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649 formulas across seven subjects, organised by chapter. 104 come with a calculator and worked steps, 100 with a model diagram: enter values and the result and picture respond straight away.
Browse by subject and chapter, or find a formula by name, symbol or keyword.
Source-linked reference tables, each with its own page, edition and review date.
Newton's Second Law
Newton’s second law relates the vector sum of external forces to mass times acceleration. An individual force equals net force only when the other forces cancel or are absent.
Newton's Law of Universal Gravitation
Newton’s law gives the magnitude of attraction between point masses or separate spherically symmetric bodies.
Kinetic Energy
Classical kinetic energy of a particle or purely translating body is . v is speed in the chosen reference frame.
Potential Energy (Gravitational)
In a uniform gravitational field, gives gravitational potential energy relative to the chosen zero level. Changes depend on vertical height differences.
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.
Uniformly Accelerated Motion (Position–Time Equation)
The position–time equation gives signed displacement at constant acceleration. Only a start from rest makes s purely proportional to .
Velocity-Time Law
The velocity–time equation describes the linear change of signed velocity under constant acceleration. Depending on signs, speed can decrease or increase.
Free Fall
Free fall from rest without air resistance: downward distance grows quadratically with time at constant g. Free fall with initial velocity also needs the v0t term.
Momentum
Classical momentum is mass times directional velocity in the selected frame; one-dimensional components carry signs.
Conservation of Momentum
A selected system conserves total momentum when external impulse over the chosen interval is zero or negligible.
Mechanical Work
Mechanical work is energy transfer by a force along its application-point displacement; only the component along displacement contributes.
Mechanical Power
Mechanical power is the rate of work transfer; gives average power over duration t.
Centripetal Force (Circular Motion)
Centripetal force is the inward radial component of net force during circular motion; at fixed radius it grows with the square of speed.
Density
Average density is mass per defined volume; compare material densities with temperature, pressure and material state specified.
Pressure
Pressure describes normal force per area; with nonuniform loading gives only average normal compressive loading.
Projectile Motion (Trajectory Parabola)
At constant g, projectile motion separates free flight into constant horizontal velocity and vertical acceleration −g. The displayed coordinates use the launch point as origin.
Buoyant Force (Archimedes)
Hydrostatic buoyancy equals the weight of actually displaced fluid; floating or motion also requires a force balance.
Sliding Friction Force
The simple dry-sliding model gives magnitude . Force opposes tangential relative sliding at the contact; static friction instead obeys .
Kepler Third Law
An unperturbed bound two-body orbit satisfies . Semimajor axis is relative; for a light companion, total mass may be approximated by central mass.
Torque
The cross product of position and force describes turning effect. ·sinθ is magnitude; torque about a selected axis is the corresponding signed projection.
Bernoulli Equation
Bernoulli relates static pressure, speed and elevation along a streamline under ideal flow conditions; at equal elevation higher speed accompanies lower static pressure.
Angular Velocity and Angular Frequency
Angular velocity is angle change per time. The shorthand gives magnitude for uniform revolution; tangential speed additionally depends on distance from the axis.
Elastic Energy of a Spring
For an ideal linear spring, is stored energy relative to its force-free natural state. Between pre-extended states use the difference of squared extensions, not the square of extra displacement.
Hydrostatic pressure
In a static liquid of constant density, local pressure rises linearly with vertical depth; add pressure at the reference surface.
Continuity equation for flow
Steady mass balance connects density, cross-sectional area and mean normal speed at one inlet and one outlet.
Moment of inertia of a particle system
Moment of inertia measures how a body’s mass is distributed relative to a chosen rotation axis.
Angular momentum of a rigid body
For pure rigid rotation about a fixed axis, the axial component obeys . Along a principal axis, scalar I also describes the entire parallel angular-momentum vector.
Rotational kinetic energy of a rigid body
Rigid rotation about a fixed axis has kinetic energy . Inertia belongs to that exact axis; squaring makes energy independent of rotation direction.
Angular velocity under constant angular acceleration
Constant signed angular acceleration about a fixed axis makes signed angular velocity change linearly with time.
Newton’s second law for rotation
Net external torque component about a fixed inertial axis equals I times signed angular acceleration when I is constant.
Rolling without slipping
For rolling without slipping on stationary ground, centre-of-mass speed equals rolling radius times angular speed.
Parallel-axis theorem for mass moment of inertia
The moment of inertia about an axis parallel to the center-of-mass axis increases by .
Gravitational potential energy in a central field
Using , the pair interaction energy is . r is center distance; binding is determined by appropriate total energy, not by U alone.
Circular-orbit speed
For a light satellite around a dominant central mass, gravity supplies circular acceleration; is the required speed.
Escape speed from a gravitational field
The minimum launch speed at radius R to reach infinity without propulsion or losses follows from energy conservation.
Weight Near a Celestial Body’s Surface
Near a surface in an approximately uniform gravitational field, weight is the magnitude of the gravitational force on a mass.
Center of Mass of a System
The center of mass is the mass-weighted mean of the position vectors of discrete components.
Stokes Drag, Terminal Settling Speed and Quadratic Drag
Fluid drag can scale linearly with relative speed for creeping flow around a sphere or quadratically for many bluff bodies at higher Reynolds numbers; a sphere’s Stokes-regime terminal speed follows from balancing effective weight against drag.
Coefficient of Restitution
The coefficient of restitution compares relative separation speed after impact with relative approach speed before impact along the collision axis.
Carnot Efficiency
Carnot efficiency bounds cyclic heat-to-work conversion between two constant-temperature reservoirs. The reversible limit has .
Heat Capacity: Q = m·c·ΔT
calculates heat for a temperature change without a phase transition using suitable approximately constant c. is heat capacity of the chosen amount.
First Law of Thermodynamics
For a closed system without relevant bulk energy changes, internal-energy change equals heat entering minus work performed by the system. Q and W have directed signs.
Stefan-Boltzmann Law
Total emitted radiant power from a black surface grows as . Emitted power and net heat exchange are different quantities.
Linear thermal expansion
Free linear length change is approximately when material direction, temperature interval and coefficient match. L0 is reference length at T0.
Latent heat during a phase change
For a pure substance at an equilibrium phase transition and constant pressure, mL gives transition energy without temperature change. Heating before or after requires additional terms.
Entropy change along a reversible path
The entropy change of a system is the integral of reversible heat divided by absolute temperature along a reversible comparison path; at constant temperature this reduces to .
Mean translational kinetic energy per gas molecule
The mean translational kinetic energy of an ideal-gas molecule depends only on absolute temperature.
Internal energy of a monatomic ideal gas
The internal energy of a monatomic ideal gas is the sum of the translational kinetic energies of its particles.
Root-mean-square molecular speed
The RMS speed is the square root of the mean squared speed of an ideal-gas molecule.
Pressure from kinetic theory
Gas pressure can be understood as momentum transfer from random molecular motion to the container walls.
Efficiency of an Energy Conversion
Efficiency compares intended energy output with complete input for the same system boundary and time interval.
Maxwell–Boltzmann Molecular Speed Distribution
At thermal equilibrium, the Maxwell–Boltzmann density describes how molecular speed magnitudes are distributed in a classical ideal gas; it is a density over .
Planck Spectral Law
Planck’s law describes blackbody equilibrium spectral radiance per wavelength interval. Power additionally requires appropriate area, angle and band integration.
Equipartition of Energy
In classical thermal equilibrium, each quadratic degree of freedom contributes an average per particle.
Work in Isothermal Expansion of an Ideal Gas
For reversible isothermal expansion of an ideal gas, the work done by the gas depends logarithmically on the volume ratio.
Boltzmann Entropy and Microstates
The Boltzmann relation assigns entropy to a macrostate with Ω accessible, equally probable microstates.
Coulomb's Law
Coulomb's law describes the electrostatic force between two point charges.
Lorentz Force
The upper equation gives the magnetic contribution to the Lorentz force on a moving charge; an electric field adds electric force.
Faraday’s law of induction
A change in oriented flux linkage produces loop emf. Terminal voltage additionally requires a defined measurement and circuit model.
Gauss's Law for Electrostatics
The electric flux through a closed surface depends only on the total charge it encloses. With sufficient symmetry, the law makes the electric field especially easy to calculate.
Ohm's Law
Ohm's law links voltage, resistance and current: the voltage across an ohmic conductor is proportional to the current.
Electric Power
applies to constant DC quantities; is the more general instantaneous relation. Resistor forms require ohmic behaviour. Sinusoidal AC rms values require the power factor.
Resistors in Series and Parallel
Ideal ohmic resistors in a lumped circuit model. Series requires common current without a branch; parallel requires the same two nodes. The strict parallel bound assumes at least two positive finite resistances.
Capacitance of a Capacitor
Capacitance relates the charge magnitude on one capacitor plate to the voltage between plates. For linear material and unchanged geometry, C is constant.
Energy Stored in a Capacitor
The energy stored in the electric field of a capacitor grows quadratically with the voltage.
Transformer turns ratio
In an ideal transformer RMS voltage follows the turns ratio and load current scales inversely. Polarity, power factor and real losses are treated separately.
Thomson formula for an LC circuit
Thomson’s formula gives the period of free undamped oscillation in an ideal LC circuit. Initial energy sets amplitude rather than ideal frequency.
Electric Field Strength
Electric field is a vector specifying electric force per signed test charge. For a uniform plate field, its magnitude also equals voltage magnitude divided by perpendicular separation.
Resistivity (Conductor Resistance)
Homogeneous conductor, uniform cross-section and specified temperature in a one-dimensional ohmic model. Contact resistance, current crowding and strong temperature profiles require separate treatment.
Magnetic Field of a Long Coil
The equation approximates interior field away from the ends of a long tightly wound air-core solenoid. Turn density and current matter, not turn count alone.
Self-Induction of a Coil
Current change alters self-flux linkage and produces emf opposing that change. Ideal passive inductor voltage uses the opposite voltage reference.
RC time constant
One ideal capacitor with constant positive capacitance and finite positive effective resistance in a linear network that is time-invariant after switching. The time constant describes voltage deviation from the final value; multiple independent storage elements can produce multiple time constants.
Motional emf
The magnitude of a straight rod’s motional emf contribution is Bℓv when field, rod and velocity are pairwise perpendicular.
Induced emf of a rotating generator
A planar coil rotating uniformly in a uniform magnetic field generates sinusoidal emf when its area normal sweeps the angle to the field uniformly.
Magnetic flux through a surface
Magnetic flux integrates the oriented normal field component over a selected surface.
Impedance of a series RLC circuit
Impedance combines resistance and frequency-dependent reactance in a series RLC circuit.
Magnetic field of a long straight wire
The external field magnitude of a long straight current-carrying wire decreases inversely with shortest axis distance; the field vector circulates tangentially around the axis.
RMS values of sinusoidal alternating quantities
The RMS value of a sinusoidal voltage or current equals the DC value that would produce the same average heating in a resistor.
Change in electric potential energy
The change in electric potential energy of an unchanged test charge equals its signed charge multiplied by the difference between destination and starting potential. Electric field work has the opposite sign.
Capacitance of a parallel-plate capacitor
For large parallel plates fully filled with a uniform linear dielectric, capacitance is proportional to overlap area and permittivity and inversely proportional to perpendicular separation.
Magnetic force on a current-carrying wire
Magnetic force on a straight current-carrying segment follows the cross product of conventional current direction and external field.
Torque on a current loop
A planar current-carrying frame can experience torque in a uniform field despite zero net force, aligning its dipole axis.
Electric-field energy density in vacuum
In vacuum, an electric field stores energy per unit volume proportional to the square of field strength.
Electric field of an infinite line charge
An ideal infinite line charge produces a radially symmetric field whose magnitude falls as .
Maxwell Equations in Integral Form
The four Maxwell equations relate electric and magnetic fluxes to charge, current and time-varying fields.
Poynting Vector and Electromagnetic Energy Flux
The Poynting vector gives the direction and instantaneous density of electromagnetic energy flow in vacuum.
Electrical Energy at Constant Voltage and Current
assumes constant voltage and current over the interval. Generally integrate the simultaneous product; separate averages cannot generally be multiplied. The sign follows the chosen passive orientation.
Hall Effect and Hall Voltage
The Hall voltage appears when a magnetic field deflects moving charge carriers sideways and builds a transverse charge separation.
Biot–Savart Law
The Biot–Savart law sums vector magnetic-field contributions from steady current elements at an observation point.
Electric Dipole: Moment, Torque and Energy
The electric dipole moment represents two opposite charges; in a uniform field, cross and dot products give torque and potential energy.
Mutual Inductance
Mutual inductance measures flux through a second coil per current in the first and determines the induced voltage.
Drift Model and Current Density
The drift model relates directed mean charge transport to current and current density in a uniform conductor cross-section.
Cyclotron Radius and Period
A charged particle entering a uniform magnetic field perpendicularly follows a circular orbit with cyclotron radius r and period T.
Force Between Parallel Currents
Two long parallel conductors attract for currents in the same direction and repel for currents in opposite directions.
Spring Pendulum (Oscillation Period)
In the ideal linear undamped mass–spring model, natural period depends on m and k; amplitude does not change it within that model.
Wave Equation (c = λ·f)
The relation connects phase speed, wavelength and frequency of the same travelling periodic component.
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.
Simple Pendulum (Period)
Ideal small-angle pendulum period depends on centre-of-mass length l and local g. Mass cancels; at larger angles period increases with amplitude.
Standing wave on a string fixed at both ends
For a uniform ideal flexible string with two fixed ends and constant v, boundary conditions give discrete wavelengths and natural frequencies.
Damped harmonic oscillation
Free underdamped motion with linear viscous resistance has an exponential envelope and damped natural angular frequency .
Sound intensity and level
Sound intensity is mean power per area; its logarithmic level is , with reference intensity in air.
Wave speed on a stretched string
The transverse-wave speed on an ideal flexible string grows with the square root of tension and decreases with the square root of linear mass density.
Bragg Condition for Crystal Diffraction
Constructive reflection from parallel crystal planes occurs when the path difference is an integer multiple of wavelength.
Phase and Group Velocity
Phase velocity follows a constant phase, while group velocity describes the motion of a narrow-band wave packet.
Radiation Pressure for Absorption and Reflection
Electromagnetic radiation transfers momentum and therefore exerts pressure on a surface.
Snell's Law of Refraction
Snell’s law connects ray angles across an interface through refractive indices. Angles are measured from the normal and indices refer to the same frequency.
Thin Lens Equation
The thin-lens relation locates real or virtual paraxial images. Signed magnification describes their orientation.
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.
Diffraction grating: principal maxima
Constructive interference from a diffraction grating occurs when the path difference between adjacent slits is an integer multiple of the wavelength.
Malus’s law
Malus gives the intensity projection of already linearly polarized input at an ideal analyzer: for field and for intensity.
Critical angle for total internal reflection
θc is incidence where the limiting propagating refracted direction becomes tangent. Above it, all normal power is reflected in the ideal two-medium model.
Brewster angle for polarization
At Brewster incidence, p reflection vanishes; remaining reflection of unpolarized input is s-polarized.
Rayleigh criterion for a circular aperture
Rayleigh’s criterion sets two incoherent point-image separation to the first-minimum radius of a circular-pupil Airy pattern.
Minima of single-slit diffraction
Minima in a single-slit pattern occur when wave contributions from different parts of the slit cancel destructively.
De Broglie Wavelength
The de Broglie wavelength describes the wave nature of matter particles, the wave-particle duality.
Photoelectric Effect (Einstein)
The photoelectric effect describes the emission of electrons from a metal surface when it is illuminated with light.
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.
Radioactive Decay Law
The decay law describes the exponential decrease of the undecayed atomic nuclei of a radioactive substance.
Mass-Energy Equivalence (E = mc²)
Einstein's most famous equation: mass and energy are equivalent, every mass corresponds to an enormous rest energy.
Photoelectric effect: stopping potential versus frequency
Above the threshold frequency, the stopping potential rises linearly with light frequency and makes photoelectron energy measurable.
Activity of a radioactive sample
Activity is the number of nuclear decays per unit time and is proportional to the number of undecayed nuclei.
Nuclear binding energy from mass defect
A nucleus’s binding energy equals the energy equivalent of its mass defect relative to separated nucleons.
Time dilation and length contraction
Relatively moving inertial observers measure a longer interval for a moving clock and a contracted length for a rod parallel to the motion.
Relativistic energy-momentum relation
The relativistic total energy of a free particle is invariantly related to its momentum and rest mass.
Time-Dependent and Stationary Schrödinger Equations
The time-dependent equation evolves a nonrelativistic quantum state; a time-independent potential also yields stationary energy eigenstates.
Energy States in a One-Dimensional Infinite Well
A particle between infinitely high walls has discrete energies and standing-wave eigenfunctions.
Compton Shift
The Compton relation gives the wavelength change when a photon scatters from a free electron.
Bohr Energy Levels and Transitions in Hydrogen
The Bohr model assigns discrete energy levels to hydrogen. A transition absorbs or emits a photon whose energy equals the level difference.
Quantized Harmonic-Oscillator Energy Levels
A quantum harmonic oscillator has discrete, equally spaced energy levels and a nonzero ground-state energy.
Tunneling: Transmission Through a Potential Barrier
The WKB approximation estimates the exponentially small transmission probability of a particle through a classically forbidden rectangular barrier.
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.
Amount of Substance (Mole)
The amount of substance n counts particles in portions of moles. Calculated from mass m and molar mass M, is the basic formula of every stoichiometric calculation.
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.
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.
Avogadro Constant (Particle Number)
The Avogadro constant NA links the amount of substance to the actual particle number: one mole contains particles, n moles correspondingly n times as many.
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.
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.
Molality
Molality gives the amount of solute per kilogram of solvent and does not change with temperature alone.
Activity and Activity Coefficient
Activity replaces concentration in thermodynamic equations with an effective dimensionless concentration corrected by the activity coefficient.
Chromatographic Retention Factor
The retention factor compares an analyte’s additional residence time with the hold-up time of the mobile phase in a column.
Mole Fraction
The mole fraction gives the amount of one component divided by the total amount of all components in a mixture.
Mass Concentration
Mass concentration is the mass of a dissolved substance divided by the final volume of the solution.
Stoichiometric Titration Balance
At the equivalence point, the titration balance equates reacted amounts after division by their stoichiometric coefficients.
Linear Calibration Line
A calibration line relates measured instrument response to known analyte concentrations within a validated linear range.
Standard Addition in Analytical Chemistry
Standard addition estimates analyte concentration by adding known amounts directly to aliquots of the sample matrix.
Limit of Detection and Limit of Quantification
Detection and quantification limits estimate when an analyte signal can be distinguished from noise and measured with suitable performance.
Chromatographic Selectivity Factor
The chromatographic selectivity factor compares the retention of two analytes after correcting for mobile-phase dead time.
Chromatographic Resolution
Chromatographic resolution compares the spacing of two peak maxima with the sum of their baseline widths.
Analytical Recovery
Analytical recovery is the measured amount attributable to a known spike divided by the amount added.
Atom Economy
Atom economy is the theoretical fraction of reactant mass incorporated into a specified desired product according to the balanced equation.
Percent Reaction Yield
Percent yield compares isolated product with the maximum amount predicted from the limiting reactant and balanced equation.
Degree of Unsaturation (DBE)
The degree of unsaturation counts the combined ring and π-bond equivalents implied by a molecular formula.
E-Factor in Green Chemistry
The E-factor divides the mass of waste generated by the mass of desired product.
Internal Standard and Relative Response Factor
An internal standard compensates for shared sample-preparation and injection variation; a relative response factor corrects for different detector responses from analyte and standard.
Ideal Gas Law
The ideal gas law relates the pressure, volume, amount of substance and temperature of an ideal gas.
Gibbs Energy (Free Enthalpy)
The Gibbs energy determines the spontaneity of chemical reactions: spontaneous, not spontaneous.
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.
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.
Henry's Law (Concentration Convention)
At a fixed temperature, Henry’s law relates the equilibrium concentration of a gas dissolved in a liquid to its partial pressure above the solution.
Osmotic Pressure (van ’t Hoff Equation)
For a dilute solution, the van ’t Hoff equation estimates osmotic pressure from the concentration of dissolved particles and the absolute temperature.
Dalton's Law of Partial Pressures
For an ideal gas mixture, total pressure is the sum of partial pressures, and each partial pressure is its mole fraction times total pressure.
Compressibility Factor for Real Gases
The compressibility factor Z measures how far a real gas molar volume at a given pressure and temperature deviates from the ideal-gas model.
van der Waals Equation
The van der Waals equation corrects the ideal-gas model for intermolecular attraction and finite molecular volume.
Chemical Potential of a Component
Chemical potential is the change in Gibbs energy when an infinitesimal amount of a component is added under specified conditions.
Gibbs–Helmholtz Equation
The Gibbs–Helmholtz equation relates the constant-pressure temperature derivative of to enthalpy.
Clausius–Clapeyron Equation for Vaporization
The integrated Clausius–Clapeyron equation estimates how a liquid’s saturation vapor pressure changes with temperature.
Kirchhoff’s Law for Reaction Enthalpy
Kirchhoff’s law corrects a reaction enthalpy from one temperature to another using the heat-capacity change.
Entropy of Mixing for an Ideal Mixture
For an ideal mixture, this equation calculates the entropy increase when components mix at constant temperature and pressure.
Boltzmann Distribution and Partition Function
The Boltzmann distribution gives equilibrium populations of energy states; the partition function normalizes their statistical weights.
Gibbs Phase Rule
The Gibbs phase rule counts independent intensive variables in a system at phase equilibrium.
Reaction Enthalpy from Average Bond Enthalpies
Average covalent bond enthalpies estimate a gas-phase reaction enthalpy by subtracting formed-bond energies from broken-bond energies.
Fugacity Coefficient for a Real Gas
The fugacity coefficient corrects a real-gas partial pressure to its thermodynamically effective fugacity.
Arrhenius Equation
The Arrhenius equation describes the temperature dependence of the rate constant of a chemical reaction.
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.
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.
Integrated First-Order Rate Law
The integrated first-order rate law calculates how reactant concentration decreases over time when the rate is proportional to its concentration.
Integrated Zero-Order Rate Law
For a zero-order process, concentration decreases linearly with time at a constant rate.
Integrated Second-Order Rate Law
For a second-order process in one reactant, reciprocal concentration changes linearly with time.
Half-Life by Reaction Order
Half-life formulas show how the time to halve concentration depends on reaction order and rate constant.
Eyring Equation
The Eyring equation relates a first-order rate constant to the Gibbs energy of activation in transition-state theory.
Steady-State Approximation for a Reaction Intermediate
The steady-state approximation treats a reactive intermediate concentration as nearly constant after a short initial transient.
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.
Solubility Product
The solubility product KL is the equilibrium constant for dissolving a sparingly soluble salt: the product of the ion concentrations, each raised to its stoichiometric coefficient.
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.
Raoult’s Law
For an ideal liquid mixture, Raoult’s law relates a component’s partial pressure to its liquid mole fraction.
Boiling-Point Elevation
For a dilute solution, boiling-point elevation estimates how much the solvent boiling temperature rises.
Freezing-Point Depression
For a dilute solution, freezing-point depression estimates how far its freezing temperature falls below that of the pure solvent.
Ionic Strength
Ionic strength combines each ion concentration with the square of its charge number to summarize the ionic environment.
Reaction Quotient
The reaction quotient compares product and reactant activities in the mixture at its current composition.
Relationship between Kp and Kc
This relation converts between dimensionless concentration-based and pressure-based equilibrium constants for an ideal-gas reaction.
Standard Gibbs Energy and Equilibrium Constant
This equation links standard reaction Gibbs energy to the thermodynamic equilibrium constant.
Reaction Gibbs Energy and Reaction Quotient
The reaction Gibbs energy at the current composition indicates the thermodynamic direction of change at fixed temperature and pressure.
Debye–Hückel Limiting Law
The Debye–Hückel limiting law estimates an ion activity coefficient from charge and ionic strength in a very dilute solution.
Van Deemter Equation
The Van Deemter equation relates mobile-phase linear velocity to band broadening in a chromatographic column.
Distribution Ratio in Liquid–Liquid Extraction
The distribution ratio compares total analytical concentrations of a substance in two immiscible liquid phases at equilibrium.
Lever Rule for Phase Fractions
The lever rule calculates the amounts of two coexisting phases from the overall composition and their equilibrium compositions.
Henderson-Hasselbalch Equation
The Henderson-Hasselbalch equation calculates the pH value of buffer solutions from the pKa of the acid and the concentration ratio.
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.
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.
Ion Product of Water
The ion product of water links the hydronium and hydroxide concentrations of every aqueous solution: at 25 °C, , from which follows.
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).
Diprotic Acid Species Fractions
Species fractions give the proportions of a diprotic acid present as H2A, HA− and at a specified pH.
Buffer Capacity of an Acid–Base Pair
Buffer capacity measures how much strong acid or base a solution can absorb per unit volume and pH change.
Titration Curve of a Weak Acid
The titration curve of a monoprotic weak acid with a strong base relates added volume to pH before, at and after the equivalence point.
Nernst Equation
The Nernst equation describes the electrode potential as a function of temperature and concentration.
Faraday's Law of Electrolysis
Faraday's law links the mass deposited in an electrolysis to the charge passed, : the Faraday constant translates charge into amount of substance.
Standard Cell Potential of a Galvanic Cell
The standard cell potential is the cathode reduction potential minus the anode reduction potential under standard conditions.
Concentration Cell as a Nernst Application
A concentration cell develops a potential because the same redox couple has different activities at its two electrodes.
Molar Conductivity
Molar conductivity normalizes a solution’s conductivity by its electrolyte concentration.
Kohlrausch’s Law of Independent Ionic Migration
Kohlrausch’s law combines limiting ionic conductivities to estimate the limiting molar conductivity of an electrolyte.
Butler–Volmer Equation
The Butler–Volmer equation gives net current density as the difference between anodic and cathodic charge-transfer currents.
Electrophoretic Mobility
Electrophoretic mobility is the drift velocity of a charged particle per unit electric field strength.
Quadratic Formula
The quadratic formula determines roots of , with . The discriminant tells whether there are two, one or no real roots.
pq Formula
The pq formula solves any quadratic equation in normalized form , that is with leading coefficient 1.
Binomial Formulas
The three binomial formulas square sums and differences and, read backwards, factor quadratic terms.
Laws of Logarithms
The laws of logarithms turn products into sums, quotients into differences and powers into multiples.
Eigenvalues and Eigenvectors
An eigenvector keeps its direction under a linear map; its eigenvalue gives the scale factor. The eigenvalues are found from the characteristic equation.
Linear Least Squares
Linear least squares chooses model parameters that minimize the sum of squared deviations from an overdetermined set of measurements.
Linear systems and matrix rank
A linear system is solvable exactly when the coefficient and augmented matrices have the same rank.
Determinant and Invertibility
The determinant of a square matrix measures oriented volume scaling and determines whether the associated linear map is invertible. For matrix it is ad−bc.
Inverse Matrix and Linear Systems
If A is square and invertible, has the unique solution obtained by multiplying by . The explicit formula here applies to matrices.
LU Factorization
LU factorization decomposes a matrix into lower and upper triangular matrices; P records any row permutations. This enables efficient solution of linear systems.
QR Factorization
QR factorization writes a matrix with linearly independent columns as the product of a matrix Q with orthonormal columns and an upper triangular matrix R.
Singular Value Decomposition
Every real matrix can be represented using orthogonal factors U and V and a diagonal matrix Σ of nonnegative singular values.
Matrix Condition Number
The condition number estimates how strongly relative input errors in a linear problem can affect the relative solution error.
Gaussian Elimination for Linear Systems
Gaussian elimination solves a linear system by using elementary row operations to transform its augmented matrix to row-echelon form.
Percentage Calculations: Part, Base, and Rate
Percentage calculations relate a part W, a reference amount G, and a rate p; enter p as a number followed by the percent sign.
Simple Interest
Simple interest is proportional to the initial principal, annual rate, and duration, without interest being added to the principal.
Polar Form of Complex Numbers and de Moivre’s Formula
Polar form describes a complex number by its modulus r and argument φ. de Moivre’s formula makes powers and roots accessible through separate operations on modulus and angle.
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 .
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.
Geometric Series
The geometric series sums terms with a constant factor q; for even the infinite series has a finite value.
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.
Fourier Series of a Periodic Function
A Fourier series represents a periodic function through sine and cosine components; its coefficients are projections onto orthogonal basis functions.
Direct Proportion
In a direct proportion, the quotient is constant, so y changes by the same factor as x.
Inverse Proportion
In an inverse proportion, the product xy stays constant; if x grows by a factor, y falls by the same factor.
Percentage Growth and Compound Interest
With the same percentage change each period, the amount is multiplied by the same growth factor at every step.
Pythagorean Theorem
The Pythagorean theorem holds for right triangles: the square of the hypotenuse equals the sum of the squares of the two legs.
Law of Cosines
The law of cosines generalizes the Pythagorean theorem to arbitrary triangles without a right angle.
Law of Sines
In a triangle the sides are proportional to the sines of their opposite angles.
Circle Area and Circumference
Area and circumference of a circle depend only on the radius r, linked by the circle number π.
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.
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.
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.
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.
Trigonometry in the Right Triangle
Sine, cosine and tangent link an acute angle of a right triangle with the ratios of its sides.
Triangle Area with Sine
The trigonometric area formula: two sides and the angle enclosed by them determine the area of the triangle.
Trigonometric identities
The fundamental identity and angle-addition formulas connect sine and cosine at different angles and enable equivalent transformations.
Radian measure, arc length, and sector area
Radian measure defines an angle as arc length divided by radius and gives direct formulas for arc length and sector area.
Areas of Parallelograms, Trapezoids and Kites
This area family uses base times perpendicular height for a parallelogram, the mean of the parallel sides times height for a trapezoid, and half the diagonal product for a kite.
Volume and Surface Area of a Right Prism
For a right prism, volume is base area times height; total surface area adds both bases and the lateral area.
Triangle Similarity and Scale Factors
Similar triangles have equal corresponding angles and proportional sides; their areas scale with the square of the side factor.
Ellipses and Hyperbolas: Axes and Foci
The standard equations distinguish an ellipse from a hyperbola and relate their semiaxes to the distance of the foci.
Euler's Formula
Euler's formula links the exponential function with the trigonometric functions in the complex plane.
Product Rule of Differentiation
The product rule allows the differentiation of products of two functions.
Chain Rule of Differentiation
The chain rule differentiates composite functions f(g(x)): outer derivative times inner derivative.
Taylor Series
The Taylor series expresses a function as a power series around an expansion point a.
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.
Derivative of the Exponential Function
The exponential function is its own derivative; for the chain rule adds the factor k.
Derivative of the Natural Logarithm
The derivative of the natural logarithm ln x is the hyperbola , valid for all .
Derivatives of Sine and Cosine
Sine and cosine differentiate into each other cyclically: sin becomes cos, cos becomes −sin, each in radians.
Quotient Rule of Differentiation
The quotient rule differentiates fractions of two functions, for example rational functions.
Antiderivatives and Basic Integrals
The power rule of integration yields antiderivatives: raise the exponent by 1 and divide by the new exponent.
Fundamental Theorem of Calculus
The fundamental theorem connects differentiation and integration: a definite integral is evaluated via an antiderivative at the limits.
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.
Integration by Parts
Integration by parts integrates products by reversing the product rule of differentiation.
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.
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).
Partial derivatives and gradient
Partial derivatives measure the change with respect to one variable while holding the others fixed. The gradient collects these rates into a vector.
Double and multiple integrals
A multiple integral sums a function over a region to calculate quantities such as volume, mass or total charge.
Lagrange multipliers
The Lagrange condition finds candidates for extrema of a function subject to an equality constraint.
First-order linear differential equation
A first-order linear differential equation relates a function to its first rate of change and models many relaxation and growth processes.
Jacobian determinant for change of variables
The Jacobian determinant measures the local area or volume scaling of a differentiable coordinate transformation.
Hessian matrix and second-derivative test
The Hessian collects second partial derivatives and can classify critical points of a function of two variables under suitable conditions.
Newton’s method for finding roots
Newton’s method approximates a root by finding where the tangent at the current estimate crosses the x-axis.
Composite trapezoidal rule
The composite trapezoidal rule approximates a definite integral by joining adjacent function values with straight line segments.
Composite Simpson's Rule
Composite Simpson's rule approximates an integral on a uniform grid by piecewise quadratic parabolas; the number n of subintervals must be even.
Multivariable Chain Rule
If z=f(x,y) depends on two quantities that themselves depend on t, the chain rule adds the contributions from both paths of change.
Directional Derivative
The directional derivative measures the instantaneous rate of change of a scalar function along a specified unit direction u.
Divergence of a Vector Field
The divergence of a vector field is a scalar local measure of the field's source or sink behavior.
Curl of a Vector Field
The curl maps a differentiable vector field to a vector field describing local rotation direction and strength.
Vector Line Integral
The vector line integral accumulates the tangential component of a vector field along an oriented curve.
Green's Theorem
Green's theorem relates a positively oriented line integral around the boundary of a planar region to an area integral of curl over its interior.
Stokes' Theorem
Stokes' theorem equates the circulation of a vector field around the boundary of an oriented surface with the flux of its curl through the surface.
Second-Order Multivariable Taylor Polynomial
The second-order Taylor polynomial approximates a smooth multivariable function near an expansion point using its value, gradient, and Hessian.
Laplace Transform
The Laplace transform maps a time function to a function of complex frequency s and converts suitable differential equations into algebraic equations.
Separable Differential Equation
For a separable differential equation, all y-terms and x-terms can be moved to separate sides and integrated.
Second-Order Linear ODE with Constant Coefficients
A homogeneous second-order linear differential equation with constant coefficients is solved through the roots of its characteristic polynomial.
Forward Euler Method
The forward Euler method approximates a solution of by extending the tangent slope as constant over a step h.
Gradient Descent
Gradient descent updates a parameter vector opposite the local gradient to reduce a differentiable objective function.
Newton's Method for Optimization
Newton's optimization method corrects parameters using the inverse Hessian and gradient to find stationary points of a sufficiently smooth function.
Classical Fourth-Order Runge-Kutta Method (RK4)
Classical RK4 approximates one step of an initial-value problem using four weighted slope evaluations.
Limits and Continuity
A limit is the value f(x) approaches near a. Continuity additionally requires it to equal the defined value f(a).
Convergence Tests for Infinite Series
The ratio and root tests decide absolute convergence for many series, while comparison and alternating-series tests handle other cases.
Improper Integrals
Improper integrals define integrals over infinite intervals or unbounded integrands as limits of ordinary finite integrals.
Fourier Transform
The Fourier transform decomposes a nonperiodic signal into its complex frequency components.
Coupled Linear Differential Equation Systems
A linear system of differential equations describes coupled state variables with a matrix and an external input.
Lagrange and Newton Polynomial Interpolation
Lagrange and Newton forms represent the same unique polynomial of degree at most n through n+1 data points with distinct nodes.
Integration by substitution
The substitution rule replaces a composite inner function with a new variable and its matching differential.
Mean Value Theorem
The mean value theorem guarantees a point between two endpoints where the instantaneous rate equals the average rate over the interval.
Implicit Differentiation
Implicit differentiation finds the local slope of a curve defined by F(x,y)=0 without first solving explicitly for y.
Derivative of an Inverse Function
The derivative of an inverse function is the reciprocal of the original derivative at the corresponding point, provided that derivative is nonzero.
Partial Fraction Decomposition of Rational Functions
Partial fraction decomposition writes a proper rational function with distinct linear denominator factors as a sum of simple fractions.
Arc Length of a Smooth Curve
Arc length adds the infinitesimal path elements along the graph .
Surface Area of a Solid of Revolution
Rotating a graph about the x-axis produces a thin surface strip whose area is circumference times infinitesimal arc length.
Gauss–Legendre Quadrature
An n-point Gauss–Legendre rule chooses nodes and weights to integrate polynomials through degree 2n−1 exactly.
Dot Product
The dot product of two vectors yields a number and measures angles and orthogonality.
Cross Product (Vector Product)
The cross product of two vectors in space yields a vector perpendicular to both.
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.
Plane Equation in Normal Form
A plane is fixed by a point and a normal vector: exactly the position vectors x⃗ with lie in it.
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.
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.
Gram–Schmidt Orthogonalization
The Gram–Schmidt procedure turns linearly independent vectors into an orthonormal basis of the same subspace by projecting onto and subtracting previous directions.
Cauchy–Schwarz Inequality
The Cauchy–Schwarz inequality bounds an inner product by the product of the two norms and underlies the angle between vectors.
Bayes' Theorem
Bayes' theorem describes the inversion of conditional probabilities.
Normal Distribution (Gaussian Bell Curve)
The normal distribution is the most important probability distribution, describing many natural phenomena.
Binomial Distribution
The binomial distribution gives the probability of exactly k successes in n independent trials with success probability p.
Expected Value
The expected value is the long-run average of a random variable: values times probabilities, summed up.
Conditional Probability
The conditional probability P(A|B) measures the probability of A given that B has already occurred.
Standard Deviation
The standard deviation σ measures the typical spread of a random variable around its expected value μ.
Linear Regression (Line of Best Fit)
Linear regression places the line of best fit through a scatter plot so that the sum of the squared deviations becomes minimal.
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.
Poisson Distribution
The Poisson distribution models the number of rare, independent events in a fixed interval with known mean rate λ.
Sigma Rules (Sigma Intervals)
The sigma rules state with which probability a normally distributed quantity lies in the intervals , and .
Confidence interval and standard error of the mean
The t confidence interval quantifies sampling uncertainty in a mean when the population standard deviation is unknown.
Sample Variance
The corrected sample variance measures spread around the sample mean and estimates the population variance.
Sample Covariance
Sample covariance describes whether two variables tend to lie together above or below their means.
Pearson Correlation Coefficient
The Pearson correlation coefficient standardizes sample covariance and measures the direction and strength of a linear association.
One-Sample t Statistic
The t statistic standardizes the distance between a sample mean and the null mean when the population standard deviation is unknown.
Chi-Square Goodness-of-Fit Test
The chi-square goodness-of-fit test compares observed frequencies with frequencies expected under an assumed distribution.
Standard Error of a Sample Proportion
The estimated standard error of a sample proportion describes its typical sampling variation for independent Bernoulli observations.
Propagation of Measurement Uncertainty
The standard uncertainty of a calculated result follows approximately from input uncertainties, sensitivities and covariances.
t, Chi-Square and F Distributions
The t distribution describes standardized means with unknown variance; chi-square comes from sums of squares and F from ratios of independent variance estimates.
Central Limit Theorem
The central limit theorem explains why standardized sample means approach a normal distribution as sample size grows.
Maximum Likelihood Estimation
Maximum likelihood selects the parameter under which the observed data have the greatest likelihood in the assumed model.
Discrete Markov Chain and Transition Matrix
A discrete Markov chain evolves states through a transition matrix; conditioned on the current state, the next state does not depend on earlier history.
Geometric, Hypergeometric, and Gamma Distributions
This reference distinguishes first-success trials, sampling without replacement, and a continuous positive Gamma variable by their probability models.
Continuous Uniform Distribution
A continuous uniform distribution assigns equal probability to subintervals of equal length within [a,b].
Multivariate Normal Distribution with Covariance Matrix
The multivariate normal distribution describes a d-dimensional random vector using a mean vector and covariance matrix.
Bienaymé–Chebyshev Inequality
The inequality guarantees a minimum fraction of values within several standard deviations for any distribution with finite variance.
Michaelis-Menten Kinetics (Enzyme Kinetics)
The Michaelis–Menten equation describes how the initial rate of a simple enzyme reaction rises with free substrate concentration and approaches a limiting rate.
qPCR ΔΔCt method
The ΔΔCt method estimates relative qPCR expression change against a reference gene and control sample.
Competitive enzyme inhibition
A competitive inhibitor raises apparent Michaelis constant while Vmax remains unchanged in the ideal model.
Langmuir binding isotherm
The Langmuir isotherm describes equilibrium occupancy of independent binding sites by a ligand.
Pfaffl model for efficiency-corrected qPCR
The Pfaffl model compares relative transcript abundance while correcting for different amplification factors in target and reference assays.
GC content of a sequence
GC content is the percentage of counted canonical bases in a nucleic acid sequence that are guanine or cytosine.
GC skew of a sequence
GC skew measures normalized cytosine-versus-guanine asymmetry in a defined sequence window.
N50 contig length in a genome assembly
N50 is the length at which at least half of all assembled bases lie in contigs of that length or longer.
Sequence Alignment: Score and Identity
Global Needleman–Wunsch alignment maximizes a sequence score; identity counts matching symbols in the aligned columns.
Absolute qPCR Quantification with a Standard Curve
A qPCR standard curve estimates starting copy number from the threshold cycle and calculates amplification efficiency from its slope.
BLAST E-value in Sequence Searches
The E-value estimates the expected number of random alignments with at least a comparable score in a database search.
Monod Growth Model
The Monod model links a microorganism’s specific growth rate to the concentration of one limiting substrate.
Fick's First Law of Diffusion
For steady diffusion, Fick’s first law relates flux per area to the concentration gradient: particles move net from high toward low concentration.
Goldman–Hodgkin–Katz Equation
The Goldman–Hodgkin–Katz equation estimates membrane potential from ion concentrations and relative permeabilities.
Hill equation for oxygen saturation
The Hill equation approximates hemoglobin oxygen saturation as a function of oxygen partial pressure.
Cardiac output
Cardiac output is the volume of blood ejected by one ventricle per minute.
Renal clearance
Clearance is the hypothetical plasma volume cleared of a substance per unit time.
Arterial oxygen content
This approximation estimates arterial blood oxygen content from hemoglobin-bound and physically dissolved oxygen.
Fick principle for cardiac output
The Fick principle calculates blood flow from oxygen consumption and the arteriovenous oxygen-content difference.
Ventricular ejection fraction
Ejection fraction is the percentage of end-diastolic ventricular volume ejected during a heartbeat.
Alveolar ventilation
Alveolar ventilation estimates the fresh air volume reaching gas-exchanging alveoli per minute.
Maximum quantum yield of photosystem II
estimates maximum photochemical quantum yield of photosystem II after dark adaptation.
Transpiration flux from conductance and vapor-pressure gradient
A gas-exchange approximation makes transpiration flux proportional to total water-vapor conductance and the leaf-to-air vapor-pressure difference.
Plant Water Potential
Water potential combines osmotic, pressure, gravitational and matric contributions to the free energy of water in plants.
Alveolar Gas Equation
The alveolar gas equation estimates alveolar oxygen partial pressure from inspired oxygen fraction, ambient pressure, carbon dioxide and respiratory quotient.
Respiratory Quotient (RQ)
The respiratory quotient compares cellular CO2 production with O2 consumption during substrate oxidation.
Bohr Equation for Physiological Dead-Space Fraction
The Bohr equation estimates the fraction of tidal volume that does not participate in alveolar gas exchange because of dead-space ventilation.
Photosynthesis Light Response and Saturation
A hyperbolic light-response model describes rising net photosynthesis at low irradiance and saturation at higher photosynthetically active radiation.
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.
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).
Recombination Frequency and Genetic Mapping
Recombination frequency estimates how often two loci are separated by crossing over and approximates map distance for short intervals.
Effective Population Size with Unequal Sex Ratio
This approximation quantifies how an unequal number of breeding males and females reduces genetically effective population size.
Selection Coefficient and Relative Fitness
The selection coefficient s expresses a genotype’s relative fitness disadvantage; h specifies how much of it appears in a heterozygote.
Linkage Disequilibrium and r-squared
D measures departure of two alleles from independent association; r-squared scales this association as a squared allele correlation.
Mutation–Selection Balance for a Recessive Allele
For a rare, fully recessive deleterious allele, a simple model balances mutation input against loss through selection.
Narrow-Sense Heritability
Narrow-sense heritability is the fraction of phenotypic variance attributable to additive genetic variance in a specified population and environment.
Inbreeding coefficient and heterozygosity
FIS summarizes relative heterozygote deficiency within a defined subpopulation against its reference expectation.
Nucleotide diversity π
Nucleotide diversity π is the mean number of pairwise sequence differences per aligned site in a sample.
FST heterozygosity-based population differentiation
An intuitive heterozygosity form of FST relates the difference between total and mean within-subpopulation heterozygosity to total heterozygosity.
Wright–Fisher allele-frequency variance
In the diploid Wright–Fisher model, the variance quantifies random allele-frequency change from finite gamete sampling per generation.
Breeder’s equation for selection response
The breeder’s equation estimates the change in a quantitative trait mean after directional selection.
Probability in Monohybrid Crosses
In a monohybrid cross between two heterozygous parents, independent combinations of equally likely gametes produce a 1:2:1 genotype ratio.
Gene Flow and Migration: Allele-Frequency Change
A single idealized migration step shifts a population allele frequency toward that of the incoming gene pool.
Mean Sequencing Coverage
The theoretical mean sequencing depth is the number of sequenced bases divided by the length of the genome or target region.
Jukes–Cantor Distance Correction
The Jukes–Cantor correction estimates substitutions per site from the observed fraction of differing bases and accounts for multiple changes at a site.
Logistic Growth
The differential equation of logistic growth describes population growth with a capacity limit and is a realistic model for biological populations.
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.
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.
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.
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.
Pielou's Evenness Index
Pielou's J normalizes Shannon entropy by the maximum possible entropy for the observed number of species, describing relative evenness.
Lotka–Volterra Predator–Prey Model
The classical Lotka–Volterra equations couple changes in prey and predator populations under simple growth and encounter assumptions.
Lotka–Volterra Interspecific Competition
The competition model extends logistic growth by including the density-dependent effect of one species on the growth of another.
Allometric scaling
A power law describes how a biological trait Y scales with body mass M.
Q₁₀ temperature coefficient
Q10 summarizes the factor by which a biological rate changes for a 10 °C temperature difference.
Relative growth rate of plants
Relative growth rate describes logarithmic biomass increase per unit time.
Discrete population growth
The geometric model describes a population with a constant multiplicative growth factor per generation.
Primary Production and Ecosystem Energy Flow
Net primary production is gross primary production minus autotrophic respiration; trophic efficiency compares production at adjacent levels.
Leslie Matrix for Age-Structured Populations
The Leslie matrix projects population sizes across age classes using age-specific fertility and survival rates.
Life-Table Net Reproductive Rate and Generation Time
Life-table quantities combine age-specific survival and reproduction into mean offspring production and generation time.
Chao1 Species-Richness Estimator
Chao1 estimates community richness from observed species and the numbers of singletons and doubletons.
Holling Type II Functional Response
The Holling Type II function describes a predator’s saturating feeding rate per predator as prey density, search rate and handling time vary.
Relative Risk (Risk Ratio)
Relative risk compares the event proportion in an exposed cohort with the proportion in a non-exposed cohort.
Diagnostic Test Sensitivity and Specificity
Sensitivity and specificity measure positive and negative test results within groups defined by a reference standard.
Wilson Score Confidence Interval for a Proportion
The Wilson score interval estimates an unknown binomial population proportion, such as prevalence, from x observed events in n independent observations.
Cohen's d Effect Size
Cohen’s d standardizes the difference between two independent group means using their pooled variability.
Odds Ratio
The odds ratio compares event odds between two groups and is especially important in case-control studies and logistic regression.
ROC area under the curve
ROC AUC summarizes score discrimination across all possible thresholds.
Kaplan–Meier estimator
The Kaplan–Meier estimator estimates event-free probability in the presence of right-censored observations.
Hazard function
The hazard is the instantaneous event rate among individuals who remain event-free up to that time.
Logistic regression
Logistic regression models the log odds of a binary outcome as a linear function of predictors.
Benjamini–Hochberg procedure
The Benjamini–Hochberg procedure controls the expected false discovery rate within a family of tests.
Positive and negative predictive values
Predictive values state how often positive or negative test results agree with reference status.
Incidence rate per person-time
The incidence rate relates new events to accumulated time at risk.
Cumulative incidence (incidence proportion)
Cumulative incidence estimates the share of initially at-risk people who develop a new outcome during a specified period.
Population attributable fraction (PAF)
PAF estimates the share of population events attributable to a binary exposure under strong causal assumptions.
Standardized mortality ratio (SMR)
The indirectly standardized mortality ratio compares observed deaths with the count expected from reference rates and the cohort’s stratum distribution.
Basic reproduction number in the simple SIR model
In the classical frequency-dependent SIR model, R0 is the expected number of secondary infections from one infectious person in a fully susceptible population.
Intraclass correlation coefficient (ICC)
In a simple random-intercept model, ICC is the share of total outcome variance that lies between clusters or groups.
Design effect for cluster samples
The design effect approximates the variance inflation of an equal-size cluster sample compared with a simple random sample of the same size.
Sample size per group for two means
A normal approximation estimates sample size per arm for two independent means with equal group sizes and variances.
Coefficient of variation
The coefficient of variation divides sample standard deviation by sample mean to express relative spread.
Cohen's Kappa for Categorical Agreement
Cohen’s kappa measures agreement between two raters on categorical data and corrects for agreement expected by chance.
Diagnostic Likelihood Ratios
Likelihood ratios compare the probability of a test result among people with a condition with the probability of the same result among those without it.
Confidence Intervals for Risk Ratios and Odds Ratios
Log-Wald confidence intervals describe sampling uncertainty in a risk ratio or odds ratio from table.
Sample Size for Estimating a Proportion
This approximation plans a simple random sample to estimate one population proportion at a specified margin of error and confidence level.
Absolute Risk Difference
The absolute risk difference quantifies the difference in event probabilities between two groups on an additive scale.
Fisher's Exact Test
For contingency table, Fisher’s exact test evaluates conditional independence using the hypergeometric distribution with fixed margins.
Stellar parallax and distance
Trigonometric parallax gives a star’s distance from its apparent annual position shift relative to very distant background stars.
Hubble–Lemaître law (low-redshift approximation)
At low redshift, the recession velocity of a sufficiently distant galaxy is approximately proportional to its distance.
Astronomical distance modulus
The distance modulus relates apparent magnitude mV, absolute magnitude MV and distance, with an explicit extinction correction AV.
Stellar diameter from angular size and distance
At small angular size, a star’s linear diameter is approximately its angular diameter multiplied by distance.
Cosmological redshift and scale factor
Cosmological redshift measures how much light wavelength stretched while the universe expanded.
Cepheids: Period–Luminosity Relation
For classical Cepheids, mean absolute magnitude is empirically linked to pulsation period; together with apparent magnitude this estimates distance.
Wien displacement law
For a blackbody spectrum expressed per wavelength, the peak wavelength is inversely proportional to absolute temperature.
Luminosity, flux and distance
For an isotropic point source in free propagation, measured flux is luminosity spread over a sphere at the observer’s distance.
Stellar radius from luminosity and effective temperature
For an approximately blackbody star, the Stefan–Boltzmann relation connects luminosity, radius and effective temperature.
Spectral redshift
Dimensionless redshift compares an observed spectral-line wavelength with its rest wavelength measured in the laboratory.
Magnitude difference from a flux ratio
Within the same photometric band, the logarithmic magnitude scale converts a flux ratio into a brightness difference.
Diffraction-limited angular resolution of a telescope
For an ideal circular aperture, the Rayleigh criterion gives the smallest angular separation at which two point sources are just resolved.
Tangential velocity from proper motion
Proper motion on the sky and geometric distance determine a star’s velocity component transverse to the line of sight.
Kelvin–Helmholtz timescale of a star
The Kelvin–Helmholtz timescale estimates how long a star could radiate at its present luminosity using gravitational binding energy.
Jeans mass of a homogeneous gas cloud
The Jeans mass estimates the mass above which self-gravity can overcome pressure support in an idealized gas.
Eddington luminosity for constant opacity
The Eddington luminosity is the ideal limit where radiation pressure on matter balances spherical gravity.
Dynamical timescale of a star
The dynamical timescale estimates how quickly a star can respond to gravitational perturbations or move across its radius.
Main-Sequence Mass–Luminosity Relation
For stars near solar mass, a rough power law describes how main-sequence luminosity increases with mass.
Main-Sequence Lifetime Approximation
Main-sequence lifetime scales roughly as available fuel divided by energy output, hence the mass-to-luminosity ratio.
Exoplanet Transit Depth
For a dark planet transiting a uniformly bright stellar disk, the relative brightness drop is approximately the area ratio.
Spectral Resolving Power
Spectral resolving power is the wavelength divided by the smallest wavelength separation the instrument can distinguish.
Hydrostatic Equilibrium in a Star
In a hydrostatic stellar model, the outward pressure gradient balances inward gravity; enclosed mass also grows with density.
Schwarzschild radius of a nonrotating black hole
The Schwarzschild radius is the event horizon of an idealized, uncharged, nonrotating, spherically symmetric mass.
Kepler’s equal-area law and areal velocity
In a central gravitational field, the radius vector sweeps out equal areas in equal times; areal velocity is constant.
Roche limit for a fluid satellite
The Roche limit estimates how close a self-gravitating, nearly strengthless satellite can approach a massive primary before tides disrupt it.
Surface gravity of a star
In the spherical Newtonian model, gravitational acceleration at a stellar surface depends on mass and radius.
Total Mass of a Binary-Star System
The generalized Kepler relation gives the sum of two bound stellar masses from orbital period and relative semimajor axis.
Vis-Viva Equation for Elliptical Orbits
The vis-viva equation gives the speed magnitude at a point on a bound Kepler ellipse from the gravitational parameter, current radius and semimajor axis.
Chandrasekhar Limit for White Dwarfs
The Chandrasekhar limit is the approximate maximum mass of a cold, non-rotating white dwarf supported by electron degeneracy pressure.
Big-O notation (asymptotic upper bound)
Big O classifies the asymptotic upper growth of a runtime or memory function up to a constant factor.
Two’s-complement signed integer
Two’s complement interprets a fixed-width N-bit word as a signed integer by assigning a negative weight to its most significant bit.
Divide-and-conquer runtime recurrence
A recurrence models a recursive algorithm through the number and size of its subproblems plus non-recursive work per call.
IEEE 754 floating-point representation and rounding
IEEE 754 represents real approximations with a sign, exponent and significand; rounding limits results to the selected format.
Boolean algebra identities and De Morgan laws
Boolean identities simplify logical expressions over truth values while preserving their truth tables.
Euclidean algorithm and modular inverse
The Euclidean algorithm computes the greatest common divisor; back-substitution finds Bézout coefficients and a modular inverse.
A* search
A* ranks open nodes by known path cost g plus estimated remaining cost h to find a least-cost goal path.
Master theorem for recurrences
The Master theorem classifies balanced divide-and-conquer recurrences by comparing f(n) with .
Amortized analysis of dynamic arrays
Amortized analysis spreads occasional expensive operations across a sequence and gives a worst-case sequence bound without probability.
Union–Find (disjoint-set union)
Union–Find maintains a partition into disjoint sets and supports finding a representative and merging sets.
Floyd–Warshall algorithm
Floyd–Warshall computes all-pairs shortest paths by successively allowing each vertex as an intermediate node.
AVL tree balance condition and height
An AVL tree is a binary search tree whose child subtree heights differ by at most one; rotations restore balance after updates.
Bellman–Ford algorithm for shortest paths
Bellman–Ford computes shortest distances from a source through repeated edge relaxation and supports negative edge weights.
Dijkstra’s algorithm for weighted shortest paths
Dijkstra computes shortest paths from one start vertex in weighted graphs when every edge weight is nonnegative.
Minimum spanning tree
A minimum spanning tree connects every vertex of a connected, undirected, weighted graph with the minimum total edge weight.
Binary search: comparison bound and running time
Binary search halves the remaining search interval in a sorted, directly indexable sequence and uses logarithmically many comparisons.
Combinatorial counting rules and inclusion–exclusion
Counting rules determine the number of possible selections; inclusion–exclusion corrects repeated counts among overlapping sets.
Graph search: BFS, DFS and shortest paths
Breadth-first search (BFS) explores an unweighted graph by layers and finds shortest paths by edge count; depth-first search (DFS) follows a path deeply before backtracking.
Diffie–Hellman key exchange (teaching model)
Diffie–Hellman lets two parties derive the same modular group value from public messages without sending their private exponents.
Bloom filter false-positive probability
This approximation estimates the false-positive probability of a classical Bloom filter.
Hash collision probability (birthday bound)
The birthday approximation estimates the probability of a collision among n independent hashes into M equally likely values.
Damped PageRank
PageRank computes a stationary importance distribution from links and random teleportation on a directed graph.
Levenshtein edit distance
Levenshtein distance is the minimum number of insertions, deletions and substitutions needed to transform one string into another.
RSA encryption: modular exponentiation
RSA maps message representatives by modular exponentiation with public exponent e and decrypts using private exponent d.
B+ tree: height and index access
A B+ tree stores records in linked leaves and uses high branching factor to limit block-oriented index access to logarithmically many page reads.
Hash table: load factor and expected search time
Under simple uniform hashing, load factor summarizes average occupancy and determines expected search work for separate chaining.
Shannon entropy
Shannon entropy is the expected information of a discrete random experiment and measures uncertainty in its probability distribution in bits.
Hamming distance between bit strings
Hamming distance counts positions at which two equal-length strings differ.
Shannon-Hartley channel capacity
The Shannon-Hartley limit gives the maximum reliably transmitted data rate of a band-limited AWGN channel.
Mutual information of discrete variables
Mutual information measures the average reduction in uncertainty about X when Y is observed.
Kullback-Leibler divergence (relative entropy)
KL divergence measures expected information loss when q is assumed instead of the true discrete distribution p.
CRC checksum as a polynomial remainder over GF(2)
A cyclic redundancy check appends the remainder of binary polynomial division to a message so that the codeword is divisible by the generator polynomial.
F₁ score
The F1 score is the harmonic mean of precision and recall for a specified positive class. It combines correctness and completeness with equal weight.
Cross-entropy (classification loss)
Cross-entropy scores how well a predicted probability distribution q matches a target distribution p over classes.
Gradient descent parameter update
Gradient descent updates parameters opposite the gradient of an objective to decrease it locally.
Balanced accuracy
Balanced accuracy averages sensitivity and specificity in binary classification, giving each class equal weight.
Precision (positive predictive value)
Precision is the fraction of positive predictions that are correct among all cases classified as positive.
Recall (sensitivity)
Recall is the fraction of actual positive cases detected by a classifier.
Matthews correlation coefficient (MCC)
MCC combines all four cells of a binary confusion matrix into a correlation-like metric.
Logistic sigmoid function
The logistic sigmoid maps a real-valued score monotonically to a value between zero and one.
Softmax function
Softmax transforms K class scores into nonnegative values that sum to one.
Fβ score (weighted precision-recall measure)
The Fβ score combines precision and recall and weights recall relative to precision through β.
Logit transformation of probability and odds
The logit maps a probability in (0,1) to the real-valued log odds.
Specificity (true negative rate)
Specificity is the fraction of actual negative cases correctly identified as negative.
Jaccard index (intersection over union)
The Jaccard index measures overlap between the predicted and actual positive sets.
ROC-AUC (area under the ROC curve)
ROC-AUC summarizes binary score discrimination across decision thresholds as an area.
Cosine similarity between vectors
Cosine similarity compares the direction of two nonzero vectors independently of their positive scale.
TF-IDF term weighting
TF-IDF weights a term by its frequency in a document and its rarity across the corpus.
Okapi BM25 ranking
BM25 scores documents for a query using term frequency, term rarity and document-length normalization.
Convolution output dimension
The convolution-size formula determines a discrete kernel’s spatial output dimension with padding, stride and dilation.
Gini impurity of a classification node
Gini impurity is the probability of misclassification when a label is drawn at random from the node’s class distribution.
Binary log loss
Binary log loss scores probabilistic predictions for labels 0 and 1 and strongly penalizes confident errors.
Information gain of a decision-tree split
Information gain scores a candidate split at a tree node by the reduction in label entropy after partitioning the training examples.
Bandwidth-delay product (BDP)
The bandwidth-delay product estimates how much data can be in flight simultaneously on a path operating at its full rate.
K-means objective (inertia)
K-means minimizes the sum of squared Euclidean distances from observations to their assigned cluster centroids.
Principal component analysis (PCA)
PCA projects centered observations onto covariance eigenvectors ordered by explained variance.
Naive Bayes classification
Bayes’ rule combines class priors and feature likelihoods; the naive model assumes conditional independence of features within each class.
Ridge regression (L2 regularization)
Ridge regression minimizes squared residual error plus an L2 penalty, shrinking coefficients to stabilize correlated or noisy predictors.
Linear support vector machine and hinge loss
For labels −1 and +1, hinge loss penalizes scores that lie inside the unit margin or on the wrong side of the separating hyperplane.
Logistic regression as a Bernoulli likelihood
Logistic regression maps a linear score to a Bernoulli probability and estimates coefficients by maximizing likelihood or minimizing negative log-likelihood.
Amdahl’s law (parallel speedup)
Amdahl’s law bounds total speedup by the fraction of program work that remains serial.
CPU time from instructions, CPI and clock rate
CPU time follows from executed instructions, average cycles per instruction and clock frequency.
Average memory access time (AMAT)
AMAT estimates mean access time from cache hit time and expected additional miss time.
Little’s law for queues
Little’s law relates mean number in a stationary system, throughput and mean time in the system.
M/M/1 queue: utilization and sojourn time
The model describes one server with Poisson arrivals and exponentially distributed service times.
Young's modulus
Young's modulus describes the ratio of tensile stress to tensile strain for a material in the linear-elastic range. Tensile stress is force per cross-sectional area; tensile strain is relative length change.
Linear and angular impulse–momentum balance
External force impulse changes a system’s total linear momentum, while external torque impulse changes its angular momentum about a specified point.
Euler–Bernoulli beam moment–curvature relation
Linear Euler–Bernoulli beam theory relates bending moment to beam curvature through flexural rigidity EI.
Static equilibrium conditions
A rigid body is in static equilibrium when its resultant force and resultant moment both vanish.
Average axial normal stress
Average normal stress is internal axial force divided by the loaded cross-sectional area; with uniform distribution it also equals local stress.
Poisson’s ratio
In a uniaxial test, Poisson’s ratio is the negative ratio of transverse strain to axial strain.
Linear thermal expansion
For a small temperature change, a homogeneous bar’s free length change is approximately proportional to its initial length and temperature change.
Bending stress in an Euler–Bernoulli beam
In elastic beam bending, longitudinal normal stress varies linearly with distance from the neutral axis.
Beam shear stress
The Jourawski relation estimates transverse shear stress from beam shear force and section geometry.
Torsional shear stress in a circular shaft
In a linearly elastic circular shaft under torsion, shear stress increases in proportion to radius.
Angle of twist of a circular shaft
A prismatic circular shaft with constant torque twists according to torque, length and torsional rigidity.
Parallel-axis theorem for area moments
The theorem shifts a second moment of area from a centroidal axis to a parallel axis.
Euler critical buckling load
Euler’s formula gives the ideal critical compression load at which a slender elastic column can buckle laterally.
Hoop stress in a thin-walled cylinder
For a thin-walled cylindrical pressure vessel, hoop stress is approximately pressure times radius divided by wall thickness.
Von Mises equivalent stress in plane stress
The von Mises stress combines a plane stress state into a scalar measure based on distortion energy.
Distributed load, shear-force and bending-moment relations
Along a beam, derivatives of internal shear and bending moment are related to distributed load and shear force.
Shear modulus and shear law
In the linear-elastic range, shear modulus is the ratio of shear stress to engineering shear strain.
Principal stresses in plane stress
Principal stresses are the extreme normal stresses obtained by rotating a plane stress element.
Elastic strain-energy density under uniaxial stress
For linear uniaxial loading, elastic energy per unit volume is the area under the stress–strain curve.
Thermal stress in a fully restrained bar
A fully restrained linear-elastic bar develops compressive stress when uniformly heated.
Mode-I stress intensity factor
The mode-I stress intensity factor describes crack-tip loading in linear-elastic fracture mechanics.
Miner rule for cumulative fatigue damage
The linear Miner rule sums consumed cycle fractions from multiple constant-amplitude fatigue blocks.
Strength factor of safety
A factor of safety relates a selected resistance or strength limit to the corresponding demand.
Damping ratio of a single-degree-of-freedom oscillator
The damping ratio normalizes viscous damping by the critical damping of a linear mass-spring system.
Undamped natural frequency of a single-degree-of-freedom system
The undamped natural frequency of a linear single-mass oscillator depends on stiffness and mass.
Logarithmic decrement of a damped oscillation
Logarithmic decrement measures amplitude loss between successive same-sign peaks of an underdamped oscillation.
Voigt rule of mixtures for axial composite modulus
For an ideal fibre composite with equal axial strain, longitudinal modulus is the volume-weighted phase modulus.
Cantilever tip deflection under an end force
A slender cantilever with an end force deflects in proportion to force and the cube of its length.
Midspan deflection of a simply supported beam with a point load
A simply supported slender beam with a central point load reaches maximum deflection at midspan.
Shear flow in a thin-walled beam section
Shear flow in a thin-walled beam is determined by transverse shear force and the first area moment at the cut.
Cantilever tip deflection under a uniform load
A cantilever under a uniformly distributed load reaches maximum deflection at the free end.
Deflection of a simply supported beam under uniform load
A simply supported beam with uniform loading reaches maximum elastic deflection at midspan.
Reuss estimate for transverse composite modulus
The ideal series model estimates transverse composite modulus under equal stress in fibre and matrix.
Differential relations between beam load, shear and bending moment
Local equilibrium of a beam element links distributed load, internal shear force and bending moment.
Area centroid and first moment of area
The area centroid is the area-weighted mean of component centroids; the first moment about an axis is area times perpendicular centroid distance.
Basquin stress–life relation for fatigue strength
In the high-cycle fatigue regime, the Basquin relation approximates the link between stress amplitude and cycles to failure.
Lamé equations for a thick-walled cylinder
Lamé’s elastic solution gives radial and hoop stress through a thick cylinder under internal and external pressure.
Goodman fatigue relation
The linear Goodman relation estimates how mean and alternating stress jointly use a fatigue-strength limit.
Castigliano’s second theorem
Castigliano’s second theorem obtains displacement in the direction of a force by differentiating elastic strain energy with respect to that force.
Tresca yield criterion (maximum shear stress)
Tresca predicts initial yielding of a ductile material when maximum shear stress reaches half the uniaxial yield strength.
Linear 1D bar-element stiffness matrix
The two-node linear bar element relates axial nodal displacements and forces for a uniform elastic member in the finite element method.
Kirsch solution: stress concentration around a circular hole
The Kirsch solution gives elastic hoop stress at the boundary of a circular hole in an infinite plate under uniaxial remote tension.
Reynolds number
The Reynolds number compares inertial and viscous effects in a flow. It is dimensionless; its interpretation depends on geometry, characteristic length and boundary conditions.
Extended Bernoulli energy equation with head loss
The engineering energy balance adds pump input, turbine extraction and irreversible head loss between two flow sections.
Volume flow rate and continuity for incompressible flow
Volume flow rate is cross-sectional area times mean normal speed; for steady incompressible pipe flow it is conserved.
Mass flow rate through a section
Mass flow rate is density times cross-sectional area times mean normal speed.
Hydraulic diameter
For a fully wetted noncircular duct, hydraulic diameter defines a characteristic length from flow area and wetted perimeter.
Darcy–Weisbach equation for pipe-friction loss
The Darcy–Weisbach equation expresses distributed friction loss in a straight pipe as head loss.
Minor-loss head at a fitting
A local loss coefficient represents additional irreversible head loss due to a fitting or change in geometry.
Hagen–Poiseuille law for laminar pipe flow
For steady laminar Newtonian flow in a long straight circular tube, Poiseuille relates volume flow to pressure drop and viscosity.
Hydraulic power of a flow machine
Hydraulic power equals pressure difference times volume flow, or density times gravity, flow and head.
Centrifugal-pump affinity laws under speed change
For similar operation of the same pump, flow, head and power approximately scale with powers of rotational speed.
Euler turbomachinery equation
Specific total-enthalpy change follows from blade speed and tangential absolute velocity at inlet and outlet.
Cavitation number
Cavitation number compares pressure margin above vapor pressure with flow dynamic pressure.
Prandtl number of a fluid
Prandtl number compares momentum diffusivity with thermal diffusivity.
Net positive suction head available
NPSHA is absolute total suction head at the pump datum minus liquid vapor-pressure head.
Control-volume mass balance
The integral continuity equation balances stored mass and mass flow across an open-system boundary.
Navier–Stokes equation for an incompressible Newtonian fluid
The momentum balance with Newton’s viscosity law describes an incompressible fluid of constant density and viscosity.
Mach number
Mach number compares a flow speed with the local speed of sound.
Froude number
The Froude number compares inertia with gravity effects and governs dynamic similarity in free-surface flows.
Colebrook–White equation
The implicit Colebrook relation gives the Darcy friction factor for turbulent round-pipe flow from Reynolds number and relative roughness.
Critical nozzle pressure ratio and choked mass flow
At the critical back pressure, an ideal converging nozzle reaches Mach 1 at its throat; lowering back pressure further does not increase the modeled mass flow.
Ergun equation for packed-bed pressure drop
The Ergun equation combines viscous and inertial contributions to the pressure gradient through a packed particle bed.
Fourier’s law of heat conduction
Conductive heat flux points opposite the temperature gradient; a plane layer gives a familiar steady one-dimensional heat rate.
Newton’s law of cooling for convection
Convective heat transfer rate is proportional to area and the temperature difference between a surface and surrounding fluid.
Thermal diffusivity
Thermal diffusivity compares heat conduction with a material’s volumetric heat-storage capacity.
Biot number for transient conduction
The Biot number compares internal conduction resistance in a body with external convective resistance.
Fourier number for transient heat conduction
The Fourier number compares elapsed time with the characteristic time for heat conduction over a length.
Lumped-capacitance model of body temperature
If internal temperature gradients are small, a body’s temperature during convective cooling follows an exponential response.
Logarithmic mean temperature difference in a heat exchanger
The logarithmic mean temperature difference combines the varying local driving temperature difference along a heat exchanger.
Isentropic state change of an ideal gas
For a quasi-static reversible adiabatic state change of an ideal gas, pVγ and TVγ−1 remain constant.
Isentropic efficiency of a compressor
Compressor isentropic efficiency compares ideal isentropic enthalpy rise with actual enthalpy rise at the same outlet pressure.
Carnot efficiency of a heat engine
Carnot efficiency is the reversible upper limit for a heat engine operating between two reservoirs.
Heat-pump coefficient of performance
The heat-pump coefficient of performance relates useful heat delivered to work supplied.
Thermal conduction resistance of a plane wall
For steady one-dimensional conduction through a homogeneous plane layer, temperature difference acts across a thermal resistance.
Sensible heat at constant heat capacity
With approximately constant specific heat capacity, sensible heat is proportional to mass and temperature change.
Overall heat transfer through a plane wall
For a single plane layer, U combines inner and outer convection with wall conduction in series.
Nusselt number for convective heat transfer
Nusselt number compares convection with conduction across a characteristic length.
Sherwood number for convective mass transfer
Sherwood number compares convective mass transfer with molecular diffusion across a characteristic length.
Lewis number for heat and mass diffusion
Lewis number is the ratio of thermal diffusivity to species diffusivity.
Heat-exchanger effectiveness
Effectiveness compares actual heat-transfer rate with the maximum possible for the inlet states.
Heat-capacity-rate ratio of a heat exchanger
The capacity-rate ratio divides the smaller by the larger heat-capacity rate of the two fluids.
Steady-flow energy balance for a control volume
The steady energy balance relates heat, shaft work, and enthalpy, kinetic and potential energy carried by flowing streams.
Control-volume entropy balance
The entropy balance accounts for storage, heat transfer, mass streams and entropy generation by irreversibility.
Fick’s laws of diffusion flux and transient concentration
Fick’s first law relates diffusive flux to a concentration gradient; for constant isotropic diffusivity, the second law gives transient concentration evolution.
Cylindrical thermal conduction resistance
Steady radial conduction through a cylindrical layer has resistance proportional to the logarithm of its radius ratio.
Net radiative heat transfer between gray surfaces
For two diffuse gray surfaces in a two-surface enclosure, net heat transfer follows the fourth-power temperature difference and a resistance network.
Kirchhoff’s junction and loop rules
Lumped network with ideal nodes and no unaccounted charge accumulation. Loop balances must include all sources, including induced voltages; spatially distributed high-frequency systems need more detailed models.
Ideal PID controller transfer function
The ideal continuous PID controller combines proportional, integral and derivative actions on the error signal.
Voltage divider with two series resistors
An unloaded voltage divider provides a fraction of its input voltage across the lower resistor R2.
Time response of an RC charging circuit
After a DC source is connected, capacitor voltage approaches the source voltage exponentially through a resistor.
Current rise in an RL circuit
After a DC source is switched on, current in a series RL circuit rises exponentially toward its steady value.
Impedance of a series RLC circuit
For sinusoidal AC, the series impedance magnitude is the quadratic combination of resistance and net reactance.
Real power in a sinusoidal AC circuit
Average real power in a single-phase sinusoidal AC circuit is the product of RMS values and power factor.
Energy stored in a capacitor
The energy of an ideal capacitor is the work required to build voltage in its electric field.
Voltage ratio of an ideal transformer
In an ideal transformer, the voltage ratio equals the turns ratio.
Magnetic energy stored in an inductor
A linear inductor stores magnetic-field energy proportional to the square of its current.
Real power in a balanced three-phase system
For balanced sinusoidal three-phase power, total real power is times line voltage, line current and power factor.
Thevenin equivalent of a linear two-terminal network
A linear network viewed at two terminals can be replaced by an ideal voltage source in series with a resistance.
Norton equivalent and Thevenin conversion
A linear two-terminal network can be represented by a current source in parallel with its internal resistance.
Transfer function of a first-order system
A linear first-order system is represented in the Laplace domain by static gain K and time constant τ.
Nyquist sampling condition for band-limited signals
A band-limited continuous signal can be reconstructed without spectral overlap when sampling exceeds twice its maximum frequency.
Closed-loop transfer function
With negative feedback, the closed-loop transfer function combines the forward path and loop gain.
Feedback sensitivity function
The sensitivity function describes how feedback attenuates disturbances and model changes in the loop.
Normalized second-order transfer function
A linear second-order system is characterized by static gain, undamped natural frequency and damping ratio.
RMS value of a sinusoidal quantity
For a pure sinusoid, RMS value equals peak value divided by .
Apparent power in a balanced three-phase system
For a balanced three-phase system, apparent power is times line voltage and line current.
Reactive power in a balanced three-phase system
For a balanced sinusoidal load, the phase-angle sine gives three-phase reactive power.
Balance condition of a Wheatstone bridge
At bridge balance, the voltage between the middle nodes is zero and the resistance ratios match.
Complementary sensitivity function
Complementary sensitivity is the closed-loop transfer contribution of loop gain.
Load resistance for maximum power transfer
A purely resistive Thevenin source transfers maximum load power when load and source resistance match.
Routh–Hurwitz stability criterion
The Routh array counts sign changes in its first column to determine the number of continuous-system poles in the right half of the s-plane without solving for roots explicitly.
Bode plot: gain and phase margins
Phase and gain margins measure, on a Bode plot, the distance of the open loop from critical feedback conditions.
Ideal operational-amplifier gain
With ideal op amps under negative feedback, resistor ratios set inverting and non-inverting voltage gain.
Shockley diode equation
The Shockley equation models current through an idealized pn diode as a function of voltage and temperature.
Rate of a cylindrical helical spring
The linear stiffness of an ideal cylindrical helical spring depends on wire shear modulus, wire diameter, mean coil diameter and active turns.
Ratio of an ideal spur-gear pair
For external spur gears, tooth-count ratio determines the inverse speed ratio and direction reversal.
Module of a spur gear
Gear module is pitch diameter divided by tooth count and sets tooth size.
Ideal torque of a lead screw
For an ideal frictionless screw, required torque equals axial force times lead divided by the circumference factor.
Belt-drive peripheral speed
With no belt slip, belt speed equals the pulley pitch-line speed.
Speed ratio of slip-free belt pulleys
Equal belt speed makes pulley speeds inversely proportional to their effective diameters.
Machine efficiency
Efficiency compares useful output power with supplied power across the same system boundary.
Basic rating life of a rolling bearing
The ISO basic L10 life scales dynamic rating C to equivalent dynamic bearing load P.
Power transmitted by a belt
Belt power equals belt speed times the difference between tight-side and slack-side tensions.
Euler–Eytelwein (capstan) equation
The capstan relation gives maximum tension ratio for a flexible line wrapped around a drum at impending slip.
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