvent: the number of dissolved particles, and so n = c·V, stays the same, hence c₁·V₁ = c₂·V₂; here the volume triples and the concentration drops to one third. https://quanta-study.de/en/formula-sheet/geschwindigkeitsgesetz-reaktionsordnung 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/geschwindigkeitsgesetz-reaktionsordnung-en-light.svg?v=d1eda9f63003b6d4 Diagram: three curves of the reaction rate v over [A] for m = 0 (flat), m = 1 (straight line) and m = 2 (parabola), meeting at [A] = 1; at [A] = 2 the factors ×1, ×2 and ×4 are highlighted. At a fixed concentration [B], the order m decides how v responds to [A]: doubling [A] leaves v unchanged for m = 0, doubles v for m = 1 and quadruples v for m = 2. https://quanta-study.de/tafelwerk/diagramme/geschwindigkeitsgesetz-reaktionsordnung-en-dark.svg?v=d1eda9f63003b6d4 Diagram: three curves of the reaction rate v over [A] for m = 0 (flat), m = 1 (straight line) and m = 2 (parabola), meeting at [A] = 1; at [A] = 2 the factors ×1, ×2 and ×4 are highlighted. At a fixed concentration [B], the order m decides how v responds to [A]: doubling [A] leaves v unchanged for m = 0, doubles v for m = 1 and quadruples v for m = 2. https://quanta-study.de/en/formula-sheet/rgt-regel 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/rgt-regel-en-light.svg?v=e170faaa0eb32315 Diagram: the reaction rate v rises exponentially with the temperature change ΔT; marked points at 0, 10, 20 and 30 K show v₁, 2·v₁, 4·v₁ and 8·v₁. With Q₁₀ = 2 the reaction rate doubles every 10 K. After ΔT = 30 K the rate is v₂ = 2³·v₁ = 8·v₁. https://quanta-study.de/tafelwerk/diagramme/rgt-regel-en-dark.svg?v=e170faaa0eb32315 Diagram: the reaction rate v rises exponentially with the temperature change ΔT; marked points at 0, 10, 20 and 30 K show v₁, 2·v₁, 4·v₁ and 8·v₁. With Q₁₀ = 2 the reaction rate doubles every 10 K. After ΔT = 30 K the rate is v₂ = 2³·v₁ = 8·v₁. https://quanta-study.de/en/formula-sheet/massenanteil-gehaltsgroessen 2026-07-09 monthly 0.82 https://quanta-study.de/en/formula-sheet/gitterenergie-born-haber 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/gitterenergie-born-haber-en-light.svg?v=cde98863533d87e1 Energy diagram: from Na(s) and ½Cl₂(g) the sublimation ΔH_Sub, ionization I and half dissociation ½D lead upward, the electron affinity EA leads a little down to Na⁺(g) and Cl⁻(g); the long arrow ΔH_lattice leads down to NaCl(s), which the direct arrow ΔH°f also reaches. Born-Haber cycle for NaCl: the detour via gaseous atoms and ions leads to the same enthalpy of formation ΔH°f, which gives ΔH_lattice = −788 kJ/mol. https://quanta-study.de/tafelwerk/diagramme/gitterenergie-born-haber-en-dark.svg?v=cde98863533d87e1 Energy diagram: from Na(s) and ½Cl₂(g) the sublimation ΔH_Sub, ionization I and half dissociation ½D lead upward, the electron affinity EA leads a little down to Na⁺(g) and Cl⁻(g); the long arrow ΔH_lattice leads down to NaCl(s), which the direct arrow ΔH°f also reaches. Born-Haber cycle for NaCl: the detour via gaseous atoms and ions leads to the same enthalpy of formation ΔH°f, which gives ΔH_lattice = −788 kJ/mol. https://quanta-study.de/en/formula-sheet/kb-pkb-basenkonstante 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/kb-pkb-basenkonstante-en-light.svg?v=1963cc57ce4fb087 Diagram: a bar from 0 to 14 is split into pKa(NH₄⁺) = 9.25 and pKb(NH₃) = 4.75, below it the sum pKa + pKb = 14. For a conjugate acid-base pair such as NH₄⁺/NH₃, pKa and pKb add up to 14 at 25 °C. The stronger the acid, the weaker its conjugate base. https://quanta-study.de/tafelwerk/diagramme/kb-pkb-basenkonstante-en-dark.svg?v=1963cc57ce4fb087 Diagram: a bar from 0 to 14 is split into pKa(NH₄⁺) = 9.25 and pKb(NH₃) = 4.75, below it the sum pKa + pKb = 14. For a conjugate acid-base pair such as NH₄⁺/NH₃, pKa and pKb add up to 14 at 25 °C. The stronger the acid, the weaker its conjugate base. https://quanta-study.de/en/formula-sheet/van-t-hoff-reaktionsisobare 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/van-t-hoff-reaktionsisobare-en-light.svg?v=2e1771596685c829 Diagram: a rising straight line of ln K over 1/T with two highlighted points, (1/T₂, ln K₂) lower left and (1/T₁, ln K₁) upper right, joined to the axes by dashed lines; the slope is −ΔH°R/R. In the van 't Hoff plot, ln K against 1/T lies on a straight line with slope −ΔH°R/R; for an exothermic reaction such as ammonia synthesis, K₂ at the higher temperature T₂ is smaller than K₁. https://quanta-study.de/tafelwerk/diagramme/van-t-hoff-reaktionsisobare-en-dark.svg?v=2e1771596685c829 Diagram: a rising straight line of ln K over 1/T with two highlighted points, (1/T₂, ln K₂) lower left and (1/T₁, ln K₁) upper right, joined to the axes by dashed lines; the slope is −ΔH°R/R. In the van 't Hoff plot, ln K against 1/T lies on a straight line with slope −ΔH°R/R; for an exothermic reaction such as ammonia synthesis, K₂ at the higher temperature T₂ is smaller than K₁. https://quanta-study.de/en/formula-sheet/henry-gesetz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/osmotischer-druck-van-t-hoff 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/molalitaet 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/aktivitaetskoeffizient 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/dalton-gesetz-partialdruck 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kompressibilitaetsfaktor 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/retentionsfaktor-chromatographie 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/integriertes-geschwindigkeitsgesetz-erste-ordnung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/molenbruch 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/massenkonzentration 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/stoffmengenbilanz-titration 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/van-der-waals-gleichung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/raoultsches-gesetz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/siedepunktserhoehung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/gefriertemperaturerniedrigung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/ionenstaerke 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/reaktionsquotient 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kp-kc-beziehung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/verteilungsanteile-diprotische-saeure 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/pufferkapazitaet 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/chemisches-potential 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/standard-gibbsenergie-gleichgewicht 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/reaktions-gibbsenergie-quotient 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/gibbs-helmholtz-gleichung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/clausius-clapeyron-verdampfung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kirchhoff-gesetz-reaktionsenthalpie 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/mischungsentropie-ideale-loesung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/integriertes-geschwindigkeitsgesetz-nullter-ordnung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/integriertes-geschwindigkeitsgesetz-zweiter-ordnung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/halbwertszeit-reaktionsordnung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/eyring-gleichung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/steady-state-approximation 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/standard-zellpotential-galvanisch 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/konzentrationszelle-nernst 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/molare-leitfaehigkeit 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kohlrausch-gesetz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/butler-volmer-gleichung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/analytische-kalibriergerade 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/standardaddition-analytik 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/nachweis-und-bestimmungsgrenze 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/chromatographische-selektivitaet 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/chromatographische-aufloesung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/analytische-wiederfindung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/atomoekonomie 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/reaktionsausbeute 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/grad-der-ungesaettigtheit 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/debye-huckel-grenzgesetz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/boltzmann-verteilung-zustandssumme 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/boltzmann-verteilung-zustandssumme-en-light.svg?v=0ae20ad5a4581642 Two energy levels E₀ and E₁ = E₀ + 5 kJ/mol; the upper level has degeneracy three. Adjacent bars show normalized occupations p₀ = 0.715 and p₁ = 0.285 at 298 K; this is an idealized two-state model. Illustrative canonical two-state model at 298 K: degeneracy g₁ = 3 raises occupation of the upper level despite ΔE = 5 kJ mol⁻¹. Bars show normalized probabilities, not measurements for a specific molecule. https://quanta-study.de/tafelwerk/diagramme/boltzmann-verteilung-zustandssumme-en-dark.svg?v=0ae20ad5a4581642 Two energy levels E₀ and E₁ = E₀ + 5 kJ/mol; the upper level has degeneracy three. Adjacent bars show normalized occupations p₀ = 0.715 and p₁ = 0.285 at 298 K; this is an idealized two-state model. Illustrative canonical two-state model at 298 K: degeneracy g₁ = 3 raises occupation of the upper level despite ΔE = 5 kJ mol⁻¹. Bars show normalized probabilities, not measurements for a specific molecule. https://quanta-study.de/en/formula-sheet/titrationskurve-schwache-saeure 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/titrationskurve-schwache-saeure-en-light.svg?v=ffd307f6530775e9 pH rises with added base volume: slowly in the buffer region, steeply around equivalence at 25 mL, then more gradually. The half-equivalence point at 12.5 mL has pH equal to pKa. Schematic curve for a monoprotic weak acid titrated with strong base. The buffer approximation applies before equivalence; at equivalence pH exceeds 7 because of the conjugate base. https://quanta-study.de/tafelwerk/diagramme/titrationskurve-schwache-saeure-en-dark.svg?v=ffd307f6530775e9 pH rises with added base volume: slowly in the buffer region, steeply around equivalence at 25 mL, then more gradually. The half-equivalence point at 12.5 mL has pH equal to pKa. Schematic curve for a monoprotic weak acid titrated with strong base. The buffer approximation applies before equivalence; at equivalence pH exceeds 7 because of the conjugate base. https://quanta-study.de/en/formula-sheet/gibbs-phasenregel 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/gibbs-phasenregel-en-light.svg?v=dac34a662ee26636 Three solid, liquid and gas regions meet along coexistence curves. The marked triple point is a single invariant state. Schematic one-component phase diagram: two intensive variables are free in regions, one along coexistence curves and none at the triple point. Curves are not to scale. https://quanta-study.de/tafelwerk/diagramme/gibbs-phasenregel-en-dark.svg?v=dac34a662ee26636 Three solid, liquid and gas regions meet along coexistence curves. The marked triple point is a single invariant state. Schematic one-component phase diagram: two intensive variables are free in regions, one along coexistence curves and none at the triple point. Curves are not to scale. https://quanta-study.de/en/formula-sheet/reaktionsenthalpie-bindungsenthalpien 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/fugazitaetskoeffizient-realgas 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/van-deemter-gleichung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/verteilungsverhaeltnis-fluessigextraktion 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/elektrophoretische-mobilitaet 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/gibbs-hebelregel-phasenanteile 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/e-faktor-gruene-chemie 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/interner-standard-relativer-responsefaktor 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/pythagorean-theorem 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/pythagorean-theorem-en-light.svg?v=fec9b014df221650 Diagram: a right triangle with a square on each side; the two leg squares a² and b² together equal the hypotenuse square c². For a right triangle: the sum of the squares on the legs a² and b² equals the square on the hypotenuse c². https://quanta-study.de/tafelwerk/diagramme/pythagorean-theorem-en-dark.svg?v=fec9b014df221650 Diagram: a right triangle with a square on each side; the two leg squares a² and b² together equal the hypotenuse square c². For a right triangle: the sum of the squares on the legs a² and b² equals the square on the hypotenuse c². https://quanta-study.de/en/formula-sheet/quadratische-formel 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/quadratische-formel-en-light.svg?v=cbc650ff1323437d Diagram: an upward-opening parabola crosses the x-axis at the two roots x₁ and x₂, the vertex lies below the axis. Parabola with two roots x₁ and x₂, the two solutions of the quadratic equation (D > 0). https://quanta-study.de/tafelwerk/diagramme/quadratische-formel-en-dark.svg?v=cbc650ff1323437d Diagram: an upward-opening parabola crosses the x-axis at the two roots x₁ and x₂, the vertex lies below the axis. Parabola with two roots x₁ and x₂, the two solutions of the quadratic equation (D > 0). https://quanta-study.de/en/formula-sheet/eulersche-formel 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/eulersche-formel-en-light.svg?v=48ca412841028143 The unit circle in the complex plane carries a pointer at the angle φ, whose dashed projections onto the axes mark cos φ and sin φ. Euler's formula on the unit circle: e^(iφ) is the point with real part cos φ and imaginary part sin φ. https://quanta-study.de/tafelwerk/diagramme/eulersche-formel-en-dark.svg?v=48ca412841028143 The unit circle in the complex plane carries a pointer at the angle φ, whose dashed projections onto the axes mark cos φ and sin φ. Euler's formula on the unit circle: e^(iφ) is the point with real part cos φ and imaginary part sin φ. https://quanta-study.de/en/formula-sheet/produktregel-ableitung 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/produktregel-ableitung-en-light.svg?v=3937f2262940b6c1 A rectangle with sides f and g has a narrow highlighted growth strip on top and on the right, the small corner piece between them is dashed. When x grows by dx, the rectangle f·g grows by the strips f′·g·dx and f·g′·dx; the corner piece becomes negligible, hence (f·g)′ = f′·g + f·g′. https://quanta-study.de/tafelwerk/diagramme/produktregel-ableitung-en-dark.svg?v=3937f2262940b6c1 A rectangle with sides f and g has a narrow highlighted growth strip on top and on the right, the small corner piece between them is dashed. When x grows by dx, the rectangle f·g grows by the strips f′·g·dx and f·g′·dx; the corner piece becomes negligible, hence (f·g)′ = f′·g + f·g′. https://quanta-study.de/en/formula-sheet/kettenregel-ableitung 2026-07-09 monthly 0.82 https://quanta-study.de/en/formula-sheet/taylor-reihe 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/taylor-reihe-en-light.svg?v=2cac58904c6ae48a The sine curve f is accompanied by three dashed approximating polynomials of order 1, 3 and 5, which agree with f over a longer and longer stretch around the expansion point a. Taylor polynomials of sin x about a = 0: the larger n, the further the polynomial hugs f. https://quanta-study.de/tafelwerk/diagramme/taylor-reihe-en-dark.svg?v=2cac58904c6ae48a The sine curve f is accompanied by three dashed approximating polynomials of order 1, 3 and 5, which agree with f over a longer and longer stretch around the expansion point a. Taylor polynomials of sin x about a = 0: the larger n, the further the polynomial hugs f. https://quanta-study.de/en/formula-sheet/bayes-theorem 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/bayes-theorem-en-light.svg?v=49813ac1d79c920c The tree diagram branches into A and ¬A at the first stage and into B and ¬B at the second; the path via A to B is highlighted. P(A|B) is the share of the highlighted path P(A)·P(B|A) among all paths ending in B; their sum is P(B). https://quanta-study.de/tafelwerk/diagramme/bayes-theorem-en-dark.svg?v=49813ac1d79c920c The tree diagram branches into A and ¬A at the first stage and into B and ¬B at the second; the path via A to B is highlighted. P(A|B) is the share of the highlighted path P(A)·P(B|A) among all paths ending in B; their sum is P(B). https://quanta-study.de/en/formula-sheet/normalverteilung 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/normalverteilung-en-light.svg?v=6575647a1e47e1bb Diagram: a symmetric bell curve peaking at μ; the distance from μ to the inflection point is marked as σ. The Gaussian bell curve is symmetric about the expected value μ; its inflection points lie at μ ± σ. https://quanta-study.de/tafelwerk/diagramme/normalverteilung-en-dark.svg?v=6575647a1e47e1bb Diagram: a symmetric bell curve peaking at μ; the distance from μ to the inflection point is marked as σ. The Gaussian bell curve is symmetric about the expected value μ; its inflection points lie at μ ± σ. https://quanta-study.de/en/formula-sheet/potenzregel 2026-07-09 monthly 0.82 https://quanta-study.de/en/formula-sheet/pq-formel 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/pq-formel-en-light.svg?v=6e82f684b24f8bc8 An upward-opening standard parabola has a dashed axis of symmetry at −p/2, from which two arrows of length √D lead to the roots x₂ and x₁. The roots x₁ and x₂ lie symmetrically about the vertex position −p/2, each at distance √D. https://quanta-study.de/tafelwerk/diagramme/pq-formel-en-dark.svg?v=6e82f684b24f8bc8 An upward-opening standard parabola has a dashed axis of symmetry at −p/2, from which two arrows of length √D lead to the roots x₂ and x₁. The roots x₁ and x₂ lie symmetrically about the vertex position −p/2, each at distance √D. https://quanta-study.de/en/formula-sheet/binomische-formeln 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/binomische-formeln-en-light.svg?v=f6cd37a86fd23d96 A large square is divided by one vertical and one horizontal line into the square a², the square b² and two highlighted rectangles ab. The square with side a + b splits into a², b² and the rectangle ab twice: (a + b)² = a² + 2ab + b². https://quanta-study.de/tafelwerk/diagramme/binomische-formeln-en-dark.svg?v=f6cd37a86fd23d96 A large square is divided by one vertical and one horizontal line into the square a², the square b² and two highlighted rectangles ab. The square with side a + b splits into a², b² and the rectangle ab twice: (a + b)² = a² + 2ab + b². https://quanta-study.de/en/formula-sheet/ableitung-e-funktion 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/ableitung-e-funktion-en-light.svg?v=e82ebc28eccb39b0 Diagram: the graph of e to the x with a tangent at one point; the tangent slope equals the function value there. The exponential is its own derivative: the tangent slope at each point equals the function value. https://quanta-study.de/tafelwerk/diagramme/ableitung-e-funktion-en-dark.svg?v=e82ebc28eccb39b0 Diagram: the graph of e to the x with a tangent at one point; the tangent slope equals the function value there. The exponential is its own derivative: the tangent slope at each point equals the function value. https://quanta-study.de/en/formula-sheet/ableitung-ln 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/ableitung-ln-en-light.svg?v=258a0d3d5f4149aa Graph of the natural logarithm with two tangents, at x = 1 with slope 1 and at x = 4 with slope 1/4. The tangent slope of ln x at x is 1/x: steep near 0, ever flatter for large x. https://quanta-study.de/tafelwerk/diagramme/ableitung-ln-en-dark.svg?v=258a0d3d5f4149aa Graph of the natural logarithm with two tangents, at x = 1 with slope 1 and at x = 4 with slope 1/4. The tangent slope of ln x at x is 1/x: steep near 0, ever flatter for large x. https://quanta-study.de/en/formula-sheet/ableitung-sinus-cosinus 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/ableitung-sinus-cosinus-en-light.svg?v=921541759fc31404 Sine curve with three short tangents and a dashed cosine curve; at the same points cos x takes the values 1, 0 and −1. The slope of sin x equals cos x at every point: slope 1 at x = 0, horizontal at π/2, slope −1 at π. https://quanta-study.de/tafelwerk/diagramme/ableitung-sinus-cosinus-en-dark.svg?v=921541759fc31404 Sine curve with three short tangents and a dashed cosine curve; at the same points cos x takes the values 1, 0 and −1. The slope of sin x equals cos x at every point: slope 1 at x = 0, horizontal at π/2, slope −1 at π. https://quanta-study.de/en/formula-sheet/quotientenregel-ableitung 2026-07-09 monthly 0.82 https://quanta-study.de/en/formula-sheet/grundintegrale-stammfunktion 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/grundintegrale-stammfunktion-en-light.svg?v=bc02c83efcd4ab73 Three parabolas shifted vertically against each other for C = 1, 0 and −1, with parallel tangents at the same point. Antiderivatives of f(x) = x: the curves x²/2 + C differ only by the constant C and have the same slope at every point. https://quanta-study.de/tafelwerk/diagramme/grundintegrale-stammfunktion-en-dark.svg?v=bc02c83efcd4ab73 Three parabolas shifted vertically against each other for C = 1, 0 and −1, with parallel tangents at the same point. Antiderivatives of f(x) = x: the curves x²/2 + C differ only by the constant C and have the same slope at every point. https://quanta-study.de/en/formula-sheet/hauptsatz-integralrechnung 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/hauptsatz-integralrechnung-en-light.svg?v=4f883b16bb6c2157 On the left the line f with a shaded area between a and b, on the right the parabola F with a highlighted arrow from the level F(a) to the level F(b). The area under f from a to b (left) equals exactly the rise of the antiderivative F from F(a) to F(b) (right). https://quanta-study.de/tafelwerk/diagramme/hauptsatz-integralrechnung-en-dark.svg?v=4f883b16bb6c2157 On the left the line f with a shaded area between a and b, on the right the parabola F with a highlighted arrow from the level F(a) to the level F(b). The area under f from a to b (left) equals exactly the rise of the antiderivative F from F(a) to F(b) (right). https://quanta-study.de/en/formula-sheet/bestimmtes-integral-flaeche 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/bestimmtes-integral-flaeche-en-light.svg?v=3cf02c0c2e6868d7 Diagram: a positive curve f(x); the area between the curve and the x-axis from a to b is shaded and labelled A. The definite integral is the area A between the graph of f and the x-axis from a to b. https://quanta-study.de/tafelwerk/diagramme/bestimmtes-integral-flaeche-en-dark.svg?v=3cf02c0c2e6868d7 Diagram: a positive curve f(x); the area between the curve and the x-axis from a to b is shaded and labelled A. The definite integral is the area A between the graph of f and the x-axis from a to b. https://quanta-study.de/en/formula-sheet/partielle-integration 2026-07-09 monthly 0.82 https://quanta-study.de/en/formula-sheet/binomialverteilung 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/binomialverteilung-en-light.svg?v=63204a933ad1feb4 Bar chart of the binomial distribution for n equal 10 and p equal 0.5; the probabilities P(X=k) form a symmetric bell shape peaking at k equal 5. Binomial distribution for n = 10 and p = 0.5: symmetric about the expected value μ = n·p = 5. https://quanta-study.de/tafelwerk/diagramme/binomialverteilung-en-dark.svg?v=63204a933ad1feb4 Bar chart of the binomial distribution for n equal 10 and p equal 0.5; the probabilities P(X=k) form a symmetric bell shape peaking at k equal 5. Binomial distribution for n = 10 and p = 0.5: symmetric about the expected value μ = n·p = 5. https://quanta-study.de/en/formula-sheet/erwartungswert 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/erwartungswert-en-light.svg?v=579cbe107e458873 Bar chart with five bars for the values 0 to 4; a triangle below the axis at 1.6 marks the expected value as the balance point. The expected value is the balance point of the distribution: with the probabilities pᵢ as weights at the values xᵢ, it balances at μ = Σ xᵢ·pᵢ = 1.6. https://quanta-study.de/tafelwerk/diagramme/erwartungswert-en-dark.svg?v=579cbe107e458873 Bar chart with five bars for the values 0 to 4; a triangle below the axis at 1.6 marks the expected value as the balance point. The expected value is the balance point of the distribution: with the probabilities pᵢ as weights at the values xᵢ, it balances at μ = Σ xᵢ·pᵢ = 1.6. https://quanta-study.de/en/formula-sheet/bedingte-wahrscheinlichkeit 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/bedingte-wahrscheinlichkeit-en-light.svg?v=cd1d615e0363d028 Two overlapping circles A and B in a frame; B has a bold outline, the intersection A∩B is highlighted. Given B, only B counts: P(A|B) is the share of the intersection A∩B within B. https://quanta-study.de/tafelwerk/diagramme/bedingte-wahrscheinlichkeit-en-dark.svg?v=cd1d615e0363d028 Two overlapping circles A and B in a frame; B has a bold outline, the intersection A∩B is highlighted. Given B, only B counts: P(A|B) is the share of the intersection A∩B within B. https://quanta-study.de/en/formula-sheet/standardabweichung 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/standardabweichung-en-light.svg?v=ff2024baa300982d Diagram: a bell curve with shaded bands; the central band μ ± σ covers about 68 percent, the wider band μ ± 2σ about 95 percent of the area. About 68 % of the values lie within μ ± σ, and about 95 % within μ ± 2σ. https://quanta-study.de/tafelwerk/diagramme/standardabweichung-en-dark.svg?v=ff2024baa300982d Diagram: a bell curve with shaded bands; the central band μ ± σ covers about 68 percent, the wider band μ ± 2σ about 95 percent of the area. About 68 % of the values lie within μ ± σ, and about 95 % within μ ± 2σ. https://quanta-study.de/en/formula-sheet/skalarprodukt 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/skalarprodukt-en-light.svg?v=1787c4f515a6958b Diagram: two vectors a and b from the same point with enclosed angle φ; the projection of a onto b is marked as the segment |a|·cos φ. The scalar product measures the projection of a onto b: a·b = |a|·|b|·cos φ. https://quanta-study.de/tafelwerk/diagramme/skalarprodukt-en-dark.svg?v=1787c4f515a6958b Diagram: two vectors a and b from the same point with enclosed angle φ; the projection of a onto b is marked as the segment |a|·cos φ. The scalar product measures the projection of a onto b: a·b = |a|·|b|·cos φ. https://quanta-study.de/en/formula-sheet/kreuzprodukt 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/kreuzprodukt-en-light.svg?v=c7a265bcb5eaa613 Diagram: vectors a and b span a parallelogram; the result vector a×b points perpendicular upward, its length equals the parallelogram area. The cross-product vector is perpendicular to a and b; its magnitude is the area of the spanned parallelogram. https://quanta-study.de/tafelwerk/diagramme/kreuzprodukt-en-dark.svg?v=c7a265bcb5eaa613 Diagram: vectors a and b span a parallelogram; the result vector a×b points perpendicular upward, its length equals the parallelogram area. The cross-product vector is perpendicular to a and b; its magnitude is the area of the spanned parallelogram. https://quanta-study.de/en/formula-sheet/logarithmengesetze 2026-07-09 monthly 0.82 https://quanta-study.de/en/formula-sheet/kosinussatz 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/kosinussatz-en-light.svg?v=e72dcbdbf6c1d59d Diagram: a triangle with sides a, b, c; the marked angle γ lies opposite side c. The law of cosines generalises the Pythagorean theorem to any triangle: c² = a² + b² − 2ab·cos γ. https://quanta-study.de/tafelwerk/diagramme/kosinussatz-en-dark.svg?v=e72dcbdbf6c1d59d Diagram: a triangle with sides a, b, c; the marked angle γ lies opposite side c. The law of cosines generalises the Pythagorean theorem to any triangle: c² = a² + b² − 2ab·cos γ. https://quanta-study.de/en/formula-sheet/sinussatz 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/sinussatz-en-light.svg?v=b8183e0ccd70b1ef Diagram: a triangle with angles α, β, γ and the sides a, b, c opposite to them. The law of sines relates sides and opposite angles: a/sin α = b/sin β = c/sin γ. https://quanta-study.de/tafelwerk/diagramme/sinussatz-en-dark.svg?v=b8183e0ccd70b1ef Diagram: a triangle with angles α, β, γ and the sides a, b, c opposite to them. The law of sines relates sides and opposite angles: a/sin α = b/sin β = c/sin γ. https://quanta-study.de/en/formula-sheet/kreisflaeche-umfang 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/kreisflaeche-umfang-en-light.svg?v=8d20ce4eabf1078f Diagram: a circle with its radius r drawn in; inside the area A = πr², around it the circumference U = 2πr. Circle with radius r: area A = πr², circumference U = 2πr. https://quanta-study.de/tafelwerk/diagramme/kreisflaeche-umfang-en-dark.svg?v=8d20ce4eabf1078f Diagram: a circle with its radius r drawn in; inside the area A = πr², around it the circumference U = 2πr. Circle with radius r: area A = πr², circumference U = 2πr. https://quanta-study.de/en/formula-sheet/kugel-volumen-oberflaeche 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/kugel-volumen-oberflaeche-en-light.svg?v=f069230ce8daa9cd Sphere with a suggested equator and its radius r drawn in; above it the surface area O = 4πr², inside the volume V = 4/3·πr³. Sphere with radius r: volume V = 4/3·πr³, surface area O = 4πr². https://quanta-study.de/tafelwerk/diagramme/kugel-volumen-oberflaeche-en-dark.svg?v=f069230ce8daa9cd Sphere with a suggested equator and its radius r drawn in; above it the surface area O = 4πr², inside the volume V = 4/3·πr³. Sphere with radius r: volume V = 4/3·πr³, surface area O = 4πr². https://quanta-study.de/en/formula-sheet/zylinder-volumen-oberflaeche 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/zylinder-volumen-oberflaeche-en-light.svg?v=194f2569f6cf57d7 On the left a cylinder with radius r and height h, on the right its unfolded net of two circles and a highlighted rectangle of width 2πr. Cylinder with radius r and height h and its net: two circles and the lateral rectangle M with sides 2πr and h. https://quanta-study.de/tafelwerk/diagramme/zylinder-volumen-oberflaeche-en-dark.svg?v=194f2569f6cf57d7 On the left a cylinder with radius r and height h, on the right its unfolded net of two circles and a highlighted rectangle of width 2πr. Cylinder with radius r and height h and its net: two circles and the lateral rectangle M with sides 2πr and h. https://quanta-study.de/en/formula-sheet/kegel-volumen-oberflaeche 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/kegel-volumen-oberflaeche-en-light.svg?v=e9b6c94bc13552a7 Cone with a dashed height h, the radius r on the base circle and a highlighted slant height s; a right angle between h and r. Cone with radius r, height h and slant height s = √(r² + h²); r, h and s form a right triangle. https://quanta-study.de/tafelwerk/diagramme/kegel-volumen-oberflaeche-en-dark.svg?v=e9b6c94bc13552a7 Cone with a dashed height h, the radius r on the base circle and a highlighted slant height s; a right angle between h and r. Cone with radius r, height h and slant height s = √(r² + h²); r, h and s form a right triangle. https://quanta-study.de/en/formula-sheet/pyramide-volumen 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/pyramide-volumen-en-light.svg?v=fff06374e4e7a428 Square pyramid in oblique projection with a shaded base G and a highlighted perpendicular height h from the apex to the centre of the base. Pyramid with base area G and perpendicular height h: it holds one third of the prism with the same base and height. https://quanta-study.de/tafelwerk/diagramme/pyramide-volumen-en-dark.svg?v=fff06374e4e7a428 Square pyramid in oblique projection with a shaded base G and a highlighted perpendicular height h from the apex to the centre of the base. Pyramid with base area G and perpendicular height h: it holds one third of the prism with the same base and height. https://quanta-study.de/en/formula-sheet/trigonometrie-rechtwinkliges-dreieck 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/trigonometrie-rechtwinkliges-dreieck-en-light.svg?v=6cc7215fb7c5fad2 Right triangle with the angle α marked; the side across from it is the opposite side, the side next to it the adjacent side, the longest the hypotenuse. Relative to the angle α: sin α = opposite/hypotenuse, cos α = adjacent/hypotenuse, tan α = opposite/adjacent. https://quanta-study.de/tafelwerk/diagramme/trigonometrie-rechtwinkliges-dreieck-en-dark.svg?v=6cc7215fb7c5fad2 Right triangle with the angle α marked; the side across from it is the opposite side, the side next to it the adjacent side, the longest the hypotenuse. Relative to the angle α: sin α = opposite/hypotenuse, cos α = adjacent/hypotenuse, tan α = opposite/adjacent. https://quanta-study.de/en/formula-sheet/geradengleichung-steigung 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/geradengleichung-steigung-en-light.svg?v=2d21d926a439faf7 Rising line through the points (x₁|y₁) and (x₂|y₂) with a slope triangle Δx and Δy; the y-intercept b is marked. The slope m = Δy/Δx is the same between any two points; b is the y-intercept. https://quanta-study.de/tafelwerk/diagramme/geradengleichung-steigung-en-dark.svg?v=2d21d926a439faf7 Rising line through the points (x₁|y₁) and (x₂|y₂) with a slope triangle Δx and Δy; the y-intercept b is marked. The slope m = Δy/Δx is the same between any two points; b is the y-intercept. https://quanta-study.de/en/formula-sheet/hesse-normalform-abstand-ebene 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/hesse-normalform-abstand-ebene-en-light.svg?v=0df6f977211e5bad Slanted plane E with normal vector n; from a point P above it a highlighted perpendicular leads down to the plane, marked as d(P;E). The distance d(P;E) is the length of the perpendicular from P to E; it runs parallel to the normal vector n. https://quanta-study.de/tafelwerk/diagramme/hesse-normalform-abstand-ebene-en-dark.svg?v=0df6f977211e5bad Slanted plane E with normal vector n; from a point P above it a highlighted perpendicular leads down to the plane, marked as d(P;E). The distance d(P;E) is the length of the perpendicular from P to E; it runs parallel to the normal vector n. https://quanta-study.de/en/formula-sheet/ebenengleichung-normalenform 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/ebenengleichung-normalenform-en-light.svg?v=d2f7d61ec88ff7bb Slanted plane E; from the origin the position vectors p and x lead to two points of the plane, the highlighted connecting vector x − p lies in the plane and is perpendicular to the normal vector n. A point X lies in E exactly when the vector x − p is perpendicular to the normal vector n: (x − p)·n = 0. https://quanta-study.de/tafelwerk/diagramme/ebenengleichung-normalenform-en-dark.svg?v=d2f7d61ec88ff7bb Slanted plane E; from the origin the position vectors p and x lead to two points of the plane, the highlighted connecting vector x − p lies in the plane and is perpendicular to the normal vector n. A point X lies in E exactly when the vector x − p is perpendicular to the normal vector n: (x − p)·n = 0. https://quanta-study.de/en/formula-sheet/abstand-punkt-gerade 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/abstand-punkt-gerade-en-light.svg?v=e4536c39a758f508 Line g with direction vector u and point Q above it; the parallelogram spanned by q − p and u is shaded, the highlighted perpendicular from Q to g is its height d(Q;g). The distance d(Q;g) is the height of the parallelogram spanned by q − p and u: area |⁠(q − p) × u⁠| divided by the base |⁠u⁠|. https://quanta-study.de/tafelwerk/diagramme/abstand-punkt-gerade-en-dark.svg?v=e4536c39a758f508 Line g with direction vector u and point Q above it; the parallelogram spanned by q − p and u is shaded, the highlighted perpendicular from Q to g is its height d(Q;g). The distance d(Q;g) is the height of the parallelogram spanned by q − p and u: area |⁠(q − p) × u⁠| divided by the base |⁠u⁠|. https://quanta-study.de/en/formula-sheet/vektor-betrag-laenge 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/vektor-betrag-laenge-en-light.svg?v=f5650e73229e892c Box in oblique projection with the edges v₁, v₂ and v₃; the dashed floor diagonal and the vertical edge v₃ form a right angle, the highlighted space diagonal is the vector v. The length |v| is the space diagonal of the box with edges v₁, v₂, v₃: Pythagoras first on the floor, then along the vertical edge. https://quanta-study.de/tafelwerk/diagramme/vektor-betrag-laenge-en-dark.svg?v=f5650e73229e892c Box in oblique projection with the edges v₁, v₂ and v₃; the dashed floor diagonal and the vertical edge v₃ form a right angle, the highlighted space diagonal is the vector v. The length |v| is the space diagonal of the box with edges v₁, v₂, v₃: Pythagoras first on the floor, then along the vertical edge. https://quanta-study.de/en/formula-sheet/differenzenquotient-h-methode 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/differenzenquotient-h-methode-en-light.svg?v=f45a844e72e2c359 Parabola with a secant through two points a distance h apart, its slope triangle, and the highlighted tangent at the point x₀. The secant through x₀ and x₀ + h has slope (f(x₀+h) − f(x₀))/h; as h → 0 it becomes the tangent with slope f′(x₀). https://quanta-study.de/tafelwerk/diagramme/differenzenquotient-h-methode-en-dark.svg?v=f45a844e72e2c359 Parabola with a secant through two points a distance h apart, its slope triangle, and the highlighted tangent at the point x₀. The secant through x₀ and x₀ + h has slope (f(x₀+h) − f(x₀))/h; as h → 0 it becomes the tangent with slope f′(x₀). https://quanta-study.de/en/formula-sheet/scheitelpunktform-parabel 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/scheitelpunktform-parabel-en-light.svg?v=d1b687cf36bf34fa Dashed standard parabola at the origin and the highlighted shifted parabola with a = 1 and vertex S(d|e); guide lines mark d on the x-axis and e on the y-axis. For a = 1 the vertex form shifts the standard parabola (dashed) by d to the right and by e upward; the vertex is S(d⁠|⁠e). https://quanta-study.de/tafelwerk/diagramme/scheitelpunktform-parabel-en-dark.svg?v=d1b687cf36bf34fa Dashed standard parabola at the origin and the highlighted shifted parabola with a = 1 and vertex S(d|e); guide lines mark d on the x-axis and e on the y-axis. For a = 1 the vertex form shifts the standard parabola (dashed) by d to the right and by e upward; the vertex is S(d⁠|⁠e). https://quanta-study.de/en/formula-sheet/eigenwerte-vektoren 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kleinste-quadrate-matrix 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/lineare-regression 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/lineare-regression-en-light.svg?v=e7c6e1d11094ff50 Seven data points with a highlighted regression line; dashed vertical lines show the deviations, and a ring marks the mean point (x̄ | ȳ) on the line. The regression line ŷ = a + b·x minimises the sum of squared vertical deviations and passes through the mean point of the data (ring), whose coordinates are the means of x and y. https://quanta-study.de/tafelwerk/diagramme/lineare-regression-en-dark.svg?v=e7c6e1d11094ff50 Seven data points with a highlighted regression line; dashed vertical lines show the deviations, and a ring marks the mean point (x̄ | ȳ) on the line. The regression line ŷ = a + b·x minimises the sum of squared vertical deviations and passes through the mean point of the data (ring), whose coordinates are the means of x and y. https://quanta-study.de/en/formula-sheet/binomialkoeffizient-kombinatorik 2026-07-09 monthly 0.82 https://quanta-study.de/en/formula-sheet/geometrische-reihe 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/geometrische-reihe-en-light.svg?v=21e4e19f921be2a1 A square, repeatedly halved into areas 1/2, 1/4, 1/8, 1/16 and ever smaller pieces that together fill the whole square. With a₁ = 1/2 and q = 1/2 the terms 1/2 + 1/4 + 1/8 + … fill the unit square: s∞ = a₁/(1 − q) = 1. https://quanta-study.de/tafelwerk/diagramme/geometrische-reihe-en-dark.svg?v=21e4e19f921be2a1 A square, repeatedly halved into areas 1/2, 1/4, 1/8, 1/16 and ever smaller pieces that together fill the whole square. With a₁ = 1/2 and q = 1/2 the terms 1/2 + 1/4 + 1/8 + … fill the unit square: s∞ = a₁/(1 − q) = 1. https://quanta-study.de/en/formula-sheet/arithmetische-reihe-gauss 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/arithmetische-reihe-gauss-en-light.svg?v=83fb8bedda131046 A highlighted staircase of 1 to 5 squares and a faint flipped copy combine into a rectangle with n columns and a₁ + aₙ rows. Two equal staircases 1 + 2 + … + n combine into the rectangle n·(a₁ + aₙ); one staircase is half of it: sₙ = n·(a₁ + aₙ)/2. https://quanta-study.de/tafelwerk/diagramme/arithmetische-reihe-gauss-en-dark.svg?v=83fb8bedda131046 A highlighted staircase of 1 to 5 squares and a faint flipped copy combine into a rectangle with n columns and a₁ + aₙ rows. Two equal staircases 1 + 2 + … + n combine into the rectangle n·(a₁ + aₙ); one staircase is half of it: sₙ = n·(a₁ + aₙ)/2. https://quanta-study.de/en/formula-sheet/rotationsvolumen 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/rotationsvolumen-en-light.svg?v=7c318bc1930738a8 The graph of a square root function and its dashed mirror image form a solid of revolution between a and b; a highlighted disc has the radius f(x). Rotating the graph of f about the x-axis gives a solid made of discs with radius f(x); summing their areas π·f(x)² from a to b gives V. https://quanta-study.de/tafelwerk/diagramme/rotationsvolumen-en-dark.svg?v=7c318bc1930738a8 The graph of a square root function and its dashed mirror image form a solid of revolution between a and b; a highlighted disc has the radius f(x). Rotating the graph of f about the x-axis gives a solid made of discs with radius f(x); summing their areas π·f(x)² from a to b gives V. https://quanta-study.de/en/formula-sheet/dreiecksflaeche-sinus 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/dreiecksflaeche-sinus-en-light.svg?v=ec88b5ff5fdb272b Triangle with sides a and b and the included angle γ; the highlighted height onto b is labelled a·sin γ. The height onto side b is a·sin γ, hence A = ½·b·(a·sin γ) = ½·a·b·sin γ. https://quanta-study.de/tafelwerk/diagramme/dreiecksflaeche-sinus-en-dark.svg?v=ec88b5ff5fdb272b Triangle with sides a and b and the included angle γ; the highlighted height onto b is labelled a·sin γ. The height onto side b is a·sin γ, hence A = ½·b·(a·sin γ) = ½·a·b·sin γ. https://quanta-study.de/en/formula-sheet/poisson-verteilung 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/poisson-verteilung-en-light.svg?v=1bedd88d88d68e5b Bar chart of the Poisson distribution for λ equal 2 with equally tall bars at k equal 1 and 2 and bars falling off quickly to the right. Poisson distribution for λ = 2: right-skewed, 1 and 2 events are most likely, large values of k quickly become rare. https://quanta-study.de/tafelwerk/diagramme/poisson-verteilung-en-dark.svg?v=1bedd88d88d68e5b Bar chart of the Poisson distribution for λ equal 2 with equally tall bars at k equal 1 and 2 and bars falling off quickly to the right. Poisson distribution for λ = 2: right-skewed, 1 and 2 events are most likely, large values of k quickly become rare. https://quanta-study.de/en/formula-sheet/sigma-regeln-umgebung 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/sigma-regeln-umgebung-en-light.svg?v=61e62c6f1a900796 Diagram: a bell curve with a highlighted area from μ − 2σ to μ + 2σ; below it three nested double arrows for the 1σ, 2σ and 3σ neighbourhoods with their percentages. Sigma rules: about 68.3 % of the values lie within μ ± σ, about 95.4 % within μ ± 2σ and about 99.7 % within μ ± 3σ. https://quanta-study.de/tafelwerk/diagramme/sigma-regeln-umgebung-en-dark.svg?v=61e62c6f1a900796 Diagram: a bell curve with a highlighted area from μ − 2σ to μ + 2σ; below it three nested double arrows for the 1σ, 2σ and 3σ neighbourhoods with their percentages. Sigma rules: about 68.3 % of the values lie within μ ± σ, about 95.4 % within μ ± 2σ and about 99.7 % within μ ± 3σ. https://quanta-study.de/en/formula-sheet/gradient-partielle-ableitungen 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/mehrfachintegral 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/lagrange-multiplikatoren 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/lineares-gleichungssystem-matrixrang 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/lineare-differentialgleichung-erster-ordnung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/konfidenzintervall-mittelwert 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/jacobi-determinante-variablentransformation 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/jacobi-determinante-variablentransformation-en-light.svg?v=485df09853ccb6e5 A unit square in uv coordinates maps under x=2u and y=3v to a rectangle twice as wide and three times as tall; area scales by a factor of six. The map x=2u, y=3v stretches the unit square into a rectangle; |J|=6 is the area scale factor. https://quanta-study.de/tafelwerk/diagramme/jacobi-determinante-variablentransformation-en-dark.svg?v=485df09853ccb6e5 A unit square in uv coordinates maps under x=2u and y=3v to a rectangle twice as wide and three times as tall; area scales by a factor of six. The map x=2u, y=3v stretches the unit square into a rectangle; |J|=6 is the area scale factor. https://quanta-study.de/en/formula-sheet/hesse-matrix-zweiter-ableitungstest 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/newton-verfahren-nullstellen 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/newton-verfahren-nullstellen-en-light.svg?v=a1510c78737fdda6 Graph of f(x)=x²−2: a point at x0 and its tangent meet the x-axis at x1; the curved function has its root close to x1. The tangent at x₀ crosses the x-axis at x₁; that intercept becomes the next approximation. https://quanta-study.de/tafelwerk/diagramme/newton-verfahren-nullstellen-en-dark.svg?v=a1510c78737fdda6 Graph of f(x)=x²−2: a point at x0 and its tangent meet the x-axis at x1; the curved function has its root close to x1. The tangent at x₀ crosses the x-axis at x₁; that intercept becomes the next approximation. https://quanta-study.de/en/formula-sheet/zusammengesetzte-trapezregel 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/zusammengesetzte-trapezregel-en-light.svg?v=b21cbf4541e8d014 Curve f(x)=x² over the interval from 0 to 2; accented chords connect three sample points and lie above this convex curve. The curved area is approximated by two trapezoids; for this convex function, each chord lies above the curve. https://quanta-study.de/tafelwerk/diagramme/zusammengesetzte-trapezregel-en-dark.svg?v=b21cbf4541e8d014 Curve f(x)=x² over the interval from 0 to 2; accented chords connect three sample points and lie above this convex curve. The curved area is approximated by two trapezoids; for this convex function, each chord lies above the curve. https://quanta-study.de/en/formula-sheet/determinante-matrix-invertierbarkeit 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/inverse-matrix-lineares-system 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/lu-zerlegung-matrix 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/qr-zerlegung-matrix 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/singulaerwertzerlegung-matrix 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/gram-schmidt-orthogonalisierung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/konditionszahl-matrix 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/simpsonregel-numerische-integration 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/mehrdimensionale-kettenregel 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/richtungsableitung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/divergenz-vektorfeld 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/rotation-curl-vektorfeld 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/vektorielles-linienintegral 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/greenscher-satz-ebene 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/stokes-satz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/taylorpolynom-mehrdimensional 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/fourierreihe-periodische-funktion 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/laplace-transformation 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/trennbare-differentialgleichung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/lineare-differentialgleichung-zweiter-ordnung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/euler-verfahren-differentialgleichung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/stichprobenvarianz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kovarianz-stichprobe 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/pearson-korrelationskoeffizient 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/t-test-statistik-einstichprobe 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/chi-quadrat-anpassungstest 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/standardfehler-anteil 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/gradientenabstieg 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/newton-verfahren-optimierung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/gauss-elimination-lineares-system 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/runge-kutta-verfahren-vierter-ordnung 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/runge-kutta-verfahren-vierter-ordnung-en-light.svg?v=c4ea85c6b6458fd7 The smooth reference curve y=e to the t shows one step from t_n to t_n+h. Samples are k1 at (t_n,y_n), k2 and k3 at the same midpoint time with different trial values, and k4 at the end time using the third trial value. The internal points are RK4 trial states and need not lie exactly on the exact curve. For y′=y and h=1, k₁…k₄ sample the start, two midpoint trial states and the end time; trial values need not lie exactly on y=eᵗ. https://quanta-study.de/tafelwerk/diagramme/runge-kutta-verfahren-vierter-ordnung-en-dark.svg?v=c4ea85c6b6458fd7 The smooth reference curve y=e to the t shows one step from t_n to t_n+h. Samples are k1 at (t_n,y_n), k2 and k3 at the same midpoint time with different trial values, and k4 at the end time using the third trial value. The internal points are RK4 trial states and need not lie exactly on the exact curve. For y′=y and h=1, k₁…k₄ sample the start, two midpoint trial states and the end time; trial values need not lie exactly on y=eᵗ. https://quanta-study.de/en/formula-sheet/prozentrechnung 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/prozentrechnung-en-light.svg?v=9f504fbb02be1b96 A bar for the whole value of 84 euros represents 100 percent. The highlighted 15 percent segment is labelled as the part value, 12.60 euros. The percentage bar shows part W as p of whole G: 15% of €84 is €12.60. https://quanta-study.de/tafelwerk/diagramme/prozentrechnung-en-dark.svg?v=9f504fbb02be1b96 A bar for the whole value of 84 euros represents 100 percent. The highlighted 15 percent segment is labelled as the part value, 12.60 euros. The percentage bar shows part W as p of whole G: 15% of €84 is €12.60. https://quanta-study.de/en/formula-sheet/direkte-proportionalitaet 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/direkte-proportionalitaet-en-light.svg?v=f467bb77408df803 Coordinate axes show an increasing straight line through the origin. When x doubles, y doubles; marked points share the same ratio y/x=k. For direct proportion, y=kx: the graph is a straight line through the origin and y/x stays constant. https://quanta-study.de/tafelwerk/diagramme/direkte-proportionalitaet-en-dark.svg?v=f467bb77408df803 Coordinate axes show an increasing straight line through the origin. When x doubles, y doubles; marked points share the same ratio y/x=k. For direct proportion, y=kx: the graph is a straight line through the origin and y/x stays constant. https://quanta-study.de/en/formula-sheet/antiproportionale-zuordnung 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/antiproportionale-zuordnung-en-light.svg?v=3c2c20366f9bcaef A hyperbola branch in the first quadrant falls from upper left to lower right. The marked points (1,4), (2,2) and (4,1) all have product xy=4. For y=k/x, y falls as x grows; along the curve the product xy=k stays constant. https://quanta-study.de/tafelwerk/diagramme/antiproportionale-zuordnung-en-dark.svg?v=3c2c20366f9bcaef A hyperbola branch in the first quadrant falls from upper left to lower right. The marked points (1,4), (2,2) and (4,1) all have product xy=4. For y=k/x, y falls as x grows; along the curve the product xy=k stays constant. https://quanta-study.de/en/formula-sheet/prozentuales-wachstum 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/prozentuales-wachstum-en-light.svg?v=f71a010a84ef501a A quantity-time graph shows a curve bending upward. Equal time steps use the same percentage factor, not the same absolute increase. Percentage growth multiplies by 1+r at every step, so G_n rises exponentially. https://quanta-study.de/tafelwerk/diagramme/prozentuales-wachstum-en-dark.svg?v=f71a010a84ef501a A quantity-time graph shows a curve bending upward. Equal time steps use the same percentage factor, not the same absolute increase. Percentage growth multiplies by 1+r at every step, so G_n rises exponentially. https://quanta-study.de/en/formula-sheet/einfache-zinsrechnung 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/einfache-zinsrechnung-en-light.svg?v=8175ad39423235dd A capital-time graph shows a straight line. Equal time intervals add the same amount; interest is not compounded. With simple interest, capital grows linearly because each period adds the same interest amount on the initial principal. https://quanta-study.de/tafelwerk/diagramme/einfache-zinsrechnung-en-dark.svg?v=8175ad39423235dd A capital-time graph shows a straight line. Equal time intervals add the same amount; interest is not compounded. With simple interest, capital grows linearly because each period adds the same interest amount on the initial principal. https://quanta-study.de/en/formula-sheet/grenzwert-stetigkeit 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/komplexe-zahlen-polarform 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/komplexe-zahlen-polarform-en-light.svg?v=f2c91485523485ef Complex plane with Re and Im axes. A vector from the origin to z has angle φ and modulus r; its projections mark x and y. Polar form separates z into modulus r and argument φ; taking a power multiplies the angle and raises the modulus. https://quanta-study.de/tafelwerk/diagramme/komplexe-zahlen-polarform-en-dark.svg?v=f2c91485523485ef Complex plane with Re and Im axes. A vector from the origin to z has angle φ and modulus r; its projections mark x and y. Polar form separates z into modulus r and argument φ; taking a power multiplies the angle and raises the modulus. https://quanta-study.de/en/formula-sheet/unsicherheitsfortpflanzung-nist 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/t-chi-quadrat-f-verteilungen 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/reihen-konvergenztests 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/uneigentliche-integrale 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/zentraler-grenzwertsatz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/fourier-transformation 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/gekoppelte-lineare-dgl-systeme 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/maximum-likelihood-schaetzung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/markov-kette-uebergangsmatrix 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/polynominterpolation-lagrange-newton 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/referenz-verteilungen-geometrisch-hypergeometrisch-gamma 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/integrationsregel-substitution 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/trigonometrische-identitaeten 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/bogenmass-bogenlaenge 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/vierecksflaechen-parallelogramm-trapez-drachen 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/prisma-volumen-oberflaeche 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/dreiecke-aehnlichkeit-skalierungsfaktoren 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/mittelwertsatz-analysis 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/implizite-ableitung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/ableitung-umkehrfunktion 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/partialbruchzerlegung-rationale-funktion 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/ellipsen-hyperbeln-brennpunkte 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/bogenlaenge-allgemeine-kurve 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/rotationsoberflaeche-integral 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/stetige-gleichverteilung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/cauchy-schwarz-ungleichung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/gauss-legendre-quadratur 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/multivariate-normalverteilung-kovarianz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/bienayme-tschebyschow-ungleichung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/michaelis-menten 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/michaelis-menten-en-light.svg?v=d38d65979cf57646 Diagram: a saturation curve v over [S] approaches a dashed line Vmax; dashed guides mark Vmax/2 and the corresponding concentration Km. The reaction rate v rises with the substrate concentration [S] and approaches the maximum rate Vmax; at [S] = Km the rate is v = Vmax/2. https://quanta-study.de/tafelwerk/diagramme/michaelis-menten-en-dark.svg?v=d38d65979cf57646 Diagram: a saturation curve v over [S] approaches a dashed line Vmax; dashed guides mark Vmax/2 and the corresponding concentration Km. The reaction rate v rises with the substrate concentration [S] and approaches the maximum rate Vmax; at [S] = Km the rate is v = Vmax/2. https://quanta-study.de/en/formula-sheet/logistisches-wachstum 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/logistisches-wachstum-en-light.svg?v=c3f81204f413799e Diagram: an S-shaped curve N over t approaching the dashed limit K; the inflection point at K/2 is marked, and a dashed exponential curve keeps rising unchecked for comparison. Logistic growth: almost exponential at first (dashed), then the carrying capacity K slows it down; the population grows fastest at N = K/2. https://quanta-study.de/tafelwerk/diagramme/logistisches-wachstum-en-dark.svg?v=c3f81204f413799e Diagram: an S-shaped curve N over t approaching the dashed limit K; the inflection point at K/2 is marked, and a dashed exponential curve keeps rising unchecked for comparison. Logistic growth: almost exponential at first (dashed), then the carrying capacity K slows it down; the population grows fastest at N = K/2. https://quanta-study.de/en/formula-sheet/hardy-weinberg 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/hardy-weinberg-en-light.svg?v=6c5772083a562cd5 Diagram: a square with side p + q, divided at p; the four fields are p² (AA), twice pq (Aa, highlighted) and q² (aa). The square (p + q)² with p = 0.6 and q = 0.4: the areas p², twice pq and q² are the genotype frequencies and together add up to 1. https://quanta-study.de/tafelwerk/diagramme/hardy-weinberg-en-dark.svg?v=6c5772083a562cd5 Diagram: a square with side p + q, divided at p; the four fields are p² (AA), twice pq (Aa, highlighted) and q² (aa). The square (p + q)² with p = 0.6 and q = 0.4: the areas p², twice pq and q² are the genotype frequencies and together add up to 1. https://quanta-study.de/en/formula-sheet/exponentielles-wachstum 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/exponentielles-wachstum-en-light.svg?v=a058428c77cba489 Diagram: a rising exponential curve N over time t, starting from the initial value N₀ on the N axis. Exponential growth: the increase is always proportional to the current size, so the curve rises ever more steeply. https://quanta-study.de/tafelwerk/diagramme/exponentielles-wachstum-en-dark.svg?v=a058428c77cba489 Diagram: a rising exponential curve N over time t, starting from the initial value N₀ on the N axis. Exponential growth: the increase is always proportional to the current size, so the curve rises ever more steeply. https://quanta-study.de/en/formula-sheet/chi-quadrat-test 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/chi-quadrat-test-en-light.svg?v=11732e094dd97d8c Diagram: a density curve over χ² falling steeply from the top left; to the right of the critical value 3.84 the narrow 5 percent tail is highlighted, a dashed line marks the computed value 1.33 to its left. Chi-squared distribution for df = 1: the computed value χ² = 1.33 lies below the critical value 3.84, only values in the upper 5 % lead to rejecting the hypothesis. https://quanta-study.de/tafelwerk/diagramme/chi-quadrat-test-en-dark.svg?v=11732e094dd97d8c Diagram: a density curve over χ² falling steeply from the top left; to the right of the critical value 3.84 the narrow 5 percent tail is highlighted, a dashed line marks the computed value 1.33 to its left. Chi-squared distribution for df = 1: the computed value χ² = 1.33 lies below the critical value 3.84, only values in the upper 5 % lead to rejecting the hypothesis. https://quanta-study.de/en/formula-sheet/shannon-index 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/shannon-index-en-light.svg?v=94524feb85a7e337 Diagram: three horizontal bars, each split into three species shares pᵢ, on top equal shares with H' = 1.10, in the middle 0.5, 0.3 and 0.2 with H' = 1.03, at the bottom one dominant species at 0.8 with H' = 0.64. Three communities with S = 3 species each: the more evenly the individuals are distributed, the larger H' becomes; the perfectly even distribution reaches the maximum ln 3 = 1.10. https://quanta-study.de/tafelwerk/diagramme/shannon-index-en-dark.svg?v=94524feb85a7e337 Diagram: three horizontal bars, each split into three species shares pᵢ, on top equal shares with H' = 1.10, in the middle 0.5, 0.3 and 0.2 with H' = 1.03, at the bottom one dominant species at 0.8 with H' = 0.64. Three communities with S = 3 species each: the more evenly the individuals are distributed, the larger H' becomes; the perfectly even distribution reaches the maximum ln 3 = 1.10. https://quanta-study.de/en/formula-sheet/simpson-index 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/simpson-index-en-light.svg?v=24b448cdda6eeceb Diagram: a square split on both sides into the species shares p₁, p₂ and p₃; the three squares on the diagonal p₁², p₂² and p₃² are highlighted and together give D. D is the probability that two randomly drawn individuals belong to the same species: the sum of the diagonal squares pᵢ², here 0.38 of the total area. https://quanta-study.de/tafelwerk/diagramme/simpson-index-en-dark.svg?v=24b448cdda6eeceb Diagram: a square split on both sides into the species shares p₁, p₂ and p₃; the three squares on the diagonal p₁², p₂² and p₃² are highlighted and together give D. D is the probability that two randomly drawn individuals belong to the same species: the sum of the diagonal squares pᵢ², here 0.38 of the total area. https://quanta-study.de/en/formula-sheet/lincoln-index 2026-07-09 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/lincoln-index-en-light.svg?v=cf6c36ec70e1031f Diagram: two bars of equal length for the population N and the second catch C; in both, the marked share M or R is highlighted with the same size, a dashed line links the two boundaries, and M/N = R/C is written below. The share of marked animals in the second catch (R of C) estimates the share in the whole population (M of N); in the example that is 8 of 50 and 40 of 250, 16 % each, which gives N = M·C/R. https://quanta-study.de/tafelwerk/diagramme/lincoln-index-en-dark.svg?v=cf6c36ec70e1031f Diagram: two bars of equal length for the population N and the second catch C; in both, the marked share M or R is highlighted with the same size, a dashed line links the two boundaries, and M/N = R/C is written below. The share of marked animals in the second catch (R of C) estimates the share in the whole population (M of N); in the example that is 8 of 50 and 40 of 250, 16 % each, which gives N = M·C/R. https://quanta-study.de/en/formula-sheet/ficksches-gesetz-erste-diffusion 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/relatives-risiko 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/goldman-hodgkin-katz-gleichung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/diagnostische-sensitivitaet-spezifitaet 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/wilson-konfidenzintervall-anteil 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/cohens-d 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/rekombinationsfrequenz-genkartierung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/odds-ratio 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/pielou-evenness-index 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/lotka-volterra-praedator-beute 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/lotka-volterra-praedator-beute-en-light.svg?v=38a5e0e044283da5 Schematic population curves over time: prey N(t) rises first and predator P(t) peaks later; both curves repeat in the idealized model. In the idealized Lotka–Volterra cycle, the predator peak follows the prey peak; the curves are schematic. https://quanta-study.de/tafelwerk/diagramme/lotka-volterra-praedator-beute-en-dark.svg?v=38a5e0e044283da5 Schematic population curves over time: prey N(t) rises first and predator P(t) peaks later; both curves repeat in the idealized model. In the idealized Lotka–Volterra cycle, the predator peak follows the prey peak; the curves are schematic. https://quanta-study.de/en/formula-sheet/lotka-volterra-konkurrenz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/effektive-populationsgroesse-geschlechterverhaeltnis 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/selektionskoeffizient-relative-fitness 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kopplungsungleichgewicht-r2 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/mutation-selektion-balance-rezessiv 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/heritabilitaet-schmal 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/hill-gleichung-sauerstoffsaettigung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/herzzeitvolumen 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/renale-clearance 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/allometrische-skalierung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/qpcr-delta-delta-ct 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kompetitive-enzymhemmung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/langmuir-bindungsisotherme 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/roc-area-under-curve 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kaplan-meier-schaetzer 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/hazard-funktion 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/logistische-regression 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/benjamini-hochberg-fdr 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/q10-temperaturkoeffizient 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/pflanzen-relatives-wachstum 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/sauerstoffgehalt-blut 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/fick-prinzip-herzzeitvolumen 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/inzuchtkoeffizient-population 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/positiver-negativer-vorhersagewert 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/inzidenzrate-personenzeit 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/diskretes-populationswachstum 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/pfaffl-qpcr-effizienzkorrektur 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/gc-gehalt-sequenz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/gc-skew-sequenz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/n50-genomassemblierung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/nukleotiddiversitaet-pi 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/fst-populationsdifferenzierung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/wright-fisher-driftvarianz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/zuechtergleichung-selektionserfolg 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kumulative-inzidenz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/populations-attributable-fraktion 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/standardisierte-mortalitaetsratio 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/basisreproduktionszahl-sir 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/ejektionsfraktion-herz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/alveolaere-ventilation 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/maximale-psii-quantenausbeute 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/transpiration-gaswechsel 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/intraklassenkorrelation 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/clusterdesign-effekt 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/stichprobengroesse-zwei-mittelwerte 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/variationskoeffizient 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/sequenzalignment-score-identitaet 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/sequenzalignment-score-identitaet-en-light.svg?v=bcd48214a2f27431 Dynamic-programming matrix for ACT against AGT. Rows and columns contain the empty prefix, A, C and A, G, T; boundary values initialize at −2 per gap. The highlighted diagonal path gives A/A, C/G, T/T, score 3, and identity two of three. Global Needleman–Wunsch alignment of ACT and AGT with match +2, mismatch −1, and gap −2. The best path scores 3; sequence identity is separately 2/3 = 66.7%. Other scoring matrices produce different scores. https://quanta-study.de/tafelwerk/diagramme/sequenzalignment-score-identitaet-en-dark.svg?v=bcd48214a2f27431 Dynamic-programming matrix for ACT against AGT. Rows and columns contain the empty prefix, A, C and A, G, T; boundary values initialize at −2 per gap. The highlighted diagonal path gives A/A, C/G, T/T, score 3, and identity two of three. Global Needleman–Wunsch alignment of ACT and AGT with match +2, mismatch −1, and gap −2. The best path scores 3; sequence identity is separately 2/3 = 66.7%. Other scoring matrices produce different scores. https://quanta-study.de/en/formula-sheet/primaerproduktion-energiefluss-oekosystem 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/primaerproduktion-energiefluss-oekosystem-en-light.svg?v=b3b7117d65f3aa8b GPP splits into RA lost through autotrophic respiration and NPP. Part of NPP appears as production at the next trophic level; efficiency is that production divided by NPP. Budget diagram: part of gross primary production is respired; the remainder is net primary production. The arrow to the next level denotes production, not all ingested food energy. https://quanta-study.de/tafelwerk/diagramme/primaerproduktion-energiefluss-oekosystem-en-dark.svg?v=b3b7117d65f3aa8b GPP splits into RA lost through autotrophic respiration and NPP. Part of NPP appears as production at the next trophic level; efficiency is that production divided by NPP. Budget diagram: part of gross primary production is respired; the remainder is net primary production. The arrow to the next level denotes production, not all ingested food energy. https://quanta-study.de/en/formula-sheet/monohybrider-erbgang-wahrscheinlichkeit 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/qpcr-standardkurve-absolute-quantifizierung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/leslie-matrix-altersstrukturierte-population 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/pflanzenwasserpotential 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/demografische-netto-reproduktionsrate-generationzeit 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/alveolaere-gasgleichung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/blast-e-value-sequenzsuche 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/chao1-artenreichtumsschaetzung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/cohens-kappa-interrater 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/genfluss-migrationsgleichgewicht-allelfrequenz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/respiratorischer-quotient 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/mittlere-sequenzierungsabdeckung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/jukes-cantor-distanzkorrektur 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/diagnostische-likelihood-ratios 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/konfidenzintervall-relatives-risiko-odds-ratio 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/stichprobenumfang-anteil 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/holling-typ-ii-funktion 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/monod-wachstumsmodell 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/bohr-totraumsfraktion 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/photosynthese-lichtantwort-saettigung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/absolute-risikodifferenz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/fisher-exakt-test 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/stellarparallaxe 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/stellarparallaxe-en-light.svg?v=75f7c9b687a7b5ae Two Earth positions above and below the Sun form two right triangles with the star. Each half-baseline is 1 au. At the star, p is marked between the Sun sightline and the upper Earth sightline. Two opposite Earth positions are approximately 2 au apart. The half-baseline from the Sun is 1 au; p is the half-angle. Not to scale. https://quanta-study.de/tafelwerk/diagramme/stellarparallaxe-en-dark.svg?v=75f7c9b687a7b5ae Two Earth positions above and below the Sun form two right triangles with the star. Each half-baseline is 1 au. At the star, p is marked between the Sun sightline and the upper Earth sightline. Two opposite Earth positions are approximately 2 au apart. The half-baseline from the Sun is 1 au; p is the half-angle. Not to scale. https://quanta-study.de/en/formula-sheet/hubble-gesetz-nahbereich 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/wien-verschiebungsgesetz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/entfernungsmodul 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/scheinbare-helligkeit-fluss 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/sternradius-leuchtkraft-temperatur 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/spektralrotverschiebung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/sternscheinbare-magnitude-flussverhaeltnis 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/teleskop-rayleigh-aufloesung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/eigenbewegung-tangentialgeschwindigkeit 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/schwarzschild-radius 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/sternwinkelgroesse-entfernung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kepler-flaechengeschwindigkeit 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kosmologische-skalenfaktor-rotverschiebung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/roche-grenze-fluidmond 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/sternoberflaechengravitation 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kelvin-helmholtz-zeitstern 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/jeans-masse-gravitationskollaps 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/eddington-leuchtkraft 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/stern-dynamische-zeitskala 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/doppelstern-massen-kepler 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/cepheiden-perioden-leuchtkraft 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/cepheiden-perioden-leuchtkraft-en-light.svg?v=89637374fb6975c3 Plot of absolute V magnitude M_V against log₁₀ period in days. The historical example line runs from about (P=1 day, M_V=−1.43) to (P=30 days, M_V=−5.58); longer periods are brighter. The specific calibration depends on population and band. Illustrative historical V-band calibration from Feast & Catchpole (1997): M_V = −2.81 log₁₀(P/day) − 1.43. Longer period means higher luminosity. Zero point and slope depend on calibration, population, and band; this is not a universal precision relation. https://quanta-study.de/tafelwerk/diagramme/cepheiden-perioden-leuchtkraft-en-dark.svg?v=89637374fb6975c3 Plot of absolute V magnitude M_V against log₁₀ period in days. The historical example line runs from about (P=1 day, M_V=−1.43) to (P=30 days, M_V=−5.58); longer periods are brighter. The specific calibration depends on population and band. Illustrative historical V-band calibration from Feast & Catchpole (1997): M_V = −2.81 log₁₀(P/day) − 1.43. Longer period means higher luminosity. Zero point and slope depend on calibration, population, and band; this is not a universal precision relation. https://quanta-study.de/en/formula-sheet/vis-viva-geschwindigkeit-elliptische-bahn 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/masse-leuchtkraft-hauptreihe 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/hauptreihenlebensdauer 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/transit-tiefe-exoplanet 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/spektrale-aufloesung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/chandrasekhar-grenze 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/hydrostatisches-gleichgewicht-sterninneres 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/shannon-entropie 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/shannon-entropie-en-light.svg?v=438d36d8a5d87ba2 Binary Shannon entropy curve from p = 0 to 1. It starts at zero, rises symmetrically to a maximum of 1 bit at 0.5, then falls to zero. Binary entropy H(p): the maximum is 1 bit at p = 0.5. Certain outcomes at p = 0 or 1 have zero entropy. https://quanta-study.de/tafelwerk/diagramme/shannon-entropie-en-dark.svg?v=438d36d8a5d87ba2 Binary Shannon entropy curve from p = 0 to 1. It starts at zero, rises symmetrically to a maximum of 1 bit at 0.5, then falls to zero. Binary entropy H(p): the maximum is 1 bit at p = 0.5. Certain outcomes at p = 0 or 1 have zero entropy. https://quanta-study.de/en/formula-sheet/f1-score 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/big-o-notation 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/big-o-notation-en-light.svg?v=d3bcdccbdbd41613 Shared axes with curves n, n log n, n², and 2 to the n. The scale is chosen for a qualitative comparison over small n, not as runtime in seconds. Comparison of n, n·log n, n², and 2ⁿ growth on the same schematic axis scale. https://quanta-study.de/tafelwerk/diagramme/big-o-notation-en-dark.svg?v=d3bcdccbdbd41613 Shared axes with curves n, n log n, n², and 2 to the n. The scale is chosen for a qualitative comparison over small n, not as runtime in seconds. Comparison of n, n·log n, n², and 2ⁿ growth on the same schematic axis scale. https://quanta-study.de/en/formula-sheet/zweierkomplement 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/laufzeitrekurrenz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kreuzentropie 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/gradientenabstieg-parameterupdate 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/balanced-accuracy 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/precision-positivvorhersagewert 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/recall-sensitivitaet 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/matthews-korrelationskoeffizient 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/hamming-distanz-bitfolgen 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/ieee-754-gleitkomma 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/ieee-754-gleitkomma-en-light.svg?v=0bb24bd892993f56 A 32-bit word is split into one sign bit s, eight exponent bits E and 23 fraction bits F. Normalized values use significand 1.F; the stored exponent is interpreted with bias 127. binary32 divides the 32-bit word into sign, exponent field and fraction; normalized values add an implicit leading 1. https://quanta-study.de/tafelwerk/diagramme/ieee-754-gleitkomma-en-dark.svg?v=0bb24bd892993f56 A 32-bit word is split into one sign bit s, eight exponent bits E and 23 fraction bits F. Normalized values use significand 1.F; the stored exponent is interpreted with bias 127. binary32 divides the 32-bit word into sign, exponent field and fraction; normalized values add an implicit leading 1. https://quanta-study.de/en/formula-sheet/boolesche-algebra-gesetze 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/euklidischer-algorithmus-modulares-inverses 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/shannon-kanalkapazitaet 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/amdahl-gesetz-speedup 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/cpu-laufzeit-instruktionen-cpi-takt 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/logistische-sigmoidfunktion 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/a-star-suche 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/master-theorem-rekurrenzen 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/amortisierungsanalyse-dynamische-arrays 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/union-find-disjunkte-mengen 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/floyd-warshall-kuerzeste-wege 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/avl-baum-balance 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/bellman-ford-kuerzeste-wege-negative-kanten 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/dijkstra-kuerzeste-wege 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/dijkstra-kuerzeste-wege-en-light.svg?v=49abf71454e90309 Directed graph with edges s→a of weight 2, s→b of 5, a→b of 1, a→t of 6, and b→t of 2. The highlighted shortest path s→a→b→t costs 5. The shortest distances from s are 0, 2, 3, and 5. With nonnegative edge weights, Dijkstra follows s→a→b→t here. The final distances are d(a)=2, d(b)=3, and d(t)=5. https://quanta-study.de/tafelwerk/diagramme/dijkstra-kuerzeste-wege-en-dark.svg?v=49abf71454e90309 Directed graph with edges s→a of weight 2, s→b of 5, a→b of 1, a→t of 6, and b→t of 2. The highlighted shortest path s→a→b→t costs 5. The shortest distances from s are 0, 2, 3, and 5. With nonnegative edge weights, Dijkstra follows s→a→b→t here. The final distances are d(a)=2, d(b)=3, and d(t)=5. https://quanta-study.de/en/formula-sheet/minimaler-spannbaum 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/minimaler-spannbaum-en-light.svg?v=e81ba3844c9e05f3 Undirected graph with four vertices and five edges. AB of weight 1, BC of 2, and CD of 3 form the highlighted minimum spanning tree. AC of 4 and BD of 5 are dashed because each would close a cycle during Kruskal’s algorithm. The total weight is 6. Kruskal selects AB=1, BC=2, and CD=3. The three edges connect four vertices without a cycle and have total weight 6. https://quanta-study.de/tafelwerk/diagramme/minimaler-spannbaum-en-dark.svg?v=e81ba3844c9e05f3 Undirected graph with four vertices and five edges. AB of weight 1, BC of 2, and CD of 3 form the highlighted minimum spanning tree. AC of 4 and BD of 5 are dashed because each would close a cycle during Kruskal’s algorithm. The total weight is 6. Kruskal selects AB=1, BC=2, and CD=3. The three edges connect four vertices without a cycle and have total weight 6. https://quanta-study.de/en/formula-sheet/binaersuche-laufzeit 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kombinatorische-zaehlregeln 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/graphsuche-bfs-dfs-kuerzeste-wege 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/graphsuche-bfs-dfs-kuerzeste-wege-en-light.svg?v=37a7c2deba2429c0 Unweighted five-vertex graph with edges s–a, s–b, a–t, b–c and c–t. The highlighted BFS route s–a–t has two edges. The dashed possible first DFS route s–b–c–t has three; discovery order depends on neighbor ordering. BFS layers vertices by edge distance and finds the shortest path s–a–t here. With a different neighbor order, DFS may first follow the longer route s–b–c–t. https://quanta-study.de/tafelwerk/diagramme/graphsuche-bfs-dfs-kuerzeste-wege-en-dark.svg?v=37a7c2deba2429c0 Unweighted five-vertex graph with edges s–a, s–b, a–t, b–c and c–t. The highlighted BFS route s–a–t has two edges. The dashed possible first DFS route s–b–c–t has three; discovery order depends on neighbor ordering. BFS layers vertices by edge distance and finds the shortest path s–a–t here. With a different neighbor order, DFS may first follow the longer route s–b–c–t. https://quanta-study.de/en/formula-sheet/diffie-hellman-schluesselaustausch 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/softmax-funktion 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/f-beta-score 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/mutuelle-information-diskret 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/logit-odds-transformation 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/spezifitaet-true-negative-rate 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/jaccard-index-klassifikation 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/durchschnittliche-speicherzugriffszeit-amat 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/bloom-filter-falschpositiv-wahrscheinlichkeit 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/roc-auc-klassifikation 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kullback-leibler-divergenz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kosinus-aehnlichkeit-vektoren 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/hashkollision-geburtstagswahrscheinlichkeit 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/tf-idf-gewichtung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/bm25-ranking 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/pagerank-iteration 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/littlesches-gesetz-warteschlange 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/mm1-warteschlange-auslastung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/levenshtein-editierdistanz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/nyquist-shannon-abtastgrenze 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/diskrete-fouriertransformation 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/rsa-modulare-potenz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/faltungsoutput-dimension 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/gini-impurity-klassifikation 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/binaerer-log-loss 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/informationsgewinn-entscheidungsbaum 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/crc-pruefsumme-polynomrest 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/bandbreite-latenz-bandbreitenverzoegerungsprodukt 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/bplus-baum-indexierung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/hash-tabellen-lastfaktor-erwartungswert 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/k-means-inertia-zielfunktion 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/pca-hauptkomponentenanalyse 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/naive-bayes-klassifikation 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/ridge-regression-l2 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/lineare-svm-hinge-loss 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/logistische-regression-bernoulli-likelihood 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/young-modul 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/reynolds-zahl 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kirchhoff-knotenmaschen 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/kirchhoff-knotenmaschen-en-light.svg?v=9a8a0da6ff4f06fa Two sources sharing a return feed an upper node through 4 and 6 ohms. A 3 ohm resistor connects that node to return; arrows mark 5/3, 1/9 and 16/9 amperes. With 12 V left and 6 V right, the shared node is at 16/3 V. Both sources supply current; the 3 ohm branch carries their sum. https://quanta-study.de/tafelwerk/diagramme/kirchhoff-knotenmaschen-en-dark.svg?v=9a8a0da6ff4f06fa Two sources sharing a return feed an upper node through 4 and 6 ohms. A 3 ohm resistor connects that node to return; arrows mark 5/3, 1/9 and 16/9 amperes. With 12 V left and 6 V right, the shared node is at 16/3 V. Both sources supply current; the 3 ohm branch carries their sum. https://quanta-study.de/en/formula-sheet/impuls-drehimpuls-bilanz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/bernoulli-verlustgleichung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/fourier-waermeleitung 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/fourier-waermeleitung-en-light.svg?v=a80bde258640ab44 Wall with a hot left side and cooler right side: a temperature gradient across thickness L and heat-flow arrows pointing right. In the steady wall model, heat flows from hot to cold; heat rate rises with area and temperature difference. https://quanta-study.de/tafelwerk/diagramme/fourier-waermeleitung-en-dark.svg?v=a80bde258640ab44 Wall with a hot left side and cooler right side: a temperature gradient across thickness L and heat-flow arrows pointing right. In the steady wall model, heat flows from hot to cold; heat rate rises with area and temperature difference. https://quanta-study.de/en/formula-sheet/euler-bernoulli-balken 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/pid-regler-uebertragungsfunktion 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/statisches-gleichgewicht 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/statisches-gleichgewicht-en-light.svg?v=f37583dc348c538d Free-body diagram of a simply supported beam: a centered load F points down and reactions RAy and RBy point up. Axes at upper right mark +x right and +y up; a curved arrow at O marks positive counterclockwise moment. Free-body diagram with sign convention: +x to the right, +y upward and positive moment counterclockwise. Force and moment sums are zero. https://quanta-study.de/tafelwerk/diagramme/statisches-gleichgewicht-en-dark.svg?v=f37583dc348c538d Free-body diagram of a simply supported beam: a centered load F points down and reactions RAy and RBy point up. Axes at upper right mark +x right and +y up; a curved arrow at O marks positive counterclockwise moment. Free-body diagram with sign convention: +x to the right, +y upward and positive moment counterclockwise. Force and moment sums are zero. https://quanta-study.de/en/formula-sheet/normalspannung-zug 2026-10-02 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/normalspannung-zug-en-light.svg?v=e139ea206c454ac3 Straight bar with equal outward tensile forces F at both ends. A cut normal to the axis marks area A. Four equal arrows schematically show a uniform tensile stress distribution. Centred axial tension: F acts normal to cut area A. Uniform stress gives local σ = F/A; the area average also holds without uniformity. https://quanta-study.de/tafelwerk/diagramme/normalspannung-zug-en-dark.svg?v=e139ea206c454ac3 Straight bar with equal outward tensile forces F at both ends. A cut normal to the axis marks area A. Four equal arrows schematically show a uniform tensile stress distribution. Centred axial tension: F acts normal to cut area A. Uniform stress gives local σ = F/A; the area average also holds without uniformity. https://quanta-study.de/en/formula-sheet/poissonzahl 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/thermische-dehnung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/biegemoment-spannung 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/biegemoment-spannung-en-light.svg?v=a1cd37fd3978b430 Rectangular beam section with the neutral axis through its centroid. With y upward, the top edge is in compression and the bottom edge in tension under σx=−M·y/I. Beside it, a linear stress profile is zero at the neutral axis, negative above and positive below. For positive bending moment M and upward y, σx=−M·y/I: compression above, zero at the neutral axis, and tension below. https://quanta-study.de/tafelwerk/diagramme/biegemoment-spannung-en-dark.svg?v=a1cd37fd3978b430 Rectangular beam section with the neutral axis through its centroid. With y upward, the top edge is in compression and the bottom edge in tension under σx=−M·y/I. Beside it, a linear stress profile is zero at the neutral axis, negative above and positive below. For positive bending moment M and upward y, σx=−M·y/I: compression above, zero at the neutral axis, and tension below. https://quanta-study.de/en/formula-sheet/balken-querkraftspannung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/torsionsschubspannung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/torsionswinkel-kreiswelle 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/flaechen-parallelachsensatz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/euler-knicklast 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/euler-knicklast-en-light.svg?v=19e02c0757488196 Slender pinned column in compression: the straight initial axis is dashed and the deflected buckling shape is accented; compressive loads act at both ends. The Euler load falls with the square of effective length; the sketch shows the ideal pinned-pinned buckling shape. https://quanta-study.de/tafelwerk/diagramme/euler-knicklast-en-dark.svg?v=19e02c0757488196 Slender pinned column in compression: the straight initial axis is dashed and the deflected buckling shape is accented; compressive loads act at both ends. The Euler load falls with the square of effective length; the sketch shows the ideal pinned-pinned buckling shape. https://quanta-study.de/en/formula-sheet/duennwandiger-druckbehaelter 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/von-mises-vergleichsspannung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/volumenstrom-kontinuitaet 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/massenstrom-kontinuitaet 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/hydraulischer-durchmesser 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/darcy-weisbach-rohrreibung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/einzelverlusthoehe 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/hagen-poiseuille-gesetz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/hydraulische-leistung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/balkenlast-querkraft-moment 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/schubmodul 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/hauptspannungen-ebener-spannungszustand 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/elastische-verformungsenergie-dichte 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/schraubenfeder-steifigkeit 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/konvektiver-waermeuebergang 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/waermediffusivitaet 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/biot-zahl 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/fourier-zahl-waermeleitung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/lumped-kapazitaetsmodell 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/logarithmische-mittlere-temperaturdifferenz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/isentropes-idealgasgesetz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/isentropenwirkungsgrad-verdichter 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/spannungsteiler 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/rc-ladeentladung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/rl-stromtransient 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/rlc-impedanz-serienschaltung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/wechselstrom-wirkleistung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/zahnrad-uebersetzung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/zahnradmodul 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/carnot-wirkungsgrad 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/waermepumpen-cop 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/waermeleitwiderstand-ebene-wand 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/thermische-spannung-behinderter-stab 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/spannungsintensitaetsfaktor-modus-i 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/miner-schadensakkumulation 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/sicherheitsfaktor-festigkeit 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kondensator-gespeicherte-energie 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/transformator-uebersetzungsverhaeltnis 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/sensible-waerme-kapazitaet 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/gesamtwaermedurchgang-ebene-wand 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/energie-spule-magnetfeld 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/dreiphasen-wirkleistung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/thevenin-aequivalentes-netzwerk 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/norton-thevenin-umwandlung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/daempfungsverhaeltnis-einmassenschwinger 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/eigenkreisfrequenz-einmassenschwinger 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/pt1-uebertragungsfunktion 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/screwtrieb-drehmoment-ideal 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/nyquist-abtastbedingung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/geschlossener-regelkreis-uebertragungsfunktion 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/regelkreis-sensitivitaetsfunktion 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/logarithmisches-dekrement 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/pt2-uebertragungsfunktion 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/riemenantrieb-umfangsgeschwindigkeit 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/riemenantrieb-drehzahlverhaeltnis 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/maschinenwirkungsgrad 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/waelzlager-nennlebensdauer 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/sinus-effektivwert 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/drehstrom-scheinleistung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/drehstrom-blindleistung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/wheatstone-bruecken-abgleich 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/pumpen-affinitaetsgesetze 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/euler-turbomaschinengleichung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kavitationszahl 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/prandtl-zahl 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/nusselt-zahl 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/sherwood-zahl 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/lewis-zahl 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/waermetauscher-wirksamkeit 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/voigt-mischungsregel-elastizitaetsmodul 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kragtraeger-durchbiegung-endkraft 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/einfeldtraeger-durchbiegung-mittellast 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/schubfluss-duennwandiger-balken 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/komplementaere-sensitivitaetsfunktion 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/leistungsanpassung-thevenin 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/riemenantrieb-uebertragene-leistung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/npsh-verfuegbar 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/waermetauscher-kapazitaetsverhaeltnis 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kragtraeger-durchbiegung-gleichlast 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/einfeldtraeger-durchbiegung-gleichlast 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/reuss-mischungsregel-quer-modul 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/balkenlast-querkraft-moment-differential 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kontrollvolumen-massenbilanz 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/kontrollvolumen-massenbilanz-en-light.svg?v=91f9622247e1fb9b A bounded control volume CV has an inlet mass flow ṁin on the left and outlet ṁout on the right. Possible mass accumulation dMcv/dt is marked inside. At steady state the two flows are equal. A fixed control volume balances stored mass against inlet and outlet mass flows. https://quanta-study.de/tafelwerk/diagramme/kontrollvolumen-massenbilanz-en-dark.svg?v=91f9622247e1fb9b A bounded control volume CV has an inlet mass flow ṁin on the left and outlet ṁout on the right. Possible mass accumulation dMcv/dt is marked inside. At steady state the two flows are equal. A fixed control volume balances stored mass against inlet and outlet mass flows. https://quanta-study.de/en/formula-sheet/kontrollvolumen-energiebilanz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kontrollvolumen-entropiebilanz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/ficksches-gesetz-diffusionsfluss 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/navier-stokes-inkompressibel 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/mach-zahl 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/froude-zahl 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/schwerpunkt-flaechenmoment 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/schwerpunkt-flaechenmoment-en-light.svg?v=78536e0553c2fc7d A T-section consists of a 60 by 20 mm base and a centered 20 by 40 mm web. It marks y1=10 mm, y2=40 mm and the composite centroid y-bar=22 mm above the bottom edge. A dashed line shows its position. T-section: component areas and centroids determine the area centroid. Qx about the bottom edge is A·ȳ, not the second moment of area I. https://quanta-study.de/tafelwerk/diagramme/schwerpunkt-flaechenmoment-en-dark.svg?v=78536e0553c2fc7d A T-section consists of a 60 by 20 mm base and a centered 20 by 40 mm web. It marks y1=10 mm, y2=40 mm and the composite centroid y-bar=22 mm above the bottom edge. A dashed line shows its position. T-section: component areas and centroids determine the area centroid. Qx about the bottom edge is A·ȳ, not the second moment of area I. https://quanta-study.de/en/formula-sheet/routh-hurwitz-stabilitaetskriterium 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/bode-verstaerkungs-phasenreserve 2026-09-30 monthly 0.82 https://quanta-study.de/tafelwerk/diagramme/bode-verstaerkungs-phasenreserve-en-light.svg?v=e6070b3a97f8fad5 Two logarithmic-frequency plots. Above, magnitude falls through 0 dB; a marker shows gain crossover and the vertical distance to critical phase. Below, phase falls through −180 degrees; horizontal distance to the 0 dB level marks gain margin. Schematic Bode response: phase margin at gain crossover, gain margin at phase crossover. Curves and margins are not to scale. https://quanta-study.de/tafelwerk/diagramme/bode-verstaerkungs-phasenreserve-en-dark.svg?v=e6070b3a97f8fad5 Two logarithmic-frequency plots. Above, magnitude falls through 0 dB; a marker shows gain crossover and the vertical distance to critical phase. Below, phase falls through −180 degrees; horizontal distance to the 0 dB level marks gain margin. Schematic Bode response: phase margin at gain crossover, gain margin at phase crossover. Curves and margins are not to scale. https://quanta-study.de/en/formula-sheet/basquin-gesetz-ermuedungsfestigkeit 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/lame-druckzylinder 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/goodman-ermuedungsnachweis 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/capstan-seilreibung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/colebrook-white-rohrreibung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/castiglianos-zweiter-satz 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/zylindrischer-waermeleitwiderstand 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/operationsverstaerker-verstaerkung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/shockley-diodengleichung 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kritisches-duesen-druckverhaeltnis-massenstrom 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/netto-waermestrahlung-sichtfaktoren 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/tresca-fliessbedingung-maximalschub 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/ergun-gleichung-druckverlust-festbett 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/stab-elementsteifigkeitsmatrix-1d 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/kirsch-spannungskonzentration-kreisloch 2026-09-30 monthly 0.82 https://quanta-study.de/en/formula-sheet/tables/si-basiseinheiten 2026-10-01 monthly 0.78 https://quanta-study.de/tafelwerk/tabellenbilder/si-basiseinheiten--main-en-light.png?v=9271210a88aece13 SI base units — Quanta SI base units https://quanta-study.de/tafelwerk/tabellenbilder/si-basiseinheiten--main-en-dark.png?v=f433730056306fc8 SI base units — Quanta SI base units https://quanta-study.de/tafelwerk/tabellenbilder/si-basiseinheiten--si-defining-constants-en-light.png?v=b4a0442055ebc9e5 The seven defining constants — Quanta The seven defining constants https://quanta-study.de/tafelwerk/tabellenbilder/si-basiseinheiten--si-defining-constants-en-dark.png?v=db7439fa9296bb1e The seven defining constants — Quanta The seven defining constants https://quanta-study.de/tafelwerk/tabellenbilder/si-basiseinheiten--si-base-dimensions-en-light.png?v=a71c0ad1bd3f7407 Base quantities: quantity and dimension symbols — Quanta Base quantities: quantity and dimension symbols https://quanta-study.de/tafelwerk/tabellenbilder/si-basiseinheiten--si-base-dimensions-en-dark.png?v=4922d2b863d1f3b2 Base quantities: quantity and dimension symbols — Quanta Base quantities: quantity and dimension symbols https://quanta-study.de/en/formula-sheet/tables/si-abgeleitete-einheiten 2026-10-01 monthly 0.78 https://quanta-study.de/tafelwerk/tabellenbilder/si-abgeleitete-einheiten--main-en-light.png?v=83eb06404fc9cdf8 Named derived SI units — Quanta Named derived SI units https://quanta-study.de/tafelwerk/tabellenbilder/si-abgeleitete-einheiten--main-en-dark.png?v=fc1366d65e74ea52 Named derived SI units — Quanta Named derived SI units https://quanta-study.de/tafelwerk/tabellenbilder/si-abgeleitete-einheiten--si-common-derived-quantities-en-light.png?v=6fa6c58f5cff004d More common derived quantities — Quanta More common derived quantities https://quanta-study.de/tafelwerk/tabellenbilder/si-abgeleitete-einheiten--si-common-derived-quantities-en-dark.png?v=3dc09e6473b279af More common derived quantities — Quanta More common derived quantities https://quanta-study.de/en/formula-sheet/tables/si-praefixe 2026-10-01 monthly 0.78 https://quanta-study.de/tafelwerk/tabellenbilder/si-praefixe--main-en-light.png?v=d460222d74d982ca SI prefixes — Quanta SI prefixes https://quanta-study.de/tafelwerk/tabellenbilder/si-praefixe--main-en-dark.png?v=50f631c5d4359d42 SI prefixes — Quanta SI prefixes https://quanta-study.de/tafelwerk/tabellenbilder/si-praefixe--si-prefix-area-volume-en-light.png?v=7f9d885123c32bc1 Prefixes on length, area and volume — Quanta Prefixes on length, area and volume https://quanta-study.de/tafelwerk/tabellenbilder/si-praefixe--si-prefix-area-volume-en-dark.png?v=88701983670d9e01 Prefixes on length, area and volume — Quanta Prefixes on length, area and volume https://quanta-study.de/tafelwerk/tabellenbilder/si-praefixe--si-prefix-compound-conversions-en-light.png?v=166f5ab541cf8760 Prefixes in mass and compound units — Quanta Prefixes in mass and compound units https://quanta-study.de/tafelwerk/tabellenbilder/si-praefixe--si-prefix-compound-conversions-en-dark.png?v=3955b1d22f4b5c1e Prefixes in mass and compound units — Quanta Prefixes in mass and compound units https://quanta-study.de/en/formula-sheet/tables/binaerpraefixe 2026-10-01 monthly 0.78 https://quanta-study.de/tafelwerk/tabellenbilder/binaerpraefixe--main-en-light.png?v=a2eb6a8a5600a74c Binary prefixes for data quantities — Quanta Binary prefixes for data quantities https://quanta-study.de/tafelwerk/tabellenbilder/binaerpraefixe--main-en-dark.png?v=8ca556b8f793f478 Binary prefixes for data quantities — Quanta Binary prefixes for data quantities https://quanta-study.de/tafelwerk/tabellenbilder/binaerpraefixe--binary-decimal-comparison-en-light.png?v=8e2ede882af22a70 Binary quantities in decimal units — Quanta Binary quantities in decimal units https://quanta-study.de/tafelwerk/tabellenbilder/binaerpraefixe--binary-decimal-comparison-en-dark.png?v=e4ce0a11f6a84985 Binary quantities in decimal units — Quanta Binary quantities in decimal units https://quanta-study.de/en/formula-sheet/tables/fundamentale-konstanten 2026-10-01 monthly 0.78 https://quanta-study.de/tafelwerk/tabellenbilder/fundamentale-konstanten--main-en-light.png?v=6ed14ccd6b3b94fd Selected fundamental constants — Quanta Selected fundamental constants https://quanta-study.de/tafelwerk/tabellenbilder/fundamentale-konstanten--main-en-dark.png?v=726153f3ad3d96f8 Selected fundamental constants — Quanta Selected fundamental constants https://quanta-study.de/tafelwerk/tabellenbilder/fundamentale-konstanten--konstanten-status-en-light.png?v=b80bf2c26e08fc76 Definition, measurement and rounding — Quanta Definition, measurement and rounding https://quanta-study.de/tafelwerk/tabellenbilder/fundamentale-konstanten--konstanten-status-en-dark.png?v=81e85cc94ed6f3cb Definition, measurement and rounding — Quanta Definition, measurement and rounding https://quanta-study.de/en/formula-sheet/tables/si-umrechnungen 2026-10-01 monthly 0.78 https://quanta-study.de/tafelwerk/tabellenbilder/si-umrechnungen--main-en-light.png?v=ba2e95bd14b9dcee Common conversions to SI — Quanta Common conversions to SI https://quanta-study.de/tafelwerk/tabellenbilder/si-umrechnungen--main-en-dark.png?v=1b254fefb2efafbc Common conversions to SI — Quanta Common conversions to SI https://quanta-study.de/tafelwerk/tabellenbilder/si-umrechnungen--si-compound-extra-en-light.png?v=e308fb264c480b5c Additional SI ratios, flows and material properties — Quanta Additional SI ratios, flows and material properties https://quanta-study.de/tafelwerk/tabellenbilder/si-umrechnungen--si-compound-extra-en-dark.png?v=e290bbdebaebfd46 Additional SI ratios, flows and material properties — Quanta Additional SI ratios, flows and material properties https://quanta-study.de/tafelwerk/tabellenbilder/si-umrechnungen--legacy-energy-viscosity-en-light.png?v=d19a91064d018f7f Legacy energy and viscosity units — Quanta Legacy energy and viscosity units https://quanta-study.de/tafelwerk/tabellenbilder/si-umrechnungen--legacy-energy-viscosity-en-dark.png?v=10622b4349057e2b Legacy energy and viscosity units — Quanta Legacy energy and viscosity units https://quanta-study.de/en/formula-sheet/tables/astronomische-laengeneinheiten 2026-10-01 monthly 0.78 https://quanta-study.de/tafelwerk/tabellenbilder/astronomische-laengeneinheiten--main-en-light.png?v=a057566a9e7bcaac Astronomical distance units — Quanta Astronomical distance units https://quanta-study.de/tafelwerk/tabellenbilder/astronomische-laengeneinheiten--main-en-dark.png?v=9c57e83cc5e539de Astronomical distance units — Quanta Astronomical distance units https://quanta-study.de/tafelwerk/tabellenbilder/astronomische-laengeneinheiten--distanz-definitionen-en-light.png?v=d2b5a3c82ae30604 Exact definitions and conversion basis — Quanta Exact definitions and conversion basis https://quanta-study.de/tafelwerk/tabellenbilder/astronomische-laengeneinheiten--distanz-definitionen-en-dark.png?v=4cb2e6a46b5a1d06 Exact definitions and conversion basis — Quanta Exact definitions and conversion basis https://quanta-study.de/en/formula-sheet/tables/chemische-referenzzustaende 2026-10-01 monthly 0.78 https://quanta-study.de/tafelwerk/tabellenbilder/chemische-referenzzustaende--main-en-light.png?v=af8da238b1c67b88 Chemical reference conventions — Quanta Chemical reference conventions https://quanta-study.de/tafelwerk/tabellenbilder/chemische-referenzzustaende--main-en-dark.png?v=1377c25d4427a696 Chemical reference conventions — Quanta Chemical reference conventions https://quanta-study.de/tafelwerk/tabellenbilder/chemische-referenzzustaende--chemical-reference-details-en-light.png?v=ded1658fcb02b8d6 Additional reference conventions and activities — Quanta Additional reference conventions and activities https://quanta-study.de/tafelwerk/tabellenbilder/chemische-referenzzustaende--chemical-reference-details-en-dark.png?v=8faf0855aa4a4383 Additional reference conventions and activities — Quanta Additional reference conventions and activities https://quanta-study.de/tafelwerk/tabellenbilder/chemische-referenzzustaende--ideal-molar-volumes-four-en-light.png?v=d4b5330d6d7e223e Four explicitly specified ideal molar gas volumes — Quanta Four explicitly specified ideal molar gas volumes https://quanta-study.de/tafelwerk/tabellenbilder/chemische-referenzzustaende--ideal-molar-volumes-four-en-dark.png?v=b5516be71f6d82e6 Four explicitly specified ideal molar gas volumes — Quanta Four explicitly specified ideal molar gas volumes https://quanta-study.de/en/formula-sheet/tables/standard-reduktionspotentiale 2026-10-01 monthly 0.78 https://quanta-study.de/tafelwerk/tabellenbilder/standard-reduktionspotentiale--main-en-light.png?v=c20c008eb9755188 Selected standard reduction potentials — Quanta Selected standard reduction potentials https://quanta-study.de/tafelwerk/tabellenbilder/standard-reduktionspotentiale--main-en-dark.png?v=c88a19562968d965 Selected standard reduction potentials — Quanta Selected standard reduction potentials https://quanta-study.de/tafelwerk/tabellenbilder/standard-reduktionspotentiale--redox-additional-conditioned-en-light.png?v=5f925f036ed38bcf Eleven further standard reductions with explicit medium — Quanta Eleven further standard reductions with explicit medium https://quanta-study.de/tafelwerk/tabellenbilder/standard-reduktionspotentiale--redox-additional-conditioned-en-dark.png?v=a49ffa4dd8b22022 Eleven further standard reductions with explicit medium — Quanta Eleven further standard reductions with explicit medium https://quanta-study.de/en/formula-sheet/tables/spektralklassen-sterne 2026-10-01 monthly 0.78 https://quanta-study.de/tafelwerk/tabellenbilder/spektralklassen-sterne--main-en-light.png?v=813a7b961d10bd55 Spectral classes of stars and brown dwarfs — Quanta Spectral classes of stars and brown dwarfs https://quanta-study.de/tafelwerk/tabellenbilder/spektralklassen-sterne--main-en-dark.png?v=044bf86468e8aee7 Spectral classes of stars and brown dwarfs — Quanta Spectral classes of stars and brown dwarfs https://quanta-study.de/tafelwerk/tabellenbilder/spektralklassen-sterne--leuchtkraftklassen-en-light.png?v=003bbd90df5995b1 Luminosity classes complement spectral type — Quanta Luminosity classes complement spectral type https://quanta-study.de/tafelwerk/tabellenbilder/spektralklassen-sterne--leuchtkraftklassen-en-dark.png?v=407b1a1d553339e1 Luminosity classes complement spectral type — Quanta Luminosity classes complement spectral type