Look up core STEM formulas with worked examples

A curated set of central STEM formulas in physics, chemistry, mathematics and biology, with background, a variable table, a worked example and sources. Learn each one as FSRS flashcards in Quanta.

Physics (52)Chemistry (24)Mathematics (49)Biology (8)

Contents

Table of contents

Subject first, then chapter, then formula. The chapters follow the common formula sheets used at school and university, so you look where you are used to looking.

118 elements · dedicated reference pages

The complete periodic table now has room to work.

Search all 118 elements by name, symbol or atomic number and open physical data, electron configurations and visible sources on dedicated reference pages.

Open the periodic table

Quality standard

What this formula sheet currently covers

The current set is deliberately curated instead of inflated: every detail page shows the formula, variables, units, background, worked example, applications and sources. New formulas are added only when they meet the same data standard.

Mechanics, electrodynamics, optics, quantum physics and thermodynamics in physics
Gases, acid-base chemistry, electrochemistry, kinetics, spectroscopy and equilibria in chemistry
Calculus, algebra, probability and complex numbers in mathematics
Enzyme kinetics and population models in biology

Physics

52 formulas

Mechanics

23 formulas

F = m \cdot a

Newton's Second Law

Newton's second law describes the relationship between the force, mass and acceleration of a body.

F_{G} = G \cdot \frac{m _{1} \cdot m _{2}}{r ^{2}}

Newton's Law of Universal Gravitation

The law of gravitation describes the attractive force between two masses as a function of their distance.

E_{kin} = \frac{1}{2} m v^{2}

Kinetic Energy

Kinetic energy is the energy of motion of a body of mass m moving at speed v.

E_{p o t} = m \cdot g \cdot h

Potential Energy (Gravitational)

The potential energy (positional energy) of a body of mass m at height h in the homogeneous gravitational field g.

F = k \cdot x

Hooke's Law (Spring Force)

Hooke's law describes the linear relationship between spring force and displacement, the basis for oscillations, spring gauges and materials mechanics.

s = \frac{1}{2} a t^{2} + v_{0} t

Uniformly Accelerated Motion (Distance-Time Law)

The distance-time law gives the distance travelled under constant acceleration; the distance grows quadratically with time.

v = a \cdot t + v_{0}

Velocity-Time Law

The velocity-time law describes how the velocity grows linearly with time under constant acceleration.

h = \frac{1}{2} g t^{2}

Free Fall

Free fall is uniformly accelerated motion without air resistance: the fall height grows quadratically with the fall time.

p = m \cdot v

Momentum

Momentum is the product of mass and velocity, the measure of the "quantity of motion" of a moving body.

m_{1} v_{1} + m_{2} v_{2} = m_{1} v_{1}^{'} + m_{2} v_{2}^{'}

Conservation of Momentum

In a closed system the total momentum is conserved, the central tool for all collision problems.

W = F \cdot s \cdot cos (α)

Mechanical Work

Mechanical work is done when a force displaces a body along a path; only the force component along the path counts.

P = \frac{W}{t}

Mechanical Power

Power states how fast work is done: the work performed divided by the time needed.

F_{z} = \frac{m \cdot v ^{2}}{r}

Centripetal Force (Circular Motion)

The centripetal force keeps a body on its circular path; it always points to the centre and grows quadratically with the orbital speed.

ρ = \frac{m}{V}

Density

Density is mass per volume, the central material constant for floating, sinking and material identification.

p = \frac{F}{A}

Pressure

Pressure is force per area: the same force acts strongly concentrated on a small area, spread out on a large one.

x = v_{0} cos α \cdot t y = v_{0} sin α \cdot t - \frac{1}{2} g t^{2}

Projectile Motion (Trajectory Parabola)

Projectile motion splits the movement into a uniform horizontal and an accelerated vertical component; together they form the trajectory parabola.

F_{A} = ρ_{Fl} \cdot V \cdot g

Buoyant Force (Archimedes)

The buoyant force after Archimedes equals the weight of the displaced fluid; it decides floating, hovering and sinking.

F_{R} = μ \cdot F_{N}

Sliding Friction Force

The sliding friction force is proportional to the normal force; the friction coefficient μ characterises the material pairing and the force always acts against the direction of motion.

\frac{T ^{2}}{a ^{3}} = \frac{4 π ^{2}}{G \cdot M}

Kepler Third Law

The squares of the orbital periods behave like the cubes of the semi-major axes; the quotient T²/a³ is the same for all bodies orbiting the same central body.

M = F \cdot r \cdot sin (θ)

Torque

Torque is the product of force and lever arm; it describes the turning and tilting effect of a force about an axis.

p + \frac{1}{2} ρ v^{2} + ρ g h = const.

Bernoulli Equation

The Bernoulli equation is energy conservation for flowing fluids: where the speed rises, the static pressure drops.

ω = 2 π f = \frac{2 π}{T} v = ω \cdot r

Angular Velocity and Angular Frequency

The angular velocity ω measures the swept angle per time; via v = ω·r it connects rotation with orbital speed.

E_{Sp ann} = \frac{1}{2} D s^{2}

Elastic Energy of a Spring

The elastic energy of a spring grows quadratically with the displacement; it is the area under the force-displacement line of Hooke law.

Heat and thermodynamics

3 formulas

η_{C a r n o t} = 1 - \frac{T _{ka lt}}{T _{w a r m}}

Carnot Efficiency

The Carnot efficiency gives the maximum possible efficiency of a heat engine operating between two temperature levels.

Q = m \cdot c \cdot Δ T

Heat Capacity: Q = m·c·ΔT

The specific-heat-capacity formula describes how much thermal energy a substance absorbs, which is essential for calorimetry, reaction enthalpies and heat transport.

P = σ \cdot A \cdot T^{4}

Stefan-Boltzmann Law

The Stefan-Boltzmann law gives the radiated power of a black body: it grows with the fourth power of the absolute temperature.

Electricity and magnetism

14 formulas

F = k_{e} \cdot \frac{q _{1} \cdot q _{2}}{r ^{2}}

Coulomb's Law

Coulomb's law describes the electrostatic force between two point charges.

F = q \cdot (v \times B)

Lorentz Force

The Lorentz force describes the force on a moving electric charge in a magnetic field.

U_{in d} = - N \cdot \frac{d Φ}{d t}

Faraday's Law of Induction

Faraday's law of induction describes the generation of a voltage by a change in the magnetic flux.

U = R \cdot I

Ohm's Law

Ohm's law links voltage, resistance and current: the voltage across an ohmic conductor is proportional to the current.

P = U \cdot I

Electric Power

Electric power is the product of voltage and current; it states how much energy a device converts per second.

R_{g es} = R_{1} + R_{2} \frac{1}{R _{g es}} = \frac{1}{R _{1}} + \frac{1}{R _{2}}

Resistors in Series and Parallel

In series the resistances add up, in parallel their reciprocals add up; the parallel resistance is always smaller than the smallest single resistor.

C = \frac{Q}{U}

Capacitance of a Capacitor

The capacitance states how much charge a capacitor stores per volt of voltage.

E = \frac{1}{2} C U^{2}

Energy Stored in a Capacitor

The energy stored in the electric field of a capacitor grows quadratically with the voltage.

\frac{U _{1}}{U _{2}} = \frac{n _{1}}{n _{2}} = \frac{I _{2}}{I _{1}}

Transformer (Turns Ratio)

In an ideal transformer the voltages behave like the numbers of turns, the currents scale inversely, and the power is conserved.

T = 2 π L \cdot C

Thomson Oscillation Formula

The Thomson formula gives the period of an electromagnetic oscillating circuit made of coil and capacitor, the electrical counterpart of the spring pendulum.

E = \frac{F}{q} = \frac{U}{d}

Electric Field Strength

The electric field strength is the force per charge; in the uniform field of a parallel-plate capacitor E = U/d additionally holds.

R = ρ \cdot \frac{l}{A}

Resistivity (Conductor Resistance)

The resistance of a wire grows with its length and falls with its cross-section; the resistivity ρ is the material constant.

B = μ_{0} \cdot \frac{N}{l} \cdot I

Magnetic Field of a Long Coil

The magnetic field inside a long coil is uniform and proportional to the current and the turn density N/l.

U_{in d} = - L \cdot \frac{d I}{d t}

Self-Induction of a Coil

Self-induction describes how a coil reacts to changes of its own current with an opposing voltage; the inductance L is its measure.

Oscillations and waves

4 formulas

T = 2 π \frac{m}{k}

Spring Pendulum (Oscillation Period)

The oscillation period of a spring pendulum depends only on mass and spring constant, not on the amplitude.

c = λ \cdot f

Wave Equation (c = λ·f)

The fundamental equation of wave physics links propagation speed, wavelength and frequency of any wave.

f_{B} = f_{Q} \cdot \frac{c \pm v _{B}}{c \mp v _{Q}}

Doppler Effect (Acoustics)

The Doppler effect describes the frequency shift when source or observer move relative to each other: approach raises, receding lowers the perceived frequency.

T = 2 π \frac{l}{g}

Simple Pendulum (Period)

The period of a simple pendulum depends only on the string length and the gravitational acceleration, not on the mass and, for small angles, not on the amplitude.

Optics

3 formulas

n_{1} \cdot sin (α_{1}) = n_{2} \cdot sin (α_{2})

Snell's Law of Refraction

Snell's law of refraction describes how light rays are refracted when crossing between two media, the basis for lenses, optical fibre and optics.

\frac{1}{f} = \frac{1}{g} + \frac{1}{b}

Thin Lens Equation

The lens equation links focal length, object distance and image distance of a thin lens, the fundamental equation of optical imaging.

d \cdot sin (α_{k}) = k \cdot λ

Double-Slit Interference

The double-slit formula gives the directions of the interference maxima: reinforcement occurs when the path difference d·sinα is an integer multiple of the wavelength.

Atomic and nuclear physics

5 formulas

λ = \frac{h}{p} = \frac{h}{m v}

De Broglie Wavelength

The de Broglie wavelength describes the wave nature of matter particles, the wave-particle duality.

E_{kin} = h \cdot f - W_{A}

Photoelectric Effect (Einstein)

The photoelectric effect describes the emission of electrons from a metal surface when it is illuminated with light.

Δ x \cdot Δ p \geq \frac{ℏ}{2}

Heisenberg Uncertainty Principle

The uncertainty principle sets a fundamental limit on how precisely position and momentum can be determined simultaneously, not a measurement problem but a property of quantum nature.

N (t) = N_{0} \cdot e^{- λ t}

Radioactive Decay Law

The decay law describes the exponential decrease of the undecayed atomic nuclei of a radioactive substance.

E = m \cdot c^{2}

Mass-Energy Equivalence (E = mc²)

Einstein's most famous equation: mass and energy are equivalent, every mass corresponds to an enormous rest energy.

Chemistry

24 formulas

Stoichiometry and concentration

7 formulas

A = ε \cdot c \cdot d

Beer-Lambert Law (Photometry)

The Beer-Lambert law relates the absorbance of a solution to its concentration and path length, the basis of UV/Vis spectroscopy and quantitative analysis.

n = \frac{m}{M}

Amount of Substance (Mole)

The amount of substance n counts particles in portions of moles. Calculated from mass m and molar mass M, n = m/M is the basic formula of every stoichiometric calculation.

M = \frac{m}{n}

Molar Mass

The molar mass M states how many grams one mole of a substance weighs. It is the quotient of mass and amount of substance and, for compounds, the sum of the atomic masses.

c = \frac{n}{V}

Molar Concentration

The molar concentration c (molarity) states how many moles of a dissolved substance are contained in one litre of solution. It connects amount of substance and solution volume.

N = n \cdot N_{A}

Avogadro Constant (Particle Number)

The Avogadro constant N_A links the amount of substance to the actual particle number: one mole contains 6.022×10²³ particles, n moles correspondingly n times as many.

c_{1} \cdot V_{1} = c_{2} \cdot V_{2}

Dilution Equation

The dilution equation uses the fact that on dilution the amount of dissolved substance stays the same: concentration times volume is equal before and after dilution.

w = \frac{m _{Stoff}}{m _{L \overset{o}{¨} sung}}

Mass Fraction (Mass Percent)

The mass fraction w states what proportion of the total mass of a solution or mixture is the dissolved substance; times 100 it gives the mass percent.

Gases and energetics

4 formulas

p \cdot V = n \cdot R \cdot T

Ideal Gas Law

The ideal gas law relates the pressure, volume, amount of substance and temperature of an ideal gas.

Δ G = Δ H - T \cdot Δ S

Gibbs Energy (Free Enthalpy)

The Gibbs energy determines the spontaneity of chemical reactions: ΔG < 0 → spontaneous, ΔG > 0 → not spontaneous.

Δ H_{R}^{0} = \sum Δ H_{f}^{0} (Produkte) - \sum Δ H_{f}^{0} (Edukte)

Reaction Enthalpy (Hess's Law)

Hess's law allows reaction enthalpies to be calculated from tabulated standard enthalpies of formation, because enthalpy is a state function: only the initial and final states count, not the path.

Δ H_{Gitter} = Δ H_{f}^{0} - Δ H_{Sub} - I - \frac{1}{2} D - E_{EA}

Lattice Energy (Born-Haber Cycle)

The Born-Haber cycle determines the lattice enthalpy of a salt, which cannot be measured directly: it decomposes the formation from the elements into measurable partial steps and applies Hess's law.

Reaction kinetics

3 formulas

k = A \cdot e^{- E_{A} / (R \cdot T)}

Arrhenius Equation

The Arrhenius equation describes the temperature dependence of the rate constant of a chemical reaction.

v = k \cdot [A]^{m} \cdot [B]^{n}

Rate Law (Reaction Order)

The rate law links the reaction rate to the reactant concentrations; the exponents m and n are the reaction orders and are determined experimentally.

v_{2} = v_{1} \cdot Q_{10}^{Δ T / (10 K)}

RGT Rule (Q10 Rule)

The RGT rule (reaction-rate-temperature rule) is a rule of thumb: a temperature increase of 10 K speeds up many reactions by roughly the factor Q₁₀, usually 2 to 4.

Chemical equilibrium

3 formulas

K_{c} = \frac{[ C ] ^{c} \cdot [ D ] ^{d}}{[ A ] ^{a} \cdot [ B ] ^{b}}

Law of Mass Action (Equilibrium Constant)

The law of mass action describes chemical equilibrium: the stoichiometrically weighted ratio of the equilibrium concentrations of products to reactants is constant.

K_{L} = [A^{m +}]^{a} \cdot [B^{n -}]^{b}

Solubility Product

The solubility product K_L is the equilibrium constant for dissolving a sparingly soluble salt: the product of the ion concentrations, each raised to its stoichiometric coefficient.

ln \frac{K _{2}}{K _{1}} = - \frac{Δ H _{R}^{0}}{R} \cdot (\frac{1}{T _{2}} - \frac{1}{T _{1}})

Van 't Hoff Equation (Reaction Isobar)

The van 't Hoff equation describes how the equilibrium constant depends on temperature: from the standard reaction enthalpy K follows at any other temperature.

Acids and bases

5 formulas

p H = p K_{a} + lo g \frac{[ A ^{-} ]}{[ H A ]}

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation calculates the pH value of buffer solutions from the pKa of the acid and the concentration ratio.

p H = - l g ([H_{3} O^{+}])

Definition of the pH Value

The pH value is the negative base-10 logarithm of the hydronium-ion concentration, the universal scale for acid and base in chemistry.

p K_{S} = - l g K_{S} = - l g \frac{[ H _{3} O ^{+} ] \cdot [ A ^{-} ]}{[ HA ]}

pKa Value (Acid Constant)

The pKa value is the negative base-10 logarithm of the acid constant Ka and measures the strength of an acid: the smaller the pKa, the stronger the acid.

K_{W} = [H_{3} O^{+}] \cdot [OH^{-}]

Ion Product of Water

The ion product of water links the hydronium and hydroxide concentrations of every aqueous solution: at 25 °C, Kw = 10⁻¹⁴ mol²/L², from which pH + pOH = 14 follows.

p K_{B} = - l g K_{B} = - l g \frac{[ BH ^{+} ] \cdot [ OH ^{-} ]}{[ B ]}

pKb Value (Base Constant)

The pKb value measures the strength of a base via the base constant Kb of its protolysis with water: the smaller the pKb, the stronger the base. For conjugate pairs pKa + pKb = 14 (25 °C).

Electrochemistry

2 formulas

E = E^{0} - \frac{RT}{z F} ln Q

Nernst Equation

The Nernst equation describes the electrode potential as a function of temperature and concentration.

m = \frac{M \cdot I \cdot t}{z \cdot F}

Faraday's Law of Electrolysis

Faraday's law links the mass deposited in an electrolysis to the charge passed, Q = I·t: the Faraday constant translates charge into amount of substance.

Mathematics

49 formulas

Algebra and equations

4 formulas

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Quadratic Formula

The quadratic formula yields all solutions of any quadratic equation ax² + bx + c = 0.

x_{1, 2} = - \frac{p}{2} \pm (\frac{p}{2})^{2} - q

pq Formula

The pq formula solves any quadratic equation in normalized form x² + px + q = 0, that is with leading coefficient 1.

(a \pm b)^{2} = a^{2} \pm 2 ab + b^{2}, (a + b) (a - b) = a^{2} - b^{2}

Binomial Formulas

The three binomial formulas square sums and differences and, read backwards, factor quadratic terms.

lo g_{b} (x \cdot y) = lo g_{b} x + lo g_{b} y, lo g_{b} (x^{r}) = r \cdot lo g_{b} x

Laws of Logarithms

The laws of logarithms turn products into sums, quotients into differences and powers into multiples.

Functions, sequences and series

4 formulas

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}, y = m \cdot x + b

Line Equation and Slope

The slope m measures the change in y per step in x; together with the y-intercept b it fixes the line y = mx + b.

y = a (x - d)^{2} + e

Vertex Form of the Parabola

The vertex form shows the vertex S(d|e) of a parabola directly in the equation; the factor a controls opening and stretching.

s_{n} = a_{1} \cdot \frac{q ^{n} - 1}{q - 1} (q \neq = 1), s_{\infty} = \frac{a _{1}}{1 - q} (∣ q ∣ < 1)

Geometric Series

The geometric series sums terms with a constant factor q; for |q| < 1 even the infinite series has a finite value.

s_{n} = \frac{n \cdot ( a _{1} + a _{n} )}{2}

Arithmetic Series (Gauss Sum Formula)

The Gauss sum formula adds terms with constant difference: number of terms times the mean of first and last term.

Geometry and trigonometry

10 formulas

a^{2} + b^{2} = c^{2}

Pythagorean Theorem

The Pythagorean theorem holds for right triangles: the square of the hypotenuse equals the sum of the squares of the two legs.

c^{2} = a^{2} + b^{2} - 2 ab \cdot cos γ

Law of Cosines

The law of cosines generalizes the Pythagorean theorem to arbitrary triangles without a right angle.

\frac{a}{s i n α} = \frac{b}{s i n β} = \frac{c}{s i n γ}

Law of Sines

In a triangle the sides are proportional to the sines of their opposite angles.

A = π r^{2}, U = 2 π r

Circle Area and Circumference

Area and circumference of a circle depend only on the radius r, linked by the circle number π.

V = \frac{4}{3} π r^{3}, O = 4 π r^{2}

Sphere: Volume and Surface Area

Volume and surface area of a sphere depend only on the radius: the volume grows with the third power of r, the surface area with the second.

V = π r^{2} h, O = 2 π r (r + h)

Cylinder: Volume and Surface Area

The cylinder volume is base area times height; the surface consists of two circular lids and the unrolled lateral rectangle.

V = \frac{1}{3} π r^{2} h, M = π rs

Cone: Volume and Surface Area

A cone holds one third of the cylinder with equal base and height; the lateral surface is computed via the slant height s.

V = \frac{1}{3} \cdot G \cdot h

Pyramid: Volume

Every pyramid holds one third of the prism with equal base area G and height h, regardless of the shape of the base.

sin α = \frac{GK}{Hyp}, cos α = \frac{AK}{Hyp}, tan α = \frac{GK}{AK}

Trigonometry in the Right Triangle

Sine, cosine and tangent link an acute angle of a right triangle with the ratios of its sides.

A = \frac{1}{2} \cdot a \cdot b \cdot sin γ

Triangle Area with Sine

The trigonometric area formula: two sides and the angle enclosed by them determine the area of the triangle.

Calculus

15 formulas

e^{i φ} = cos φ + i \cdot sin φ

Euler's Formula

Euler's formula links the exponential function with the trigonometric functions in the complex plane.

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

Product Rule of Differentiation

The product rule allows the differentiation of products of two functions.

(f \circ g)^{'} (x) = f^{'} (g (x)) \cdot g^{'} (x)

Chain Rule of Differentiation

The chain rule differentiates composite functions f(g(x)): outer derivative times inner derivative.

f (x) = \sum_{n = 0}^{\infty} \frac{f ^{(n)} ( a )}{n !} (x - a)^{n}

Taylor Series

The Taylor series expresses a function as a power series around an expansion point a.

\frac{d}{d x} (x^{n}) = n \cdot x^{n - 1}

Power Rule of Differentiation

The power rule is the most fundamental differentiation rule: it allows the immediate differentiation of any monomial, serving as the entry point to all of differential calculus.

\frac{d}{d x} e^{x} = e^{x}, \frac{d}{d x} e^{k x} = k \cdot e^{k x}

Derivative of the Exponential Function

The exponential function is its own derivative; for e^(kx) the chain rule adds the factor k.

\frac{d}{d x} ln x = \frac{1}{x} (x > 0)

Derivative of the Natural Logarithm

The derivative of the natural logarithm ln x is the hyperbola 1/x, valid for all x > 0.

(sin x)^{'} = cos x, (cos x)^{'} = - sin x

Derivatives of Sine and Cosine

Sine and cosine differentiate into each other cyclically: sin becomes cos, cos becomes −sin, each in radians.

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - f \cdot g ^{'}}{g ^{2}}

Quotient Rule of Differentiation

The quotient rule differentiates fractions of two functions, for example rational functions.

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C (n \neq = - 1)

Antiderivatives and Basic Integrals

The power rule of integration yields antiderivatives: raise the exponent by 1 and divide by the new exponent.

\int_{a}^{b} f (x) d x = F (b) - F (a)

Fundamental Theorem of Calculus

The fundamental theorem connects differentiation and integration: a definite integral is evaluated via an antiderivative at the limits.

A = \int_{a}^{b} f (x) d x (f (x) \geq 0)

Area under a Curve (Definite Integral)

The definite integral measures the area between graph and x-axis as long as the graph runs above the axis.

\int u \cdot v^{'} d x = u \cdot v - \int u^{'} \cdot v d x

Integration by Parts

Integration by parts integrates products by reversing the product rule of differentiation.

f^{'} (x_{0}) = lim_{h \to 0} \frac{f ( x _{0} + h ) - f ( x _{0} )}{h}

Difference Quotient and h-Method

The difference quotient measures the average rate of change; in the limit h towards 0 it becomes the derivative, the local rate of change.

V = π \int_{a}^{b} f (x)^{2} d x

Volume of Revolution

If the graph of f rotates around the x-axis, a solid of revolution arises; its volume sums circular discs with radius f(x).

Analytic geometry and vectors

6 formulas

a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3} = ∣ a ∣ \cdot ∣ b ∣ \cdot cos φ

Dot Product

The dot product of two vectors yields a number and measures angles and orthogonality.

a \times b = a_{2} b_{3} - a_{3} b_{2} a_{3} b_{1} - a_{1} b_{3} a_{1} b_{2} - a_{2} b_{1}

Cross Product (Vector Product)

The cross product of two vectors in space yields a vector perpendicular to both.

d (P; E) = \frac{∣ a p _{1} + b p _{2} + c p _{3} - d ∣}{a ^{2} + b ^{2} + c ^{2}}

Hesse Normal Form (Distance Point-Plane)

The Hesse normal form gives the distance of a point from a plane: insert the point into the coordinate equation and divide by the length of the normal vector.

E : (x - p) \cdot n = 0

Plane Equation in Normal Form

A plane is fixed by a point and a normal vector: exactly the position vectors x⃗ with (x⃗ − p⃗)·n⃗ = 0 lie in it.

d (Q; g) = \frac{∣ ( q - p ) \times u ∣}{∣ u ∣}

Distance Point-Line

The distance of a point from a line in space is the length of the perpendicular; the cross product formula yields it in one step.

∣ v ∣ = v_{1}^{2} + v_{2}^{2} + v_{3}^{2}

Magnitude of a Vector (Length)

The magnitude of a vector is its length: the square root of the sum of the squared coordinates, a twice-applied Pythagoras in space.

Probability and statistics

10 formulas

P (A ∣ B) = \frac{P ( B ∣ A ) \cdot P ( A )}{P ( B )}

Bayes' Theorem

Bayes' theorem describes the inversion of conditional probabilities.

f (x) = \frac{1}{σ 2 π} e^{- \frac{( x - μ ) ^{2}}{2 σ ^{2}}}

Normal Distribution (Gaussian Bell Curve)

The normal distribution is the most important probability distribution, describing many natural phenomena.

P (X = k) = (k n) \cdot p^{k} \cdot (1 - p)^{n - k}

Binomial Distribution

The binomial distribution gives the probability of exactly k successes in n independent trials with success probability p.

E (X) = μ = \sum_{i} x_{i} \cdot p_{i}

Expected Value

The expected value is the long-run average of a random variable: values times probabilities, summed up.

P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )} (P (B) > 0)

Conditional Probability

The conditional probability P(A|B) measures the probability of A given that B has already occurred.

σ = Var (X) = E ((X - μ)^{2})

Standard Deviation

The standard deviation σ measures the typical spread of a random variable around its expected value μ.

b = \frac{\sum ( x _{i} - x ˉ ) ( y _{i} - y ˉ )}{\sum ( x _{i} - x ˉ ) ^{2}}, a = \overset{y}{ˉ} - b \overset{x}{ˉ}

Linear Regression (Line of Best Fit)

Linear regression places the line of best fit ŷ = a + bx through a scatter plot so that the sum of the squared deviations becomes minimal.

(k n) = \frac{n !}{k ! ( n - k )!}

Binomial Coefficient (n choose k)

The binomial coefficient n choose k counts how many k-element subsets can be selected from n objects, without regard to order.

P (X = k) = \frac{λ ^{k}}{k !} \cdot e^{- λ}

Poisson Distribution

The Poisson distribution models the number of rare, independent events in a fixed interval with known mean rate λ.

P (μ - 2 σ \leq X \leq μ + 2 σ) \approx 95, 4 %

Sigma Rules (Sigma Intervals)

The sigma rules state with which probability a normally distributed quantity lies in the intervals μ ± σ, μ ± 2σ and μ ± 3σ.

Biology

8 formulas

v = \frac{V _{m a x} \cdot [ S ]}{K _{m} + [ S ]}

Michaelis-Menten Kinetics (Enzyme Kinetics)

The Michaelis-Menten equation describes the dependence of the enzyme reaction rate on substrate concentration. This model is a central model of enzyme kinetics.

\frac{d N}{d t} = r \cdot N \cdot (1 - \frac{N}{K})

Logistic Growth

The differential equation of logistic growth describes population growth with a capacity limit and is a realistic model for biological populations.

p^{2} + 2 pq + q^{2} = 1

Hardy-Weinberg Equilibrium

The Hardy-Weinberg equilibrium links the allele frequencies p and q in an ideal population with the genotype frequencies and stays constant from generation to generation as long as no evolutionary factors act.

N (t) = N_{0} \cdot e^{r \cdot t}

Exponential Growth

The exponential growth model describes a population that grows at a constant rate under unlimited resources, so that the increase is always proportional to the current size.

χ^{2} = \sum \frac{( B - E ) ^{2}}{E}

Chi-Square Test (Goodness of Fit)

The chi-square goodness-of-fit test checks whether observed frequencies (e.g. the phenotypes of a cross) are compatible with the frequencies expected under a hypothesis (e.g. a Mendelian segregation ratio).

H^{'} = - \sum_{i = 1}^{S} p_{i} \cdot ln (p_{i})

Shannon Index (Biodiversity)

The Shannon index (Shannon-Wiener index) measures the diversity of a community. It accounts for both species richness and the evenness with which individuals are distributed among the species.

D = \sum_{i = 1}^{S} p_{i}^{2}

Simpson Index (Dominance and Diversity)

The Simpson index D measures the dominance in a community: the probability that two randomly drawn individuals belong to the same species. Its complement (1 − D) is the Simpson diversity index and rises with diversity.

N = \frac{M \cdot C}{R}

Lincoln Index (Mark-Recapture Method)

The Lincoln-Petersen index estimates the size of a population that cannot be counted completely. A first catch is marked and released; from the fraction of marked animals in the second catch the total number follows.

Tools

Build what you just looked up

Three tools for chemistry and formulas, free and usable without an account. They run in the browser, and your input never leaves the device.

3D molecule builderPlace atoms and bonds, rotate the structure in 3D, and read the molecular formula and molar mass straight off it.Open tool Structural formula generatorEnter SMILES notation, render the 2D structural formula instantly and download it as a PNG. Client-side and privacy-friendly.Open tool LaTeX formula editorWrite formulas in LaTeX and see them typeset live, with fractions, indices, integrals and reaction arrows.Open tool

Learn formulas for good

Spaced-repetition apps

Compare scheduling workflows

Study evidence

Read methods, results and limits

Compare flashcard apps

FSRS support and study features

Start Quanta

Free · FSRS flashcards