MW SysArcKnowledge Platform
Menu

Learn by calculating

Start with numbers. Build toward quantum mathematics.

Follow a focused route through the mathematics quantum mechanics actually depends on. Every step has calculators, visible working, assumptions, meaning and limits.

Start the QM path

The learning loop

Predict → calculate → explain → change

  1. Read the problem and predict what should happen.
  2. Enter values and calculate locally in your browser.
  3. Inspect the formula and every calculation step.
  4. Explain the answer's sign, size, units and assumptions.
  5. Change one input and predict the new result before recalculating.

The quantum-mechanics spine

71 essential calculator lessons in dependency order

Finish Step 10 and you will have practised the algebra, probability, calculus, linear algebra, waves, complex numbers and physics used by introductory quantum mechanics.

Step 1

Numbers and algebra for physics

Rearrange equations and work confidently with fractions, powers, roots and logarithms.

Before starting: Comfort reading whole numbers and decimals.
  1. Step 1 · Lesson 1Fraction simplifierReduce a fraction to lowest terms using the greatest common divisor.
  2. Step 1 · Lesson 2Ratio simplifierSimplify a two-number whole-number ratio and calculate its decimal relationship.
  3. Step 1 · Lesson 3ExponentRaise a base to a selected power and see the repeated-multiplication meaning.
  4. Step 1 · Lesson 4Square rootFind the principal square root and verify it by squaring the result.
  5. Step 1 · Lesson 5LogarithmCalculate a logarithm with any valid positive base.
  6. Step 1 · Lesson 6Linear equationSolve a linear equation in the form ax + b = c and see the isolation steps.
  7. Step 1 · Lesson 7Quadratic formulaSolve ax² + bx + c = 0 and inspect the discriminant.
Step 2

Functions and graphs

Read linear, quadratic, exponential and periodic relationships as changing physical models.

Before starting: Understand equations, powers, roots and coordinates.
  1. Step 2 · Lesson 1SlopeCalculate the slope of the line through two coordinate points.
  2. Step 2 · Lesson 2Linear functionEvaluate f(x) = mx + b and identify its slope and vertical intercept.
  3. Step 2 · Lesson 3Quadratic functionEvaluate ax² + bx + c and calculate the parabola's vertex coordinates.
  4. Step 2 · Lesson 4Exponential functionEvaluate exponential growth or decay in the form f(x)=abˣ.
  5. Step 2 · Lesson 5Sine, cosine and tangentCalculate sine, cosine and tangent for an angle entered in degrees.
  6. Step 2 · Lesson 6Sine waveEvaluate a sinusoid from amplitude, frequency, time and phase.
Step 3

Geometry, angles and trigonometry

Resolve lengths, angles, rotations and oscillations using radians, sine and cosine.

Before starting: Understand algebra, functions and square roots.
  1. Step 3 · Lesson 1Pythagorean theoremFind the hypotenuse of a right triangle from its two legs.
  2. Step 3 · Lesson 2Degrees and radiansConvert an angle entered in degrees to radians and common turn measures.
  3. Step 3 · Lesson 3Right triangleSolve a right triangle from its two perpendicular legs.
  4. Step 3 · Lesson 4Cosine law sideFind a triangle side from two sides and their included angle.
  5. Step 3 · Lesson 5Polar to CartesianConvert a radius and angle into x and y coordinates.
  6. Step 3 · Lesson 6Cartesian to polarConvert x and y coordinates into radius and direction.
  7. Step 3 · Lesson 7Unit circleConvert an angle into its unit-circle coordinates, sine, cosine and reference angle.
  8. Step 3 · Lesson 8Inverse trig functionsCalculate arcsine, arccosine and arctangent for a real ratio and show their principal angles.
  9. Step 3 · Lesson 9Trig identityEvaluate sine, cosine and the identity sin²θ+cos²θ=1 at any angle.
Step 4

Probability and distributions

Interpret averages, spread and probability before treating a wavefunction as a probability amplitude.

Before starting: Understand fractions, percentages, functions and square roots.
  1. Step 4 · Lesson 1Mean and medianCalculate the arithmetic mean and median of six values.
  2. Step 4 · Lesson 2Standard deviationCalculate population and sample standard deviation for six values.
  3. Step 4 · Lesson 3Z-scoreStandardise a value by measuring how many standard deviations it lies from the mean.
  4. Step 4 · Lesson 4ProbabilityCalculate event probability from favourable and total equally likely outcomes.
  5. Step 4 · Lesson 5Expected valueCalculate the weighted mean of three possible outcomes.
  6. Step 4 · Lesson 6Normal CDFEstimate the probability below a value in a normal distribution.
Step 5

Calculus and continuous change

Use derivatives and integrals to describe motion, accumulation and changing wavefunctions.

Before starting: Understand algebra, functions, graphs, exponents and trigonometry.
  1. Step 5 · Lesson 1Power-rule derivativeDifferentiate a monomial axⁿ and evaluate its derivative at a selected x-value.
  2. Step 5 · Lesson 2Quadratic derivativeDifferentiate ax²+bx+c and evaluate its slope.
  3. Step 5 · Lesson 3Exponential derivativeDifferentiate Ae^(kx) at a selected x.
  4. Step 5 · Lesson 4Trigonometric derivativeDifferentiate A sin(kx+φ) and evaluate both the function and derivative at x.
  5. Step 5 · Lesson 5Chain ruleCombine an outer derivative and inner derivative to evaluate a composite function's derivative at a point.
  6. Step 5 · Lesson 6Product ruleEvaluate the derivative of a product from f, g and their derivatives at the same point.
  7. Step 5 · Lesson 7Quotient ruleEvaluate the derivative of f/g from both function values and derivatives at the same point.
  8. Step 5 · Lesson 8Power-rule integralIntegrate a monomial axⁿ and evaluate its antiderivative at a selected x-value.
  9. Step 5 · Lesson 9Definite power integralEvaluate the definite integral of axⁿ between lower and upper bounds.
  10. Step 5 · Lesson 10Simpson's ruleApproximate an integral from endpoint and midpoint function values.
Step 6

Vectors and linear algebra

Treat states and transformations as vectors, matrices, inner products and eigenvalue problems.

Before starting: Understand algebra, coordinates, trigonometry and functions.
  1. Step 6 · Lesson 1Vector magnitudeCalculate the length of a three-dimensional vector from its components.
  2. Step 6 · Lesson 2Vector additionAdd two three-dimensional vectors component by component.
  3. Step 6 · Lesson 3Dot productCalculate the scalar dot product of two three-dimensional vectors.
  4. Step 6 · Lesson 4Angle between vectorsFind the angle between two three-dimensional vectors using their dot product.
  5. Step 6 · Lesson 5Matrix–vector productApply a 2×2 linear transformation to a two-component vector.
  6. Step 6 · Lesson 62×2 matrix multiplicationMultiply two 2×2 matrices in order A times B.
  7. Step 6 · Lesson 72×2 eigenvaluesCalculate the real eigenvalues of a 2×2 matrix from its trace and determinant.
Step 7

Differential equations and waves

Connect rates of change to growth, oscillation, damping, frequency and wavelength.

Before starting: Understand calculus, exponentials, trigonometry and vectors.
  1. Step 7 · Lesson 1Exponential growth ODESolve y′=ky from an initial value and evaluate the solution at time t.
  2. Step 7 · Lesson 2Euler method stepApproximate one step for y′=ay+b using the slope at the current point.
  3. Step 7 · Lesson 3Harmonic oscillatorEvaluate displacement and velocity for x″+ω²x=0 with amplitude and phase.
  4. Step 7 · Lesson 4Damped oscillatorEvaluate an underdamped oscillation x(t)=Ae⁻ᵝᵗcos(ωt+φ) and its decaying envelope.
  5. Step 7 · Lesson 5Wave speed and wavelengthCalculate wave speed and period from frequency and wavelength.
  6. Step 7 · Lesson 6Angular to linear speedConvert angular speed at a radius into tangential speed.
Step 8

Complex numbers and Fourier ideas

Represent phase and interference with complex amplitudes, then decompose sampled waves into frequencies.

Before starting: Understand radians, trigonometry, vectors, exponentials and oscillation.
  1. Step 8 · Lesson 1Complex additionAdd two complex numbers by combining their real and imaginary components.
  2. Step 8 · Lesson 2Complex multiplicationMultiply two complex numbers and display the rectangular-form result.
  3. Step 8 · Lesson 3Complex divisionDivide one complex number by another using the denominator's conjugate.
  4. Step 8 · Lesson 4Complex polar formConvert z=a+bi into magnitude and phase angle.
  5. Step 8 · Lesson 5Euler formulaEvaluate eⁱᶿ as cosine and sine components on the unit circle.
  6. Step 8 · Lesson 6Phasor additionAdd two magnitude-phase phasors and return the combined magnitude and phase.
  7. Step 8 · Lesson 7Four-sample DFTCalculate one discrete Fourier transform bin from four real-valued samples.
Step 9

The physics bridge

Connect the mathematics to motion, momentum, energy, forces, photons and matter waves.

Before starting: Understand calculus, vectors, waves and differential equations.
  1. Step 9 · Lesson 1Constant accelerationCalculate final velocity and displacement under constant acceleration.
  2. Step 9 · Lesson 2Kinetic energy and momentumCalculate classical linear momentum and kinetic energy from mass and velocity.
  3. Step 9 · Lesson 3Work and force angleCalculate mechanical work done by a constant force over a displacement.
  4. Step 9 · Lesson 4Newton's second lawCalculate net force and weight from mass, acceleration and gravitational field strength.
  5. Step 9 · Lesson 5Coulomb forceCalculate the signed electrostatic force magnitude between two point charges.
  6. Step 9 · Lesson 6Photon energyConvert a photon wavelength into frequency, energy in joules and energy in electronvolts.
  7. Step 9 · Lesson 7De Broglie wavelengthCalculate the matter wavelength associated with a non-relativistic particle's momentum.
Step 10

Quantum-mechanics foundations

Calculate with uncertainty, complex probability amplitudes, normalized states, qubits and quantized energies.

Before starting: Complete the preceding physics, probability, linear-algebra, wave and complex-number steps.
  1. Step 10 · Lesson 1Uncertainty principleCalculate the minimum momentum and velocity uncertainty implied by a position uncertainty.
  2. Step 10 · Lesson 2Wavefunction probabilityConvert a complex wavefunction amplitude into magnitude, phase and probability density.
  3. Step 10 · Lesson 3Two-state normalizationNormalize two real state amplitudes and calculate their measurement probabilities.
  4. Step 10 · Lesson 4Bloch sphere stateConvert Bloch-sphere angles into qubit amplitudes and measurement probabilities.
  5. Step 10 · Lesson 5Infinite well energyCalculate a particle's quantized energy level in a one-dimensional infinite square well.
  6. Step 10 · Lesson 6Hydrogen transitionCalculate hydrogen energy levels and the photon energy and wavelength between two principal quantum numbers.

Optional mathematics library

Explore without blocking your QM progress

3929 additional calculators remain available for deeper study, schoolwork and practical problems. They are useful, but they are not required checkpoints on the main path.

Number foundations5 optional lessons
Absolute valueGCF and LCMPercentage of a numberPercentage changeCube root
Equations and algebra2 optional lessons
Proportion solverTwo-equation system
Coordinate and practical geometry13 optional lessons
MidpointDistance between pointsRectangleTriangle areaCircleTrapezoid areaEllipseCube dimensionsRectangular prismCylinderCone volumeSphereSquare pyramid
Statistics and probability4 optional lessons
Weighted meanData rangeOdds to probabilityCombinations and permutations
Precalculus and functions4 optional lessons
Arithmetic sequenceArithmetic seriesGeometric sequenceGeometric series
Linear algebra3 optional lessons
Cross product2×2 determinant2×2 matrix inverse
Differential equations4 optional lessons
Half-life decayLogistic growth ODENewton coolingFirst-order linear ODE
Complex and Fourier mathematics1 optional lessons
Principal complex root
Mathematical physics2 optional lessons
Projectile motionGravitational force
Number patterns and practical algebra10 optional lessons
FactorialFibonacci numberTriangular numberSum 1 to nModuloPercentage differenceNth rootChange-of-base logarithmDistance, rate and timeRatio allocation
Extended geometry10 optional lessons
Annulus areaCircular segment areaArc lengthHeron's triangle areaParallelogram areaRhombus areaRegular polygonPolygon interior angles3D distanceTorus geometry
Extended trigonometry3 optional lessons
Cosine law angleSine law sideBearing components
Extended statistics10 optional lessons
Population varianceQuartilesInterquartile rangeRoot mean squareGeometric meanHarmonic meanCoefficient of variationCovariancePearson correlationLinear regression
Extended probability7 optional lessons
Probability complementIndependent event unionConditional probabilityBinomial exact probabilityBinomial mean and variancePoisson probabilityBayes theorem
Applied derivatives and integrals6 optional lessons
Logarithm derivativeQuadratic tangent lineQuadratic average rateDefinite linear integralDefinite quadratic integralTrapezoidal rule
Growth, decay and compounding6 optional lessons
Simple interestCompound interestContinuous compoundingDoubling timeExponential half-lifeRequired growth rate
Computational number theory6 optional lessons
Prime checkerPrime factorizationDivisor countSum of divisorsEuler totientModular exponentiation
Analytic algebra and geometry6 optional lessons
Quadratic vertexQuadratic discriminantCubic equationInverse linear functionEllipse eccentricityGreat-circle distance
Matrix computation3 optional lessons
2×2 matrix additionMatrix traceMatrix scalar multiplication
Statistical inference and numerical calculus3 optional lessons
Normal interval probabilityMean confidence intervalNewton square-root step
Algebra formula relationships156 optional lessons
Direct VariationDirect Variation: solve constant of variation kDirect Variation: solve input xInverse VariationInverse Variation: solve variation constant kInverse Variation: solve input xIndexed RadicalIndexed Radical: solve radicandIndexed Radical: solve root indexPercentage ProportionPercentage Proportion: solve percentage ratePercentage Proportion: solve whole quantityScale-Factor OutputScale-Factor Output: solve original quantityScale-Factor Output: solve scale factorUnit Rate TotalUnit Rate Total: solve quantity countUnit Rate Total: solve amount per unitMarkup Percentage AmountMarkup Percentage Amount: solve markup percentageMarkup Percentage Amount: solve base priceDiscount Percentage AmountDiscount Percentage Amount: solve discount percentageDiscount Percentage Amount: solve original priceSales-Tax Percentage AmountSales-Tax Percentage Amount: solve tax percentageSales-Tax Percentage Amount: solve taxable subtotalCommission Percentage AmountCommission Percentage Amount: solve commission percentageCommission Percentage Amount: solve eligible sales amountRatio Unit RateRatio Unit Rate: solve represented quantityRatio Unit Rate: solve reference-unit countMultiplicative Formula TermMultiplicative Formula Term: solve first factorMultiplicative Formula Term: solve second factorPercent Change RatePercent Change Rate: solve original valuePercent Change Rate: solve new valueScientific-Notation Power ScaleScientific-Notation Power Scale: solve radix baseScientific-Notation Power Scale: solve integer scale exponentPolynomial Coefficient CountPolynomial Coefficient Count: solve polynomial degreePolynomial Coefficient Count: solve constant-term slotPolynomial Quotient DegreePolynomial Quotient Degree: solve dividend polynomial degreePolynomial Quotient Degree: solve divisor polynomial degreePolynomial Monic Coefficient NormalizationPolynomial Monic Coefficient Normalization: solve selected polynomial coefficientPolynomial Monic Coefficient Normalization: solve nonzero leading coefficientEqual-Degree Rational Horizontal AsymptoteEqual-Degree Rational Horizontal Asymptote: solve numerator leading coefficientEqual-Degree Rational Horizontal Asymptote: solve denominator leading coefficientLaboratory Dilution FactorLaboratory Dilution Factor: solve final diluted volumeLaboratory Dilution Factor: solve sample aliquot volumeSerial Dilution Total FactorSerial Dilution Total Factor: solve dilution factor per stageSerial Dilution Total Factor: solve number of equal dilution stagesFlight Endurance from Usable FuelFlight Endurance from Usable Fuel: solve usable fuel quantityFlight Endurance from Usable Fuel: solve average fuel-flow rateAircraft Wing LoadingAircraft Wing Loading: solve aircraft weightAircraft Wing Loading: solve reference wing areaAircraft Power LoadingAircraft Power Loading: solve aircraft weightAircraft Power Loading: solve available propulsive powerField Seed RequirementField Seed Requirement: solve seed rate per areaField Seed Requirement: solve planned planting areaField Fertilizer RequirementField Fertilizer Requirement: solve fertilizer application rate per areaField Fertilizer Requirement: solve treated field areaHydraulic Residence TimeHydraulic Residence Time: solve effective stored water volumeHydraulic Residence Time: solve average volumetric outflow rateMusic Tempo Beat PeriodMusic Tempo Beat Period: solve elapsed musical durationMusic Tempo Beat Period: solve beat countMusical Measure DurationMusical Measure Duration: solve duration per beatMusical Measure Duration: solve beats per measureAudio Amplitude Gain in DecibelsAudio Amplitude Gain in Decibels: solve output amplitudeAudio Amplitude Gain in Decibels: solve input amplitudeAudio Power Gain in DecibelsAudio Power Gain in Decibels: solve output powerAudio Power Gain in Decibels: solve input powerMeteorological Vapor-Pressure DeficitMeteorological Vapor-Pressure Deficit: solve saturation vapor pressureMeteorological Vapor-Pressure Deficit: solve actual vapor pressureWet-Bulb Temperature DepressionWet-Bulb Temperature Depression: solve dry-bulb air temperatureWet-Bulb Temperature Depression: solve wet-bulb temperatureDiurnal Air-Temperature RangeDiurnal Air-Temperature Range: solve daily maximum air temperatureDiurnal Air-Temperature Range: solve daily minimum air temperatureRobot Gear Speed RatioRobot Gear Speed Ratio: solve motor input angular speedRobot Gear Speed Ratio: solve joint output angular speedRobot Commanded Position ErrorRobot Commanded Position Error: solve commanded joint or axis positionRobot Commanded Position Error: solve measured joint or axis positionRobot Control-Cycle Time MarginRobot Control-Cycle Time Margin: solve control-cycle deadlineRobot Control-Cycle Time Margin: solve measured computation and communication timeRobot Gripper Force MarginRobot Gripper Force Margin: solve available gripping forceRobot Gripper Force Margin: solve required gripping forceFood Recipe Scaling FactorFood Recipe Scaling Factor: solve desired recipe yieldFood Recipe Scaling Factor: solve original recipe yieldBattery Load RuntimeBattery Load Runtime: solve usable battery energyBattery Load Runtime: solve average load powerBattery Energy Reserve MarginBattery Energy Reserve Margin: solve available usable battery energyBattery Energy Reserve Margin: solve required mission or load energyGeophysical Gravity AnomalyGeophysical Gravity Anomaly: solve corrected observed gravityGeophysical Gravity Anomaly: solve reference-model gravityActivated-Sludge Solids Retention TimeActivated-Sludge Solids Retention Time: solve biological solids inventoryActivated-Sludge Solids Retention Time: solve solids mass leaving per timeWastewater Membrane Transmembrane PressureWastewater Membrane Transmembrane Pressure: solve mean feed-channel pressureWastewater Membrane Transmembrane Pressure: solve permeate-side pressureSemiconductor Gate-Charge Switching TimeSemiconductor Gate-Charge Switching Time: solve effective gate charge movedSemiconductor Gate-Charge Switching Time: solve average gate-drive currentX-Ray Diffraction Peak ShiftX-Ray Diffraction Peak Shift: solve measured diffraction peak positionX-Ray Diffraction Peak Shift: solve reference peak positionElectrode OverpotentialElectrode Overpotential: solve operating electrode potentialElectrode Overpotential: solve equilibrium electrode potentialElectrochemical Cell PotentialElectrochemical Cell Potential: solve cathode reduction potentialElectrochemical Cell Potential: solve anode reduction potentialRadio Received Link MarginRadio Received Link Margin: solve predicted received signal levelRadio Received Link Margin: solve receiver sensitivity thresholdRadio Link Path LossRadio Link Path Loss: solve transmitted isotropic-equivalent levelRadio Link Path Loss: solve received isotropic-equivalent levelSurvey Grid Scale Corrected DistanceSurvey Grid Scale Corrected Distance: solve measured or reduced ground distanceSurvey Grid Scale Corrected Distance: solve applicable grid scale factorSurvey Differential-Level ElevationSurvey Differential-Level Elevation: solve known reference elevationSurvey Differential-Level Elevation: solve signed net backsight-minus-foresight changeMeasurement Guard-Band LimitMeasurement Guard-Band Limit: solve specification acceptance limitMeasurement Guard-Band Limit: solve selected guard-band width
Precalculus formula relationships27 optional lessons
Power LawPower Law: solve base xPower Law: solve exponent nContinuous Growth FactorContinuous Growth Factor: solve initial valueContinuous Growth Factor: solve growth exponent rtContinuous Decay FactorContinuous Decay Factor: solve initial valueContinuous Decay Factor: solve decay exponent ktContinuous AmplificationContinuous Amplification: solve initial amplitudeContinuous Amplification: solve dimensionless gain exponentContinuous AttenuationContinuous Attenuation: solve initial amplitudeContinuous Attenuation: solve dimensionless loss exponentChange-of-Base LogarithmChange-of-Base Logarithm: solve positive argumentChange-of-Base Logarithm: solve positive logarithm baseArithmetic Sequence Accumulated ChangeArithmetic Sequence Accumulated Change: solve starting termArithmetic Sequence Accumulated Change: solve accumulated common-difference changeFinite Arithmetic Series from AverageFinite Arithmetic Series from Average: solve term countFinite Arithmetic Series from Average: solve average term valueCompound-Period Growth FactorCompound-Period Growth Factor: solve growth factor per periodCompound-Period Growth Factor: solve period count
Statistics formula relationships1270 optional lessons
Two-Value Arithmetic MeanTwo-Value Arithmetic Mean: solve first valueTwo-Value Arithmetic Mean: solve second valueTwo-Value Geometric MeanTwo-Value Geometric Mean: solve first positive valueTwo-Value Geometric Mean: solve second positive valueTwo-Value Harmonic MeanTwo-Value Harmonic Mean: solve first positive rateTwo-Value Harmonic Mean: solve second positive rateMean–Total RelationshipMean–Total Relationship: solve arithmetic meanMean–Total Relationship: solve observation countStandard Error–Sample SizeStandard Error–Sample Size: solve standard deviationStandard Error–Sample Size: solve sample sizeZ-Score DisplacementZ-Score Displacement: solve z-scoreZ-Score Displacement: solve standard deviationMargin–Sample Size RatioMargin–Sample Size Ratio: solve critical spread product zσMargin–Sample Size Ratio: solve target marginVariance–Squared-Deviation TotalVariance–Squared-Deviation Total: solve population varianceVariance–Squared-Deviation Total: solve observation countCoefficient of Variation RatioCoefficient of Variation Ratio: solve standard deviationCoefficient of Variation Ratio: solve mean magnitudeF Variance RatioF Variance Ratio: solve first variance estimateF Variance Ratio: solve second variance estimateWeighted Score ContributionWeighted Score Contribution: solve weight percentageWeighted Score Contribution: solve criterion scoreRisk RatioRisk Ratio: solve risk in exposed groupRisk Ratio: solve risk in comparison groupStandardized Residual SquareStandardized Residual Square: solve residual magnitudeStandardized Residual Square: solve reference scaleMeasurement Error AmplificationMeasurement Error Amplification: solve input error magnitudeMeasurement Error Amplification: solve sensitivity multiplierMonte Carlo Standard ErrorMonte Carlo Standard Error: solve sample standard deviationMonte Carlo Standard Error: solve independent simulation countTwo-Signal Root Mean SquareTwo-Signal Root Mean Square: solve first signal levelTwo-Signal Root Mean Square: solve second signal levelVariance from First Two MomentsVariance from First Two Moments: solve mean of squared valuesVariance from First Two Moments: solve arithmetic meanPooled Observation CountPooled Observation Count: solve first sample countPooled Observation Count: solve second sample countPooled Squared-Deviation TotalPooled Squared-Deviation Total: solve within-group squared deviationsPooled Squared-Deviation Total: solve between-group adjustmentChi-Square Expected FrequencyChi-Square Expected Frequency: solve row proportionChi-Square Expected Frequency: solve column totalResidual Degrees of FreedomResidual Degrees of Freedom: solve observation countResidual Degrees of Freedom: solve fitted parameter countSignal Energy from RMSSignal Energy from RMS: solve sample countSignal Energy from RMS: solve RMS signal levelIndependent Error Standard DeviationIndependent Error Standard Deviation: solve first error standard deviationIndependent Error Standard Deviation: solve second error standard deviationVariance Component TotalVariance Component Total: solve first independent variance componentVariance Component Total: solve second independent variance componentFinite-Population Prevalence PercentageFinite-Population Prevalence Percentage: solve population casesFinite-Population Prevalence Percentage: solve population membersSampling Fraction PercentageSampling Fraction Percentage: solve sample sizeSampling Fraction Percentage: solve population sizeFano FactorFano Factor: solve count varianceFano Factor: solve count meanSurvey Design EffectSurvey Design Effect: solve actual estimator varianceSurvey Design Effect: solve simple-random-sample varianceEffective Sample Size from Design EffectEffective Sample Size from Design Effect: solve nominal sample sizeEffective Sample Size from Design Effect: solve design effectHistogram Bin DensityHistogram Bin Density: solve bin relative frequencyHistogram Bin Density: solve bin widthMean Square from Signal EnergyMean Square from Signal Energy: solve sum of squared observationsMean Square from Signal Energy: solve observation countPrecision from VariancePrecision from Variance: solve unit scalePrecision from Variance: solve positive varianceStandardized Deviation ScoreStandardized Deviation Score: solve deviation from meanStandardized Deviation Score: solve standard deviationConfidence Margin from Critical ValueConfidence Margin from Critical Value: solve critical valueConfidence Margin from Critical Value: solve standard errorStratified Proportional Sample AllocationStratified Proportional Sample Allocation: solve total sample sizeStratified Proportional Sample Allocation: solve stratum population sharePooled Event RatePooled Event Rate: solve total observed eventsPooled Event Rate: solve total exposureExplained Variation PercentageExplained Variation Percentage: solve explained sum of squaresExplained Variation Percentage: solve total sum of squaresDesign-Inflated Sampling VarianceDesign-Inflated Sampling Variance: solve simple-random-sample varianceDesign-Inflated Sampling Variance: solve design effectGeometric-Mean Log RatioGeometric-Mean Log Ratio: solve positive first geometric meanGeometric-Mean Log Ratio: solve positive reference geometric meanChi-Square Standardized Cell ContributionChi-Square Standardized Cell Contribution: solve observed-minus-expected residualChi-Square Standardized Cell Contribution: solve square root of expected countRelative Standard Error PercentageRelative Standard Error Percentage: solve standard errorRelative Standard Error Percentage: solve estimate magnitudeStandardized Mean DifferenceStandardized Mean Difference: solve difference between group meansStandardized Mean Difference: solve pooled standard deviationVariance Ratio StatisticVariance Ratio Statistic: solve first sample varianceVariance Ratio Statistic: solve second sample varianceReduced Chi-SquareReduced Chi-Square: solve chi-square statisticReduced Chi-Square: solve model degrees of freedomInformation Criterion per ObservationInformation Criterion per Observation: solve total information criterion valueInformation Criterion per Observation: solve observation countSurvey Expansion WeightSurvey Expansion Weight: solve represented population countSurvey Expansion Weight: solve sampled unit countRelative Estimator Bias PercentageRelative Estimator Bias Percentage: solve estimator bias magnitudeRelative Estimator Bias Percentage: solve target parameter magnitudeMissing Observation PercentageMissing Observation Percentage: solve missing observation countMissing Observation Percentage: solve expected observation countProcess Acceptance PercentageProcess Acceptance Percentage: solve accepted item countProcess Acceptance Percentage: solve inspected item countRetained Sample Count from FractionRetained Sample Count from Fraction: solve original sample sizeRetained Sample Count from Fraction: solve retention fractionContingency Table Degrees of FreedomContingency Table Degrees of Freedom: solve row count minus oneContingency Table Degrees of Freedom: solve column count minus oneIntraclass Variance RatioIntraclass Variance Ratio: solve between-cluster variance componentIntraclass Variance Ratio: solve total varianceEta-Squared Effect PercentageEta-Squared Effect Percentage: solve effect sum of squaresEta-Squared Effect Percentage: solve total sum of squaresSample Variance from Corrected Sum of SquaresSample Variance from Corrected Sum of Squares: solve sum of squared deviationsSample Variance from Corrected Sum of Squares: solve sample degrees of freedomMean Estimator VarianceMean Estimator Variance: solve individual-observation varianceMean Estimator Variance: solve independent observation countBootstrap Bias EstimateBootstrap Bias Estimate: solve mean bootstrap estimateBootstrap Bias Estimate: solve original-sample estimatePooled Model Degrees of FreedomPooled Model Degrees of Freedom: solve first independent degrees of freedomPooled Model Degrees of Freedom: solve second independent degrees of freedomAverage Log-LikelihoodAverage Log-Likelihood: solve total log-likelihoodAverage Log-Likelihood: solve observation countDeviance per Degree of FreedomDeviance per Degree of Freedom: solve model devianceDeviance per Degree of Freedom: solve residual degrees of freedomVariance-Based Estimator Efficiency PercentageVariance-Based Estimator Efficiency Percentage: solve reference estimator varianceVariance-Based Estimator Efficiency Percentage: solve candidate estimator varianceCovariance from Cross-Deviation SumCovariance from Cross-Deviation Sum: solve sum of paired cross-deviationsCovariance from Cross-Deviation Sum: solve covariance degrees of freedomMean Absolute ErrorMean Absolute Error: solve sum of absolute errorsMean Absolute Error: solve observation countRoot Mean Square Error from Squared-Error SumRoot Mean Square Error from Squared-Error Sum: solve sum of squared errorsRoot Mean Square Error from Squared-Error Sum: solve observation countCorrelation Explained-Variance PercentageCorrelation Explained-Variance Percentage: solve percentage scaleCorrelation Explained-Variance Percentage: solve correlation coefficientFalse Discovery Proportion PercentageFalse Discovery Proportion Percentage: solve false positive discoveriesFalse Discovery Proportion Percentage: solve total reported discoveriesDiagnostic Sensitivity PercentageDiagnostic Sensitivity Percentage: solve true positive countDiagnostic Sensitivity Percentage: solve actual positive countDiagnostic Specificity PercentageDiagnostic Specificity Percentage: solve true negative countDiagnostic Specificity Percentage: solve actual negative countPositive Predictive Value PercentagePositive Predictive Value Percentage: solve true positive countPositive Predictive Value Percentage: solve all positive-test countNegative Predictive Value PercentageNegative Predictive Value Percentage: solve true negative countNegative Predictive Value Percentage: solve all negative-test countBalanced Classification AccuracyBalanced Classification Accuracy: solve sensitivity percentageBalanced Classification Accuracy: solve specificity percentageF1 Harmonic Classification ScoreF1 Harmonic Classification Score: solve positive predictive valueF1 Harmonic Classification Score: solve sensitivity or recallIncidence Rate RatioIncidence Rate Ratio: solve first person-time incidence rateIncidence Rate Ratio: solve reference person-time incidence rateHazard Rate RatioHazard Rate Ratio: solve first instantaneous hazardHazard Rate Ratio: solve reference instantaneous hazardPositive Diagnostic Likelihood RatioPositive Diagnostic Likelihood Ratio: solve sensitivity as decimalPositive Diagnostic Likelihood Ratio: solve false-positive rate as decimalRegression Coefficient of DeterminationRegression Coefficient of Determination: solve residual sum of squaresRegression Coefficient of Determination: solve total corrected sum of squaresRegression Residual VarianceRegression Residual Variance: solve residual sum of squaresRegression Residual Variance: solve residual degrees of freedomRegression Coefficient t StatisticRegression Coefficient t Statistic: solve estimated coefficientRegression Coefficient t Statistic: solve coefficient standard errorVariance Inflation Factor from ToleranceVariance Inflation Factor from Tolerance: solve regression toleranceVariance Inflation Factor from Tolerance: solve unit reciprocal scaleMean Regression LeverageMean Regression Leverage: solve fitted parameter countMean Regression Leverage: solve observation countCook Influence Distance RatioCook Influence Distance Ratio: solve scaled squared residual-leverage contributionCook Influence Distance Ratio: solve parameter-count variance scaleInformation-Criterion DeltaInformation-Criterion Delta: solve candidate model information criterionInformation-Criterion Delta: solve best model information criterionBIC Complexity PenaltyBIC Complexity Penalty: solve fitted parameter countBIC Complexity Penalty: solve natural logarithm of observation countAkaike Relative LikelihoodAkaike Relative Likelihood: solve best-model relative scaleAkaike Relative Likelihood: solve half criterion deltaDurbin–Watson Residual StatisticDurbin–Watson Residual Statistic: solve sum of squared successive residual differencesDurbin–Watson Residual Statistic: solve residual sum of squaresBinomial Proportion Standard ErrorBinomial Proportion Standard Error: solve Bernoulli variance factor p(1-p)Binomial Proportion Standard Error: solve sample sizeFinite Population CorrectionFinite Population Correction: solve population-minus-sample countFinite Population Correction: solve population-minus-one countTwo-by-Two Contingency Odds RatioTwo-by-Two Contingency Odds Ratio: solve product of concordant diagonal countsTwo-by-Two Contingency Odds Ratio: solve product of discordant diagonal countsYouden Diagnostic IndexYouden Diagnostic Index: solve sensitivity-plus-specificityYouden Diagnostic Index: solve unit probability scaleClassification Accuracy PercentageClassification Accuracy Percentage: solve correct classification countClassification Accuracy Percentage: solve total classified countCohen Kappa Chance-Corrected AgreementCohen Kappa Chance-Corrected Agreement: solve observed agreement proportionCohen Kappa Chance-Corrected Agreement: solve chance-expected agreement proportionMatthews Correlation from Grouped TermsMatthews Correlation from Grouped Terms: solve confusion-matrix determinant numeratorMatthews Correlation from Grouped Terms: solve positive geometric denominatorSurvey Response Rate PercentageSurvey Response Rate Percentage: solve completed response countSurvey Response Rate Percentage: solve eligible contacted countPooled Standard Deviation from Sum of SquaresPooled Standard Deviation from Sum of Squares: solve pooled within-group squared-deviation sumPooled Standard Deviation from Sum of Squares: solve pooled degrees of freedomKish Effective Sample Size from Weight SumsKish Effective Sample Size from Weight Sums: solve sum of survey weightsKish Effective Sample Size from Weight Sums: solve square root of summed squared weightsModel Generalization Loss GapModel Generalization Loss Gap: solve validation or test lossModel Generalization Loss Gap: solve training lossTraining Example ThroughputTraining Example Throughput: solve processed example countTraining Example Throughput: solve elapsed training timeAssociation-Rule Support PercentageAssociation-Rule Support Percentage: solve transactions containing the complete itemsetAssociation-Rule Support Percentage: solve total analyzed transactionsAssociation-Rule Confidence PercentageAssociation-Rule Confidence Percentage: solve transactions containing antecedent and consequentAssociation-Rule Confidence Percentage: solve transactions containing antecedentAssociation-Rule LiftAssociation-Rule Lift: solve rule confidence as a decimalAssociation-Rule Lift: solve consequent marginal support as a decimalAssociation-Rule LeverageAssociation-Rule Leverage: solve joint antecedent-consequent supportAssociation-Rule Leverage: solve product of marginal supportsAssociation-Rule ConvictionAssociation-Rule Conviction: solve one minus consequent supportAssociation-Rule Conviction: solve one minus rule confidenceClassification False-Positive Rate PercentageClassification False-Positive Rate Percentage: solve false positive countClassification False-Positive Rate Percentage: solve actual negative countClassification False-Negative Rate PercentageClassification False-Negative Rate Percentage: solve false negative countClassification False-Negative Rate Percentage: solve actual positive countNegative Diagnostic Likelihood RatioNegative Diagnostic Likelihood Ratio: solve false-negative rate as a decimalNegative Diagnostic Likelihood Ratio: solve specificity as a decimalFowlkes–Mallows Classification IndexFowlkes–Mallows Classification Index: solve positive predictive value as a decimalFowlkes–Mallows Classification Index: solve sensitivity as a decimalBalanced Classification Error RateBalanced Classification Error Rate: solve false-positive rateBalanced Classification Error Rate: solve false-negative rateKaplan–Meier Conditional Survival FactorKaplan–Meier Conditional Survival Factor: solve events at the time pointKaplan–Meier Conditional Survival Factor: solve individuals at risk immediately beforeKaplan–Meier Survival Probability UpdateKaplan–Meier Survival Probability Update: solve survival probability before time pointKaplan–Meier Survival Probability Update: solve conditional survival factorNelson–Aalen Cumulative-Hazard IncrementNelson–Aalen Cumulative-Hazard Increment: solve events at the time pointNelson–Aalen Cumulative-Hazard Increment: solve individuals at risk immediately beforeNelson–Aalen Cumulative-Hazard UpdateNelson–Aalen Cumulative-Hazard Update: solve cumulative hazard before time pointNelson–Aalen Cumulative-Hazard Update: solve current hazard incrementSingle-Event Cumulative Incidence from SurvivalSingle-Event Cumulative Incidence from Survival: solve unit probability totalSingle-Event Cumulative Incidence from Survival: solve event-free survival probabilityStandardized Mortality RatioStandardized Mortality Ratio: solve observed event countStandardized Mortality Ratio: solve expected event count under standard ratesRestricted Mean Survival-Time PercentageRestricted Mean Survival-Time Percentage: solve area under survival curve to horizonRestricted Mean Survival-Time Percentage: solve restriction time horizonMedian Survival Bracket WidthMedian Survival Bracket Width: solve upper time bracketing survival one-halfMedian Survival Bracket Width: solve lower time bracketing survival one-halfCompeting-Risk Incidence RemainderCompeting-Risk Incidence Remainder: solve total cumulative incidence across causesCompeting-Risk Incidence Remainder: solve sum of known cause-specific incidencesAverage Person-Time Follow-UpAverage Person-Time Follow-Up: solve total observed person-timeAverage Person-Time Follow-Up: solve contributing participant countMeta-Analysis Normalized Study Weight PercentageMeta-Analysis Normalized Study Weight Percentage: solve study inverse-variance weightMeta-Analysis Normalized Study Weight Percentage: solve sum of all study weightsRandom-Effects Meta-Analysis Total VarianceRandom-Effects Meta-Analysis Total Variance: solve study within-sampling varianceRandom-Effects Meta-Analysis Total Variance: solve between-study variance tau squaredCochran Q Heterogeneity ExcessCochran Q Heterogeneity Excess: solve Cochran Q statisticCochran Q Heterogeneity Excess: solve heterogeneity degrees of freedomMeta-Analysis I-Squared Heterogeneity FractionMeta-Analysis I-Squared Heterogeneity Fraction: solve heterogeneity degrees of freedomMeta-Analysis I-Squared Heterogeneity Fraction: solve Cochran Q statisticFixed-Effect Pooled-Mean VarianceFixed-Effect Pooled-Mean Variance: solve unit reciprocal scaleFixed-Effect Pooled-Mean Variance: solve sum of inverse-variance weightsBayesian Credible-Interval WidthBayesian Credible-Interval Width: solve upper credible boundBayesian Credible-Interval Width: solve lower credible boundMCMC Proposal Acceptance Rate PercentageMCMC Proposal Acceptance Rate Percentage: solve accepted proposal countMCMC Proposal Acceptance Rate Percentage: solve total proposal countMCMC Integrated Autocorrelation FactorMCMC Integrated Autocorrelation Factor: solve post-warmup draw countMCMC Integrated Autocorrelation Factor: solve effective sample sizeMCMC Effective Sample Fraction PercentageMCMC Effective Sample Fraction Percentage: solve effective sample sizeMCMC Effective Sample Fraction Percentage: solve post-warmup draw countMonte Carlo Draw ThroughputMonte Carlo Draw Throughput: solve completed simulation drawsMonte Carlo Draw Throughput: solve elapsed runtimeMCMC R-Hat Excess above Convergence BaselineMCMC R-Hat Excess above Convergence Baseline: solve estimated rank-normalized R-hatMCMC R-Hat Excess above Convergence Baseline: solve unit convergence baselineMCMC Post-Warm-Up Draw CountMCMC Post-Warm-Up Draw Count: solve total generated drawsMCMC Post-Warm-Up Draw Count: solve discarded warm-up drawsMCMC Thinned Draw CountMCMC Thinned Draw Count: solve post-warmup draw countMCMC Thinned Draw Count: solve thinning intervalEqual-Cluster Sampling Design EffectEqual-Cluster Sampling Design Effect: solve cluster size minus oneEqual-Cluster Sampling Design Effect: solve intraclass correlation coefficientLincoln–Petersen Capture–Recapture PopulationLincoln–Petersen Capture–Recapture Population: solve product of first and second capture countsLincoln–Petersen Capture–Recapture Population: solve recaptured marked countHorvitz–Thompson Weighted ContributionHorvitz–Thompson Weighted Contribution: solve observed sampled-unit valueHorvitz–Thompson Weighted Contribution: solve unit inclusion probabilityContinuous-Uniform Distribution RangeContinuous-Uniform Distribution Range: solve upper support endpointContinuous-Uniform Distribution Range: solve lower support endpointContinuous-Uniform Standard DeviationContinuous-Uniform Standard Deviation: solve support range widthContinuous-Uniform Standard Deviation: solve square root of twelveNormal Variance from Standard DeviationNormal Variance from Standard Deviation: solve normal standard deviationNormal Variance from Standard Deviation: solve unit variance scaleLognormal Distribution MedianLognormal Distribution Median: solve unit positive scaleLognormal Distribution Median: solve underlying normal location muLognormal Geometric Standard DeviationLognormal Geometric Standard Deviation: solve unit positive scaleLognormal Geometric Standard Deviation: solve underlying normal scale sigmaTwo-Source Combined Standard UncertaintyTwo-Source Combined Standard Uncertainty: solve first independent standard uncertaintyTwo-Source Combined Standard Uncertainty: solve second independent standard uncertaintyRelative Standard Uncertainty PercentageRelative Standard Uncertainty Percentage: solve standard uncertainty magnitudeRelative Standard Uncertainty Percentage: solve measured-value magnitudeExpanded Measurement UncertaintyExpanded Measurement Uncertainty: solve coverage factorExpanded Measurement Uncertainty: solve combined standard uncertaintySensitivity-Coefficient Uncertainty ContributionSensitivity-Coefficient Uncertainty Contribution: solve absolute sensitivity coefficientSensitivity-Coefficient Uncertainty Contribution: solve input standard uncertaintyInput Variance Contribution to Uncertainty BudgetInput Variance Contribution to Uncertainty Budget: solve squared sensitivity coefficientInput Variance Contribution to Uncertainty Budget: solve input standard uncertaintyCalibration Indication CorrectionCalibration Indication Correction: solve reference standard valueCalibration Indication Correction: solve instrument indicated valueMeasurement Tolerance Utilization PercentageMeasurement Tolerance Utilization Percentage: solve absolute measurement error or uncertaintyMeasurement Tolerance Utilization Percentage: solve allowed tolerance magnitudeGuard-Banded Acceptance LimitGuard-Banded Acceptance Limit: solve specification tolerance magnitudeGuard-Banded Acceptance Limit: solve measurement guard-band magnitudeSignal-to-Uncertainty RatioSignal-to-Uncertainty Ratio: solve signal or measurand magnitudeSignal-to-Uncertainty Ratio: solve standard uncertainty magnitudeRepeatability Type-A Degrees of FreedomRepeatability Type-A Degrees of Freedom: solve independent repeat measurement countRepeatability Type-A Degrees of Freedom: solve estimated-mean parameter countNormalized Discounted Cumulative GainNormalized Discounted Cumulative Gain: solve discounted cumulative gainNormalized Discounted Cumulative Gain: solve ideal discounted cumulative gainMean Average PrecisionMean Average Precision: solve sum of per-query average precisionsMean Average Precision: solve evaluated query countMean Reciprocal RankMean Reciprocal Rank: solve sum of first-relevant reciprocal ranksMean Reciprocal Rank: solve evaluated query countDiscounted Cumulative-Gain Rank ContributionDiscounted Cumulative-Gain Rank Contribution: solve relevance gain at selected rankDiscounted Cumulative-Gain Rank Contribution: solve rank discount denominatorBinary-Log Ranking DiscountBinary-Log Ranking Discount: solve rank plus oneBinary-Log Ranking Discount: solve binary logarithm base twoFirst-Relevant Reciprocal-Rank ScoreFirst-Relevant Reciprocal-Rank Score: solve unit relevance scoreFirst-Relevant Reciprocal-Rank Score: solve first relevant result rankAverage-Precision Relevant-Rank ContributionAverage-Precision Relevant-Rank Contribution: solve precision at selected rankAverage-Precision Relevant-Rank Contribution: solve binary relevance indicatorRank-Biased Precision Persistence WeightRank-Biased Precision Persistence Weight: solve persistence probabilityRank-Biased Precision Persistence Weight: solve zero-based rank exponentExpected Reciprocal-Rank Continuation ContributionExpected Reciprocal-Rank Continuation Contribution: solve probability user reaches selected rankExpected Reciprocal-Rank Continuation Contribution: solve relevance or satisfaction probability at rankPairwise Ranking Concordance PercentagePairwise Ranking Concordance Percentage: solve correctly ordered comparable pairsPairwise Ranking Concordance Percentage: solve total comparable pairsTime-Series Simple Moving AverageTime-Series Simple Moving Average: solve sum of values in current windowTime-Series Simple Moving Average: solve observations in windowMean Forecast ErrorMean Forecast Error: solve sum of signed forecast errorsMean Forecast Error: solve forecast observation countForecast Mean Absolute Percentage ErrorForecast Mean Absolute Percentage Error: solve sum of absolute percentage errorsForecast Mean Absolute Percentage Error: solve eligible forecast countSymmetric Mean Absolute Percentage ErrorSymmetric Mean Absolute Percentage Error: solve one-hundred times summed absolute errorsSymmetric Mean Absolute Percentage Error: solve sum of absolute actual-plus-forecast magnitudesForecast Tracking SignalForecast Tracking Signal: solve cumulative signed forecast errorForecast Tracking Signal: solve mean absolute deviationTheil Forecast U RatioTheil Forecast U Ratio: solve model forecast RMSETheil Forecast U Ratio: solve naive benchmark RMSEMultiplicative Seasonal IndexMultiplicative Seasonal Index: solve season-specific averageMultiplicative Seasonal Index: solve overall series averageLag Autocorrelation CoefficientLag Autocorrelation Coefficient: solve lagged autocovarianceLag Autocorrelation Coefficient: solve zero-lag varianceForecast Skill Score against ReferenceForecast Skill Score against Reference: solve model error scoreForecast Skill Score against Reference: solve reference forecast error scoreTime-Series Resampling Interval FactorTime-Series Resampling Interval Factor: solve new sampling intervalTime-Series Resampling Interval Factor: solve original sampling intervalMoran Spatial Autocorrelation IndexMoran Spatial Autocorrelation Index: solve normalized weighted cross-deviation numeratorMoran Spatial Autocorrelation Index: solve squared-deviation denominatorGeary Spatial Autocorrelation RatioGeary Spatial Autocorrelation Ratio: solve normalized weighted pairwise squared-difference sumGeary Spatial Autocorrelation Ratio: solve squared-deviation reference sumGetis–Ord Spatial Standard ScoreGetis–Ord Spatial Standard Score: solve observed-minus-expected spatial statisticGetis–Ord Spatial Standard Score: solve null standard deviationNearest-Neighbor Spatial Pattern IndexNearest-Neighbor Spatial Pattern Index: solve observed mean nearest-neighbor distanceNearest-Neighbor Spatial Pattern Index: solve complete-spatial-randomness expected distanceSpatial Point DensitySpatial Point Density: solve observed point countSpatial Point Density: solve study-region areaEmpirical Semivariogram ValueEmpirical Semivariogram Value: solve mean squared paired difference at lagEmpirical Semivariogram Value: solve unit semivariance scaleVariogram Nugget-to-Sill PercentageVariogram Nugget-to-Sill Percentage: solve variogram nugget varianceVariogram Nugget-to-Sill Percentage: solve total sill varianceVariogram Range Utilization PercentageVariogram Range Utilization Percentage: solve selected separation distanceVariogram Range Utilization Percentage: solve modeled variogram rangeRipley K from Normalized Pair CountRipley K from Normalized Pair Count: solve study area times weighted neighbor-pair sumRipley K from Normalized Pair Count: solve ordered point-pair countLocal Spatial Cross-Product ContributionLocal Spatial Cross-Product Contribution: solve standardized focal observationLocal Spatial Cross-Product Contribution: solve weighted neighbor-score sumMachine-Time Utilization PercentageMachine-Time Utilization Percentage: solve busy processing timeMachine-Time Utilization Percentage: solve available scheduled machine timeSchedule Productive-Time Efficiency PercentageSchedule Productive-Time Efficiency Percentage: solve value-adding processing timeSchedule Productive-Time Efficiency Percentage: solve total elapsed flow timeSchedule Average Job Flow TimeSchedule Average Job Flow Time: solve sum of job flow timesSchedule Average Job Flow Time: solve completed job countCache Miss Rate PercentageCache Miss Rate Percentage: solve cache miss countCache Miss Rate Percentage: solve total cache access countSharpe Risk-Adjusted Return RatioSharpe Risk-Adjusted Return Ratio: solve portfolio return minus risk-free returnSharpe Risk-Adjusted Return Ratio: solve portfolio return standard deviationSortino Downside-Risk RatioSortino Downside-Risk Ratio: solve portfolio return minus target returnSortino Downside-Risk Ratio: solve downside deviationActive Return Information RatioActive Return Information Ratio: solve portfolio minus benchmark active returnActive Return Information Ratio: solve tracking-error standard deviationTreynor Systematic-Risk RatioTreynor Systematic-Risk Ratio: solve portfolio return minus risk-free returnTreynor Systematic-Risk Ratio: solve portfolio betaJensen Performance AlphaJensen Performance Alpha: solve observed portfolio returnJensen Performance Alpha: solve CAPM-implied expected returnSquare-Root-of-Time Value-at-Risk ScaleSquare-Root-of-Time Value-at-Risk Scale: solve one-period VaR magnitudeSquare-Root-of-Time Value-at-Risk Scale: solve square root of target period countExpected-Shortfall-to-VaR RatioExpected-Shortfall-to-VaR Ratio: solve expected shortfall magnitudeExpected-Shortfall-to-VaR Ratio: solve value-at-risk magnitude at same levelDownside Market Capture RatioDownside Market Capture Ratio: solve portfolio compounded return in benchmark-down periodsDownside Market Capture Ratio: solve benchmark compounded return in down periodsUpside Market Capture RatioUpside Market Capture Ratio: solve portfolio compounded return in benchmark-up periodsUpside Market Capture Ratio: solve benchmark compounded return in up periodsMaximum Drawdown PercentageMaximum Drawdown Percentage: solve peak value minus subsequent trough valueMaximum Drawdown Percentage: solve peak portfolio valueMeasurement Process BiasMeasurement Process Bias: solve mean measured valueMeasurement Process Bias: solve accepted reference valueInstrument Full-Scale Error PercentageInstrument Full-Scale Error Percentage: solve absolute indication errorInstrument Full-Scale Error Percentage: solve instrument full-scale spanRepeatability-to-Tolerance PercentageRepeatability-to-Tolerance Percentage: solve repeatability variation magnitudeRepeatability-to-Tolerance Percentage: solve allowed tolerance widthReproducibility-to-Tolerance PercentageReproducibility-to-Tolerance Percentage: solve reproducibility variation magnitudeReproducibility-to-Tolerance Percentage: solve allowed tolerance widthGauge R&R Variance ShareGauge R&R Variance Share: solve measurement-system varianceGauge R&R Variance Share: solve total observed varianceMeasurement Resolution-to-Tolerance RatioMeasurement Resolution-to-Tolerance Ratio: solve instrument resolution incrementMeasurement Resolution-to-Tolerance Ratio: solve allowed tolerance widthCalibration Sensitivity CoefficientCalibration Sensitivity Coefficient: solve output indication changeCalibration Sensitivity Coefficient: solve reference input changeCalibration Zero OffsetCalibration Zero Offset: solve indicated zero-level outputCalibration Zero Offset: solve ideal zero-level outputManufacturing Defect RateManufacturing Defect Rate: solve defective unit countManufacturing Defect Rate: solve inspected unit countManufacturing First-Pass YieldManufacturing First-Pass Yield: solve units passing without reworkManufacturing First-Pass Yield: solve units entering processCarbon Emissions IntensityCarbon Emissions Intensity: solve greenhouse-gas emissionsCarbon Emissions Intensity: solve useful output quantityEnergy Use IntensityEnergy Use Intensity: solve site or source energy consumedEnergy Use Intensity: solve served floor area or output quantityWater Use IntensityWater Use Intensity: solve water withdrawn or consumedWater Use Intensity: solve served output quantityWaste Diversion RateWaste Diversion Rate: solve waste diverted from disposalWaste Diversion Rate: solve total generated wasteManufacturing Material YieldManufacturing Material Yield: solve material mass in conforming productManufacturing Material Yield: solve total material input massManufacturing Scrap RateManufacturing Scrap Rate: solve scrapped material massManufacturing Scrap Rate: solve total material input massRenewable Energy ShareRenewable Energy Share: solve renewable energy suppliedRenewable Energy Share: solve total energy suppliedLand-Use Output DensityLand-Use Output Density: solve useful annual outputLand-Use Output Density: solve occupied land areaAvoided Emissions PercentageAvoided Emissions Percentage: solve baseline minus project emissionsAvoided Emissions Percentage: solve baseline emissionsCircular Material Use RateCircular Material Use Rate: solve secondary material inputCircular Material Use Rate: solve total material inputSolution Mass ConcentrationSolution Mass Concentration: solve solute massSolution Mass Concentration: solve solution volumeSolution Volume Concentration PercentageSolution Volume Concentration Percentage: solve solute volumeSolution Volume Concentration Percentage: solve final solution volumeSolution Mass Fraction PercentageSolution Mass Fraction Percentage: solve solute massSolution Mass Fraction Percentage: solve total solution massCell Suspension Number ConcentrationCell Suspension Number Concentration: solve counted cell quantityCell Suspension Number Concentration: solve counted sample volumeLaboratory Plating Efficiency PercentageLaboratory Plating Efficiency Percentage: solve colony-forming units recoveredLaboratory Plating Efficiency Percentage: solve viable units platedAnalytical Assay Recovery PercentageAnalytical Assay Recovery Percentage: solve measured recovered amountAnalytical Assay Recovery Percentage: solve known added amountSpectrophotometer Transmittance PercentageSpectrophotometer Transmittance Percentage: solve transmitted radiant intensitySpectrophotometer Transmittance Percentage: solve incident radiant intensityLaboratory Replicate Relative RangeLaboratory Replicate Relative Range: solve replicate result rangeLaboratory Replicate Relative Range: solve replicate result meanAircraft Payload FractionAircraft Payload Fraction: solve payload massAircraft Payload Fraction: solve takeoff massAircraft Fuel-Specific RangeAircraft Fuel-Specific Range: solve air distance travelledAircraft Fuel-Specific Range: solve fuel quantity consumedField Plant Population DensityField Plant Population Density: solve established plant countField Plant Population Density: solve field areaCrop Row Plant Linear DensityCrop Row Plant Linear Density: solve plants counted along rowsCrop Row Plant Linear Density: solve sampled row lengthField Seeding Rate by AreaField Seeding Rate by Area: solve seed quantity sownField Seeding Rate by Area: solve planted field areaGermination-Adjusted Seed CountGermination-Adjusted Seed Count: solve target established plant countGermination-Adjusted Seed Count: solve expected germination percentageFertilizer Nutrient MassFertilizer Nutrient Mass: solve fertilizer product massFertilizer Nutrient Mass: solve labelled nutrient percentageCrop Harvest IndexCrop Harvest Index: solve economic harvested yieldCrop Harvest Index: solve total above-ground crop biomassCrop Water ProductivityCrop Water Productivity: solve crop yieldCrop Water Productivity: solve water consumed or suppliedFabric Area DensityFabric Area Density: solve fabric specimen massFabric Area Density: solve fabric specimen areaWarp End DensityWarp End Density: solve warp ends countedWarp End Density: solve fabric width sampledWeft Pick DensityWeft Pick Density: solve weft picks countedWeft Pick Density: solve fabric length sampledFabric Cutting YieldFabric Cutting Yield: solve pattern-piece area cutFabric Cutting Yield: solve laid fabric areaFabric Roll Linear MassFabric Roll Linear Mass: solve fabric roll net massFabric Roll Linear Mass: solve fabric length on rollPaper Sheet GrammagePaper Sheet Grammage: solve paper specimen massPaper Sheet Grammage: solve paper specimen areaPrintable Sheet Area YieldPrintable Sheet Area Yield: solve usable printable areaPrintable Sheet Area Yield: solve total sheet areaPackaging-to-Product Mass RatioPackaging-to-Product Mass Ratio: solve packaging system massPackaging-to-Product Mass Ratio: solve contained product massWatershed Drainage DensityWatershed Drainage Density: solve mapped channel lengthWatershed Drainage Density: solve watershed areaStorm Runoff CoefficientStorm Runoff Coefficient: solve direct runoff depthStorm Runoff Coefficient: solve storm rainfall depthAquifer Specific DischargeAquifer Specific Discharge: solve groundwater volumetric dischargeAquifer Specific Discharge: solve flow cross-sectional areaStream Baseflow IndexStream Baseflow Index: solve estimated baseflow volumeStream Baseflow Index: solve total streamflow volumeTraffic Lane Capacity UtilizationTraffic Lane Capacity Utilization: solve observed or forecast lane demandTraffic Lane Capacity Utilization: solve practical lane capacityTraffic Time HeadwayTraffic Time Headway: solve observation timeTraffic Time Headway: solve vehicle passage countTraffic Space HeadwayTraffic Space Headway: solve occupied road lengthTraffic Space Headway: solve vehicle count on segmentIntersection Degree of SaturationIntersection Degree of Saturation: solve demand flow during analysis periodIntersection Degree of Saturation: solve available saturation-flow capacityTraffic Signal Effective Green RatioTraffic Signal Effective Green Ratio: solve effective green timeTraffic Signal Effective Green Ratio: solve signal cycle lengthRoad Travel Time IndexRoad Travel Time Index: solve congested travel timeRoad Travel Time Index: solve free-flow travel timeTraffic Buffer Time PercentageTraffic Buffer Time Percentage: solve extra time allowanceTraffic Buffer Time Percentage: solve typical trip timeFreight Vehicle Load FactorFreight Vehicle Load Factor: solve carried freight massFreight Vehicle Load Factor: solve rated freight capacityRunning Race PaceRunning Race Pace: solve elapsed running timeRunning Race Pace: solve distance completedCycling Power-to-Mass RatioCycling Power-to-Mass Ratio: solve sustained cycling powerCycling Power-to-Mass Ratio: solve rider-system massSwimming Stroke RateSwimming Stroke Rate: solve stroke countSwimming Stroke Rate: solve elapsed swimming timeSwimming Distance per StrokeSwimming Distance per Stroke: solve distance swumSwimming Distance per Stroke: solve stroke countSession RPE Training LoadSession RPE Training Load: solve session durationSession RPE Training Load: solve session rating of perceived exertionTeam Possession ShareTeam Possession Share: solve team possession timeTeam Possession Share: solve total measured possession timeForest Stand Tree DensityForest Stand Tree Density: solve live tree countForest Stand Tree Density: solve sampled land areaForest Basal Area DensityForest Basal Area Density: solve summed tree basal areaForest Basal Area Density: solve sampled land areaForest Canopy Cover PercentageForest Canopy Cover Percentage: solve ground area vertically covered by canopyForest Canopy Cover Percentage: solve surveyed ground areaRock Porosity PercentageRock Porosity Percentage: solve connected and isolated pore volumeRock Porosity Percentage: solve bulk specimen volumeRock and Soil Void RatioRock and Soil Void Ratio: solve void volumeRock and Soil Void Ratio: solve solid-grain volumeGeotechnical Dry Bulk DensityGeotechnical Dry Bulk Density: solve oven-dry solid massGeotechnical Dry Bulk Density: solve bulk specimen volumeOre Mass Grade PercentageOre Mass Grade Percentage: solve contained valuable-material massOre Mass Grade Percentage: solve total ore massDrill Core Recovery PercentageDrill Core Recovery Percentage: solve recovered core lengthDrill Core Recovery Percentage: solve drilled core-run lengthAudio Crest FactorAudio Crest Factor: solve peak signal magnitudeAudio Crest Factor: solve root-mean-square magnitudeAudio Data Compression RatioAudio Data Compression Ratio: solve uncompressed audio sizeAudio Data Compression Ratio: solve compressed audio sizeVessel Deadweight UtilizationVessel Deadweight Utilization: solve carried deadweight itemsVessel Deadweight Utilization: solve rated deadweight capacityPort Berth Occupancy PercentagePort Berth Occupancy Percentage: solve occupied berth timePort Berth Occupancy Percentage: solve available berth timeCargo Stowage FactorCargo Stowage Factor: solve cargo occupied volumeCargo Stowage Factor: solve cargo massContainer Slot UtilizationContainer Slot Utilization: solve occupied container slotsContainer Slot Utilization: solve available container slotsPort Cargo Handling RatePort Cargo Handling Rate: solve cargo quantity transferredPort Cargo Handling Rate: solve active handling timeVessel Fuel Consumption per Nautical MileVessel Fuel Consumption per Nautical Mile: solve voyage fuel consumedVessel Fuel Consumption per Nautical Mile: solve nautical miles travelledAtmospheric Water-Vapor Mixing RatioAtmospheric Water-Vapor Mixing Ratio: solve water-vapor massAtmospheric Water-Vapor Mixing Ratio: solve dry-air massSpecific Humidity PercentageSpecific Humidity Percentage: solve water-vapor massSpecific Humidity Percentage: solve total moist-air massMeteorological Snow-to-Liquid RatioMeteorological Snow-to-Liquid Ratio: solve fresh snow depthMeteorological Snow-to-Liquid Ratio: solve melted liquid-water-equivalent depthLighting Minimum-to-Average UniformityLighting Minimum-to-Average Uniformity: solve minimum measured illuminanceLighting Minimum-to-Average Uniformity: solve average measured illuminanceArchitectural Daylight FactorArchitectural Daylight Factor: solve indoor daylight illuminanceArchitectural Daylight Factor: solve simultaneous unobstructed outdoor illuminanceLuminaire Utilization CoefficientLuminaire Utilization Coefficient: solve luminous flux reaching workplaneLuminaire Utilization Coefficient: solve luminous flux emitted by luminairesLuminaire Lumen MaintenanceLuminaire Lumen Maintenance: solve maintained luminous fluxLuminaire Lumen Maintenance: solve initial luminous fluxBuilding Lighting Power DensityBuilding Lighting Power Density: solve installed lighting electrical powerBuilding Lighting Power Density: solve served floor areaDough Hydration PercentageDough Hydration Percentage: solve water mass in dough formulaDough Hydration Percentage: solve total flour massFrozen Dessert Volume OverrunFrozen Dessert Volume Overrun: solve initial liquid-mix volumeFrozen Dessert Volume Overrun: solve final aerated product volumeFood Wet-Basis Moisture ContentFood Wet-Basis Moisture Content: solve water mass in sampleFood Wet-Basis Moisture Content: solve total wet sample massFood Dry-Basis Moisture ContentFood Dry-Basis Moisture Content: solve water mass in sampleFood Dry-Basis Moisture Content: solve dry-solids massFood Water ActivityFood Water Activity: solve food equilibrium vapor pressureFood Water Activity: solve pure-water vapor pressure at same temperatureFood Ingredient Inclusion RateFood Ingredient Inclusion Rate: solve selected ingredient massFood Ingredient Inclusion Rate: solve total finished batch massCooking Mass RetentionCooking Mass Retention: solve cooked product massCooking Mass Retention: solve raw product massFood Concentration FactorFood Concentration Factor: solve final marker concentrationFood Concentration Factor: solve initial marker concentrationPhotovoltaic Array Nameplate Power DensityPhotovoltaic Array Nameplate Power Density: solve array nameplate powerPhotovoltaic Array Nameplate Power Density: solve occupied module aperture areaSolar-Energy Specific YieldSolar-Energy Specific Yield: solve generated electrical energySolar-Energy Specific Yield: solve installed photovoltaic nameplate powerSolar Photovoltaic Performance RatioSolar Photovoltaic Performance Ratio: solve measured final system yieldSolar Photovoltaic Performance Ratio: solve reference irradiance yieldSolar Inverter Loading RatioSolar Inverter Loading Ratio: solve photovoltaic DC nameplate capacitySolar Inverter Loading Ratio: solve inverter AC nameplate capacityWind-Turbine Tip-Speed RatioWind-Turbine Tip-Speed Ratio: solve rotor blade-tip linear speedWind-Turbine Tip-Speed Ratio: solve undisturbed wind speedWind-Turbine Capacity FactorWind-Turbine Capacity Factor: solve actual electrical energy generatedWind-Turbine Capacity Factor: solve nameplate energy over same periodBattery Charge or Discharge C-RateBattery Charge or Discharge C-Rate: solve charge or discharge currentBattery Charge or Discharge C-Rate: solve rated ampere-hour capacityBattery Depth of DischargeBattery Depth of Discharge: solve charge or energy withdrawnBattery Depth of Discharge: solve full rated charge or energyBattery State of ChargeBattery State of Charge: solve estimated remaining charge or energyBattery State of Charge: solve matched full-capacity estimateBattery Round-Trip Energy EfficiencyBattery Round-Trip Energy Efficiency: solve electrical energy returned on dischargeBattery Round-Trip Energy Efficiency: solve electrical energy supplied during chargeBattery Gravimetric Specific EnergyBattery Gravimetric Specific Energy: solve stored or usable battery energyBattery Gravimetric Specific Energy: solve battery-system massLivestock Dry-Matter Intake as Body-Mass PercentageLivestock Dry-Matter Intake as Body-Mass Percentage: solve daily feed dry-matter intakeLivestock Dry-Matter Intake as Body-Mass Percentage: solve animal live body massLivestock Water Intake per AnimalLivestock Water Intake per Animal: solve measured herd water consumptionLivestock Water Intake per Animal: solve animals representedAquaculture Biomass Stocking DensityAquaculture Biomass Stocking Density: solve live aquatic-animal biomassAquaculture Biomass Stocking Density: solve effective culture-water volumeDairy Milk Yield per Lactation DayDairy Milk Yield per Lactation Day: solve milk volume or mass producedDairy Milk Yield per Lactation Day: solve represented lactation daysDairy Milk-Fat YieldDairy Milk-Fat Yield: solve milk mass producedDairy Milk-Fat Yield: solve milk fat percentageLayer-Hen Egg Production RateLayer-Hen Egg Production Rate: solve saleable eggs producedLayer-Hen Egg Production Rate: solve available hen-daysPasture Animal-Unit Stocking RatePasture Animal-Unit Stocking Rate: solve animal units carriedPasture Animal-Unit Stocking Rate: solve grazeable pasture areaAquaculture Daily Feed-to-Biomass RateAquaculture Daily Feed-to-Biomass Rate: solve daily feed offeredAquaculture Daily Feed-to-Biomass Rate: solve standing live biomassWarehouse Storage-Location OccupancyWarehouse Storage-Location Occupancy: solve occupied storage locationsWarehouse Storage-Location Occupancy: solve usable storage locationsWarehouse Pallet-Position DensityWarehouse Pallet-Position Density: solve usable pallet-position countWarehouse Pallet-Position Density: solve warehouse floor area assignedWarehouse Order-Line Pick RateWarehouse Order-Line Pick Rate: solve completed accurate order linesWarehouse Order-Line Pick Rate: solve active picking labour timeWarehouse Dock Load ThroughputWarehouse Dock Load Throughput: solve inbound or outbound loads handledWarehouse Dock Load Throughput: solve active dock operating timeWarehouse Travel Distance per PickWarehouse Travel Distance per Pick: solve total picker or vehicle travel distanceWarehouse Travel Distance per Pick: solve completed pick countWarehouse Immediate Storage SelectivityWarehouse Immediate Storage Selectivity: solve loads directly accessible without moving another loadWarehouse Immediate Storage Selectivity: solve total stored loadsWarehouse Cross-Dock Flow ShareWarehouse Cross-Dock Flow Share: solve received units transferred directly toward outbound flowWarehouse Cross-Dock Flow Share: solve total received unitsWarehouse Order Consolidation FactorWarehouse Order Consolidation Factor: solve picked handling units combinedWarehouse Order Consolidation Factor: solve completed outbound order countRoom Mean Sound-Absorption CoefficientRoom Mean Sound-Absorption Coefficient: solve total equivalent sound-absorption areaRoom Mean Sound-Absorption Coefficient: solve total room boundary surface areaAcoustic Band Edge Frequency RatioAcoustic Band Edge Frequency Ratio: solve upper band-edge frequencyAcoustic Band Edge Frequency Ratio: solve lower band-edge frequencyOccupational Noise Dose PercentageOccupational Noise Dose Percentage: solve accumulated weighted noise exposureOccupational Noise Dose Percentage: solve allowable weighted noise exposureSeismic P-to-S Wave Velocity RatioSeismic P-to-S Wave Velocity Ratio: solve compressional P-wave speedSeismic P-to-S Wave Velocity Ratio: solve shear S-wave speedSeismic Resonance Quality FactorSeismic Resonance Quality Factor: solve resonance centre frequencySeismic Resonance Quality Factor: solve resonance bandwidthWastewater Hydraulic Surface LoadingWastewater Hydraulic Surface Loading: solve wastewater volumetric flow rateWastewater Hydraulic Surface Loading: solve treatment surface areaWastewater Organic Volumetric LoadingWastewater Organic Volumetric Loading: solve influent organic mass load per timeWastewater Organic Volumetric Loading: solve active reactor volumeWastewater Treatment Removal FractionWastewater Treatment Removal Fraction: solve effluent constituent concentrationWastewater Treatment Removal Fraction: solve influent constituent concentrationActivated-Sludge Food-to-Microorganism RatioActivated-Sludge Food-to-Microorganism Ratio: solve biodegradable substrate mass load per timeActivated-Sludge Food-to-Microorganism Ratio: solve active biomass inventoryWastewater Oxygen Transfer EfficiencyWastewater Oxygen Transfer Efficiency: solve oxygen mass transferred to liquidWastewater Oxygen Transfer Efficiency: solve oxygen mass suppliedReturn Activated-Sludge Flow RatioReturn Activated-Sludge Flow Ratio: solve return activated-sludge flow rateReturn Activated-Sludge Flow Ratio: solve influent wastewater flow rateWastewater Sludge Volume IndexWastewater Sludge Volume Index: solve settled sludge volume per sample volumeWastewater Sludge Volume Index: solve mixed-liquor suspended-solids concentrationMembrane Bioreactor Permeate FluxMembrane Bioreactor Permeate Flux: solve permeate volumetric flow rateMembrane Bioreactor Permeate Flux: solve active membrane areaHVAC Outdoor-Air FractionHVAC Outdoor-Air Fraction: solve outdoor-air volume flowHVAC Outdoor-Air Fraction: solve total supply-air volume flowHVAC Ventilation Airflow per PersonHVAC Ventilation Airflow per Person: solve outdoor ventilation airflowHVAC Ventilation Airflow per Person: solve represented occupant countHVAC Cooling Load DensityHVAC Cooling Load Density: solve design sensible and latent cooling loadHVAC Cooling Load Density: solve conditioned floor areaHVAC Coefficient of PerformanceHVAC Coefficient of Performance: solve useful heating or cooling rateHVAC Coefficient of Performance: solve equipment input powerHVAC Heat-Exchanger EffectivenessHVAC Heat-Exchanger Effectiveness: solve actual heat-transfer rateHVAC Heat-Exchanger Effectiveness: solve maximum thermodynamically possible rateHVAC Fan Specific PowerHVAC Fan Specific Power: solve fan-system electrical input powerHVAC Fan Specific Power: solve delivered air volume flow rateBipolar Transistor DC Current GainBipolar Transistor DC Current Gain: solve collector currentBipolar Transistor DC Current Gain: solve base currentSemiconductor Die Power DensitySemiconductor Die Power Density: solve die power dissipationSemiconductor Die Power Density: solve active die areaCrystal Lattice Relative StrainCrystal Lattice Relative Strain: solve lattice-parameter changeCrystal Lattice Relative Strain: solve reference lattice parameterCrystalline Phase Mass FractionCrystalline Phase Mass Fraction: solve selected crystalline-phase massCrystalline Phase Mass Fraction: solve total characterized specimen massCrystal Defect Number DensityCrystal Defect Number Density: solve counted or estimated crystal defectsCrystal Defect Number Density: solve characterized crystal volumeElectrochemical Coulombic EfficiencyElectrochemical Coulombic Efficiency: solve recoverable discharge chargeElectrochemical Coulombic Efficiency: solve charge supplied during chargingElectrode Specific Charge CapacityElectrode Specific Charge Capacity: solve reversible electrode charge capacityElectrode Specific Charge Capacity: solve active electrode-material massElectrolysis Specific Energy ConsumptionElectrolysis Specific Energy Consumption: solve electrical energy consumedElectrolysis Specific Energy Consumption: solve useful product massElectrochemical Faradaic Product YieldElectrochemical Faradaic Product Yield: solve measured target-product amountElectrochemical Faradaic Product Yield: solve theoretical Faradaic product amountAntenna Front-to-Back Power RatioAntenna Front-to-Back Power Ratio: solve radiated or received power in forward reference directionAntenna Front-to-Back Power Ratio: solve power in backward reference directionDigital Radio Spectral EfficiencyDigital Radio Spectral Efficiency: solve net information bit rateDigital Radio Spectral Efficiency: solve occupied channel bandwidthAntenna Aperture EfficiencyAntenna Aperture Efficiency: solve effective receiving aperture areaAntenna Aperture Efficiency: solve physical projected aperture areaRadio Transmitter Duty FactorRadio Transmitter Duty Factor: solve transmitter active on-timeRadio Transmitter Duty Factor: solve total observation timeRadionuclide Specific ActivityRadionuclide Specific Activity: solve sample radioactive activityRadionuclide Specific Activity: solve radionuclide or sample massRadioactive Activity ConcentrationRadioactive Activity Concentration: solve sample radioactive activityRadioactive Activity Concentration: solve represented sample volumeRadiation Half-Value-Layer CountRadiation Half-Value-Layer Count: solve shield material thicknessRadiation Half-Value-Layer Count: solve material half-value-layer thicknessCT Table Travel per RotationCT Table Travel per Rotation: solve total helical table travelCT Table Travel per Rotation: solve completed gantry rotationsPET Standardized Uptake ValuePET Standardized Uptake Value: solve decay-corrected tissue activity concentrationPET Standardized Uptake Value: solve injected activity per body-size measureMedical Image Contrast-to-Noise RatioMedical Image Contrast-to-Noise Ratio: solve absolute signal difference between target and backgroundMedical Image Contrast-to-Noise Ratio: solve noise standard deviationVibration Amplitude TransmissibilityVibration Amplitude Transmissibility: solve transmitted response amplitudeVibration Amplitude Transmissibility: solve input excitation amplitudeStructural Dynamic Amplification FactorStructural Dynamic Amplification Factor: solve peak or steady dynamic responseStructural Dynamic Amplification Factor: solve corresponding static responseModal Effective-Mass ParticipationModal Effective-Mass Participation: solve selected mode effective massModal Effective-Mass Participation: solve total physical mass in excitation directionStructural Strain-Energy DensityStructural Strain-Energy Density: solve stored elastic strain energyStructural Strain-Energy Density: solve material volume representedVibration Displacement-to-Clearance UtilizationVibration Displacement-to-Clearance Utilization: solve peak vibration displacement amplitudeVibration Displacement-to-Clearance Utilization: solve available mechanical clearanceMachinery Vibration Severity Reference RatioMachinery Vibration Severity Reference Ratio: solve measured vibration severity valueMachinery Vibration Severity Reference Ratio: solve selected reference or alarm limitAcoustic Equivalent Absorption AreaAcoustic Equivalent Absorption Area: solve represented surface areaAcoustic Equivalent Absorption Area: solve surface absorption coefficientAcoustic Crest FactorAcoustic Crest Factor: solve peak acoustic waveform magnitudeAcoustic Crest Factor: solve RMS acoustic waveform magnitudeWatershed Stream FrequencyWatershed Stream Frequency: solve count of mapped stream segmentsWatershed Stream Frequency: solve watershed areaWatershed Runoff CoefficientWatershed Runoff Coefficient: solve event runoff depthWatershed Runoff Coefficient: solve matched catchment precipitation depthStreamflow Flashiness IndexStreamflow Flashiness Index: solve sum of absolute consecutive flow changesStreamflow Flashiness Index: solve sum of flow observationsCatchment Sediment YieldCatchment Sediment Yield: solve exported sediment mass over periodCatchment Sediment Yield: solve catchment areaWater-Well Specific CapacityWater-Well Specific Capacity: solve stabilized pumping dischargeWater-Well Specific Capacity: solve associated drawdownChromatography Relative Retention TimeChromatography Relative Retention Time: solve analyte adjusted retention timeChromatography Relative Retention Time: solve reference adjusted retention timeChromatography Peak ResolutionChromatography Peak Resolution: solve retention-time separation between two peaksChromatography Peak Resolution: solve sum of the two baseline peak widthsChromatography Theoretical Plate HeightChromatography Theoretical Plate Height: solve effective column lengthChromatography Theoretical Plate Height: solve estimated theoretical plate countAnalytical Signal-to-Noise RatioAnalytical Signal-to-Noise Ratio: solve measured analytical signal magnitudeAnalytical Signal-to-Noise Ratio: solve matched noise magnitudeAnalytical Method RecoveryAnalytical Method Recovery: solve measured recovered analyte amountAnalytical Method Recovery: solve known analyte amount introducedMass-Spectrometry Resolving PowerMass-Spectrometry Resolving Power: solve mass-to-charge value of peakMass-Spectrometry Resolving Power: solve peak width at stated height criterionAnalytical Internal-Standard Response RatioAnalytical Internal-Standard Response Ratio: solve analyte instrument responseAnalytical Internal-Standard Response Ratio: solve internal-standard instrument responseRunning Vertical Oscillation RatioRunning Vertical Oscillation Ratio: solve vertical torso oscillation per strideRunning Vertical Oscillation Ratio: solve forward stride lengthRunning Ground-Contact Duty FactorRunning Ground-Contact Duty Factor: solve foot ground-contact durationRunning Ground-Contact Duty Factor: solve complete stride-cycle durationGait Step LengthGait Step Length: solve forward distance travelledGait Step Length: solve counted gait stepsGait Cycle DurationGait Cycle Duration: solve elapsed observation durationGait Cycle Duration: solve completed gait cyclesSwimming Stroke IndexSwimming Stroke Index: solve mean swimming speedSwimming Stroke Index: solve distance travelled per stroke cycleTraining Load MonotonyTraining Load Monotony: solve mean daily training loadTraining Load Monotony: solve standard deviation of daily training loadMuscle Specific TensionMuscle Specific Tension: solve estimated muscle forceMuscle Specific Tension: solve physiological cross-sectional areaEarthquake Focal-Depth FractionEarthquake Focal-Depth Fraction: solve earthquake focal depthEarthquake Focal-Depth Fraction: solve source-to-station hypocentral distanceSeismic Ray SlownessSeismic Ray Slowness: solve seismic travel timeSeismic Ray Slowness: solve represented ray-path distanceSeismic Damping Loss FactorSeismic Damping Loss Factor: solve energy dissipated per oscillation cycleSeismic Damping Loss Factor: solve twice-pi-scaled peak stored energySeismic Average Shear-Wave VelocitySeismic Average Shear-Wave Velocity: solve profile depth representedSeismic Average Shear-Wave Velocity: solve summed shear-wave travel time through profileMaterial Linear ShrinkageMaterial Linear Shrinkage: solve measured length reductionMaterial Linear Shrinkage: solve original reference lengthMaterial Surface-to-Volume RatioMaterial Surface-to-Volume Ratio: solve accessible material surface areaMaterial Surface-to-Volume Ratio: solve represented material volumeIndentation Mean Contact PressureIndentation Mean Contact Pressure: solve applied indentation loadIndentation Mean Contact Pressure: solve projected contact areaMaterial Fracture Energy DensityMaterial Fracture Energy Density: solve energy dissipated in crack formationMaterial Fracture Energy Density: solve new fracture surface areaCrystal Grain Aspect RatioCrystal Grain Aspect Ratio: solve characteristic grain major dimensionCrystal Grain Aspect Ratio: solve characteristic grain minor dimensionMaterial Directional Anisotropy RatioMaterial Directional Anisotropy Ratio: solve property measured in selected primary directionMaterial Directional Anisotropy Ratio: solve same property in comparison directionNaval Hull Block CoefficientNaval Hull Block Coefficient: solve moulded displacement volumeNaval Hull Block Coefficient: solve length-beam-draft bounding-box volumeNaval Prismatic CoefficientNaval Prismatic Coefficient: solve moulded displacement volumeNaval Prismatic Coefficient: solve maximum-section-area times selected lengthNaval Waterplane Area CoefficientNaval Waterplane Area Coefficient: solve actual waterplane areaNaval Waterplane Area Coefficient: solve waterline-length times maximum waterline beamNaval Froude NumberNaval Froude Number: solve vessel speedNaval Froude Number: solve square-root gravity-length reference speedNaval Reynolds NumberNaval Reynolds Number: solve vessel speed-length productNaval Reynolds Number: solve water kinematic viscosityPropeller Advance per RevolutionPropeller Advance per Revolution: solve vessel advance speed through waterPropeller Advance per Revolution: solve propeller revolutions per unit timeVessel Power-to-Displacement RatioVessel Power-to-Displacement Ratio: solve delivered or installed vessel powerVessel Power-to-Displacement Ratio: solve vessel displacement massSeed Application RateSeed Application Rate: solve seed mass appliedSeed Application Rate: solve seeded field areaCrop Field EmergenceCrop Field Emergence: solve emerged crop plants countedCrop Field Emergence: solve viable seeds placed in represented areaSoil Void RatioSoil Void Ratio: solve soil void-space volumeSoil Void Ratio: solve soil solid-particle volumeIrrigation Distribution UniformityIrrigation Distribution Uniformity: solve mean applied depth in low-quarter sampleIrrigation Distribution Uniformity: solve overall mean applied depthFertilizer Agronomic EfficiencyFertilizer Agronomic Efficiency: solve crop yield increase above unfertilized controlFertilizer Agronomic Efficiency: solve applied nutrient massSoil Dry Bulk DensitySoil Dry Bulk Density: solve oven-dry soil massSoil Dry Bulk Density: solve sample bulk volumeSoil Gravimetric Water ContentSoil Gravimetric Water Content: solve soil water mass lost on dryingSoil Gravimetric Water Content: solve oven-dry soil massSoil Volumetric Water ContentSoil Volumetric Water Content: solve water volume in represented sampleSoil Volumetric Water Content: solve total soil bulk volumeCrop Residue CoverCrop Residue Cover: solve sampling points intersecting crop residueCrop Residue Cover: solve total valid sampling pointsAircraft Horizontal-Tail Volume CoefficientAircraft Horizontal-Tail Volume Coefficient: solve horizontal-tail area-moment-arm productAircraft Horizontal-Tail Volume Coefficient: solve wing area-mean-chord productAircraft Wing Taper RatioAircraft Wing Taper Ratio: solve wing tip chordAircraft Wing Taper Ratio: solve wing root chordAircraft Specific Air RangeAircraft Specific Air Range: solve air distance flownAircraft Specific Air Range: solve fuel mass consumedAircraft Fuel Mass FractionAircraft Fuel Mass Fraction: solve usable fuel mass representedAircraft Fuel Mass Fraction: solve aircraft reference massRock Core RecoveryRock Core Recovery: solve total recovered core lengthRock Core Recovery: solve drilled core-run lengthRock Quality DesignationRock Quality Designation: solve sum of qualifying sound core-piece lengthsRock Quality Designation: solve drilled core-run lengthMine Stripping RatioMine Stripping Ratio: solve waste material quantityMine Stripping Ratio: solve ore material quantityBlast Powder FactorBlast Powder Factor: solve explosive charge massBlast Powder Factor: solve broken rock mass or volumeEcological Biomass DensityEcological Biomass Density: solve living biomass representedEcological Biomass Density: solve surveyed habitat areaPopulation Finite Growth FactorPopulation Finite Growth Factor: solve population size at end of intervalPopulation Finite Growth Factor: solve population size at start of intervalHabitat Edge DensityHabitat Edge Density: solve mapped habitat-edge lengthHabitat Edge Density: solve landscape area representedEcosystem Carbon Stock DensityEcosystem Carbon Stock Density: solve ecosystem carbon stock representedEcosystem Carbon Stock Density: solve land or water area representedFabric Areal DensityFabric Areal Density: solve conditioned fabric specimen massFabric Areal Density: solve measured specimen areaYarn Linear DensityYarn Linear Density: solve conditioned yarn massYarn Linear Density: solve measured yarn lengthTextile Thread DensityTextile Thread Density: solve counted warp or weft threadsTextile Thread Density: solve measured fabric widthTextile Process EfficiencyTextile Process Efficiency: solve accepted textile outputTextile Process Efficiency: solve theoretical or scheduled outputMeasurement Relative ErrorMeasurement Relative Error: solve signed measurement errorMeasurement Relative Error: solve reference quantity magnitudeMetrology Resolution-to-Tolerance RatioMetrology Resolution-to-Tolerance Ratio: solve instrument display or discrimination resolutionMetrology Resolution-to-Tolerance Ratio: solve specified tolerance-zone widthMeasurement Uncertainty Contribution ShareMeasurement Uncertainty Contribution Share: solve selected variance contributionMeasurement Uncertainty Contribution Share: solve combined measurement varianceEcological Capture-Recapture Population Estimate
Geometry formula relationships435 optional lessons
Ellipse Semiaxes AreaEllipse Semiaxes Area: solve semimajor axis aEllipse Semiaxes Area: solve semiminor axis bTriangle Base–Height AreaTriangle Base–Height Area: solve base lengthTriangle Base–Height Area: solve perpendicular heightPyramid Base-Area VolumePyramid Base-Area Volume: solve base areaPyramid Base-Area Volume: solve perpendicular heightCylinder Radius–Height VolumeCylinder Radius–Height Volume: solve base radiusCylinder Radius–Height Volume: solve perpendicular heightCone Radius–Height VolumeCone Radius–Height Volume: solve base radiusCone Radius–Height Volume: solve perpendicular heightAnnulus Radii AreaAnnulus Radii Area: solve outer radiusAnnulus Radii Area: solve inner radiusRight-Triangle Leg RelationshipRight-Triangle Leg Relationship: solve first legRight-Triangle Leg Relationship: solve second legCircle Sector AreaCircle Sector Area: solve radiusCircle Sector Area: solve central angle in degreesCircle Arc LengthCircle Arc Length: solve radiusCircle Arc Length: solve central angle in degreesCircle Chord LengthCircle Chord Length: solve circle radiusCircle Chord Length: solve central angle in degreesRight-Triangle Hypotenuse AltitudeRight-Triangle Hypotenuse Altitude: solve first legRight-Triangle Hypotenuse Altitude: solve second legRegular Polygon PerimeterRegular Polygon Perimeter: solve number of sidesRegular Polygon Perimeter: solve side lengthParallelogram Base–Height AreaParallelogram Base–Height Area: solve base lengthParallelogram Base–Height Area: solve perpendicular heightRhombus Diagonal-Product AreaRhombus Diagonal-Product Area: solve first diagonal lengthRhombus Diagonal-Product Area: solve second diagonal lengthKite Diagonal AreaKite Diagonal Area: solve first diagonal lengthKite Diagonal Area: solve second diagonal lengthTrapezoid Average-Base AreaTrapezoid Average-Base Area: solve average of parallel basesTrapezoid Average-Base Area: solve perpendicular heightPrism Base-Area VolumePrism Base-Area Volume: solve base areaPrism Base-Area Volume: solve perpendicular prism heightRight-Prism Lateral AreaRight-Prism Lateral Area: solve base perimeterRight-Prism Lateral Area: solve prism heightRectangular Face DiagonalRectangular Face Diagonal: solve rectangle lengthRectangular Face Diagonal: solve rectangle widthDimensional Area ScalingDimensional Area Scaling: solve original areaDimensional Area Scaling: solve linear scale factorDimensional Volume ScalingDimensional Volume Scaling: solve original volumeDimensional Volume Scaling: solve linear scale factorSimilar-Figure Length ScaleSimilar-Figure Length Scale: solve original corresponding lengthSimilar-Figure Length Scale: solve linear scale factorSimilar-Figure Area ScaleSimilar-Figure Area Scale: solve original figure areaSimilar-Figure Area Scale: solve linear scale factorSimilar-Solid Volume ScaleSimilar-Solid Volume Scale: solve original solid volumeSimilar-Solid Volume Scale: solve linear scale factorEqual Circles Total CircumferenceEqual Circles Total Circumference: solve diameter of each circleEqual Circles Total Circumference: solve equal-circle countCylinder Curved Surface AreaCylinder Curved Surface Area: solve cylinder radiusCylinder Curved Surface Area: solve perpendicular heightCone Curved Surface AreaCone Curved Surface Area: solve cone radiusCone Curved Surface Area: solve slant heightRight-Triangle Hypotenuse SquareRight-Triangle Hypotenuse Square: solve first leg lengthRight-Triangle Hypotenuse Square: solve second leg lengthPythagorean Leg-Square DifferencePythagorean Leg-Square Difference: solve hypotenuse lengthPythagorean Leg-Square Difference: solve known leg lengthTangential Polygon Area from ApothemTangential Polygon Area from Apothem: solve polygon perimeterTangential Polygon Area from Apothem: solve apothem lengthEllipse Semiaxis RatioEllipse Semiaxis Ratio: solve semimajor axisEllipse Semiaxis Ratio: solve semiminor axisPrism Lateral Surface AreaPrism Lateral Surface Area: solve base perimeterPrism Lateral Surface Area: solve perpendicular prism heightRegular Pyramid Lateral AreaRegular Pyramid Lateral Area: solve base perimeterRegular Pyramid Lateral Area: solve common slant heightSolid Surface-to-Volume RatioSolid Surface-to-Volume Ratio: solve total surface areaSolid Surface-to-Volume Ratio: solve solid volumeRegular Polygon Central AngleRegular Polygon Central Angle: solve full-turn angle in degreesRegular Polygon Central Angle: solve side countSphere Surface Area ScaleSphere Surface Area Scale: solve surface coefficientSphere Surface Area Scale: solve sphere radiusSphere Volume ScaleSphere Volume Scale: solve volume coefficientSphere Volume Scale: solve sphere radiusEllipse Focal Distance SquaredEllipse Focal Distance Squared: solve semimajor axisEllipse Focal Distance Squared: solve semiminor axisRectangle Area–Perimeter CompactnessRectangle Area–Perimeter Compactness: solve rectangle areaRectangle Area–Perimeter Compactness: solve perimeterThin Annular Strip AreaThin Annular Strip Area: solve mean strip radiusThin Annular Strip Area: solve radial strip widthHydraulic DiameterHydraulic Diameter: solve flow cross-sectional areaHydraulic Diameter: solve wetted perimeterGeometric Slenderness RatioGeometric Slenderness Ratio: solve effective member lengthGeometric Slenderness Ratio: solve radius of gyrationElastic Section ModulusElastic Section Modulus: solve second moment of areaElastic Section Modulus: solve extreme-fibre distanceSquared Radius of GyrationSquared Radius of Gyration: solve second moment of areaSquared Radius of Gyration: solve cross-sectional areaCyclic Quadrilateral Opposite AngleCyclic Quadrilateral Opposite Angle: solve supplement angle in degreesCyclic Quadrilateral Opposite Angle: solve known opposite angle in degreesCircle Chord SagittaCircle Chord Sagitta: solve circle radiusCircle Chord Sagitta: solve half-chord lengthConical Taper RatioConical Taper Ratio: solve diameter changeConical Taper Ratio: solve axial lengthProjected-to-Surface Area RatioProjected-to-Surface Area Ratio: solve projected areaProjected-to-Surface Area Ratio: solve actual surface areaVolume-to-Bounding-Box RatioVolume-to-Bounding-Box Ratio: solve object volumeVolume-to-Bounding-Box Ratio: solve bounding-box volumeRectangle Polar Second MomentRectangle Polar Second Moment: solve centroidal second moment about first axisRectangle Polar Second Moment: solve centroidal second moment about second axisTwo-Part Composite Section AreaTwo-Part Composite Section Area: solve first component areaTwo-Part Composite Section Area: solve second nonoverlapping component areaNet Section Area after OpeningNet Section Area after Opening: solve gross section areaNet Section Area after Opening: solve opening areaPacking Fraction PercentagePacking Fraction Percentage: solve occupied object volumePacking Fraction Percentage: solve container volumeMaterial Porosity PercentageMaterial Porosity Percentage: solve void volumeMaterial Porosity Percentage: solve bulk material volumeBody Fineness RatioBody Fineness Ratio: solve body lengthBody Fineness Ratio: solve maximum body diameterPlanar Circularity IndexPlanar Circularity Index: solve planar region areaPlanar Circularity Index: solve region perimeterSurface Sphericity PercentageSurface Sphericity Percentage: solve equal-volume sphere surface areaSurface Sphericity Percentage: solve actual particle surface areaMean Chord Length from Volume and SurfaceMean Chord Length from Volume and Surface: solve body volumeMean Chord Length from Volume and Surface: solve body surface areaAverage Path CurvatureAverage Path Curvature: solve total turning angle in radiansAverage Path Curvature: solve path arc lengthPlanar Isoperimetric DeficitPlanar Isoperimetric Deficit: solve perimeter squaredPlanar Isoperimetric Deficit: solve four-pi-times-area benchmarkParallel-Axis Transfer TermParallel-Axis Transfer Term: solve transferred area or massParallel-Axis Transfer Term: solve axis offsetSquared Torsional RadiusSquared Torsional Radius: solve polar second momentSquared Torsional Radius: solve cross-sectional areaSection First Moment of AreaSection First Moment of Area: solve section component areaSection First Moment of Area: solve centroid distance from reference axisTwo-Part Composite Solid VolumeTwo-Part Composite Solid Volume: solve first nonoverlapping solid volumeTwo-Part Composite Solid Volume: solve second nonoverlapping solid volumeNet Excavation VolumeNet Excavation Volume: solve gross excavated volumeNet Excavation Volume: solve retained or excluded volumeMaterial Void RatioMaterial Void Ratio: solve void volumeMaterial Void Ratio: solve solid-material volumeMass-Specific Surface AreaMass-Specific Surface Area: solve total surface areaMass-Specific Surface Area: solve material massVolume-to-Area Characteristic LengthVolume-to-Area Characteristic Length: solve body volumeVolume-to-Area Characteristic Length: solve representative surface areaConvex Polygon Triangulation CountConvex Polygon Triangulation Count: solve polygon vertex countConvex Polygon Triangulation Count: solve two-vertex boundary offsetSelf-Similarity DimensionSelf-Similarity Dimension: solve self-similar piece countSelf-Similarity Dimension: solve linear magnification factorProjective Cross-Ratio from Directed Segment ProductsProjective Cross-Ratio from Directed Segment Products: solve first directed-segment productProjective Cross-Ratio from Directed Segment Products: solve second directed-segment productRemaining Barycentric Coordinate WeightRemaining Barycentric Coordinate Weight: solve total affine weightRemaining Barycentric Coordinate Weight: solve sum of known barycentric weightsHomogeneous Coordinate NormalizationHomogeneous Coordinate Normalization: solve homogeneous coordinate componentHomogeneous Coordinate Normalization: solve nonzero projective scaleTriangle Area from InradiusTriangle Area from Inradius: solve triangle inradiusTriangle Area from Inradius: solve triangle semiperimeterTriangle Circumradius from Side ProductTriangle Circumradius from Side Product: solve product of three side lengthsTriangle Circumradius from Side Product: solve four times triangle areaBrahmagupta Cyclic-Quadrilateral AreaBrahmagupta Cyclic-Quadrilateral Area: solve product (s-a)(s-b)Brahmagupta Cyclic-Quadrilateral Area: solve product (s-c)(s-d)Frustum Volume from Mean Cross-SectionFrustum Volume from Mean Cross-Section: solve axial heightFrustum Volume from Mean Cross-Section: solve appropriate mean cross-sectional areaHydraulic RadiusHydraulic Radius: solve flow cross-sectional areaHydraulic Radius: solve wetted perimeterStereographic Projection Scale FactorStereographic Projection Scale Factor: solve unit sphere diameter scaleStereographic Projection Scale Factor: solve one plus squared planar radiusHyperbolic Distance from Positive Endpoint RatioHyperbolic Distance from Positive Endpoint Ratio: solve positive larger endpoint expressionHyperbolic Distance from Positive Endpoint Ratio: solve positive smaller endpoint expressionOriented Triangle Area from Shoelace DeterminantOriented Triangle Area from Shoelace Determinant: solve signed two-by-two shoelace determinantOriented Triangle Area from Shoelace Determinant: solve orientation unit scaleBarycentric Subsimplex Volume PercentageBarycentric Subsimplex Volume Percentage: solve opposite subsimplex volumeBarycentric Subsimplex Volume Percentage: solve whole simplex volumeSolid Angle from Spherical Patch AreaSolid Angle from Spherical Patch Area: solve spherical patch areaSolid Angle from Spherical Patch Area: solve sphere radiusMap Ground-Distance ScaleMap Ground-Distance Scale: solve map distanceMap Ground-Distance Scale: solve representative-fraction scale denominatorMap Ground-Area ScaleMap Ground-Area Scale: solve map areaMap Ground-Area Scale: solve linear scale denominatorRaster Ground Sample DistanceRaster Ground Sample Distance: solve represented ground swath widthRaster Ground Sample Distance: solve pixel columns across swathRaster Pixel Ground-Footprint AreaRaster Pixel Ground-Footprint Area: solve along-track ground sample distanceRaster Pixel Ground-Footprint Area: solve cross-track ground sample distanceMap Ground Positional ErrorMap Ground Positional Error: solve measured map-position errorMap Ground Positional Error: solve map scale denominatorRoute Sinuosity IndexRoute Sinuosity Index: solve distance measured along routeRoute Sinuosity Index: solve straight endpoint distanceContour Map Slope PercentageContour Map Slope Percentage: solve vertical elevation changeContour Map Slope Percentage: solve horizontal ground runLongitude Parallel DistanceLongitude Parallel Distance: solve equatorial longitude-arc distanceLongitude Parallel Distance: solve latitude in degreesMap Linear Enlargement FactorMap Linear Enlargement Factor: solve enlarged map lengthMap Linear Enlargement Factor: solve original map lengthMap Area Enlargement FactorMap Area Enlargement Factor: solve enlarged linear dimensionMap Area Enlargement Factor: solve original linear dimensionDigital Image Pixels per InchDigital Image Pixels per Inch: solve pixel count along one dimensionDigital Image Pixels per Inch: solve physical length in inchesDigital Image Total Pixel CountDigital Image Total Pixel Count: solve image width in pixelsDigital Image Total Pixel Count: solve image height in pixelsCamera Sensor Crop FactorCamera Sensor Crop Factor: solve reference-format diagonalCamera Sensor Crop Factor: solve camera-sensor diagonalEquivalent Camera Focal LengthEquivalent Camera Focal Length: solve actual lens focal lengthEquivalent Camera Focal Length: solve sensor crop factorCamera Lens F-NumberCamera Lens F-Number: solve lens focal lengthCamera Lens F-Number: solve entrance-pupil diameterOptical Linear MagnificationOptical Linear Magnification: solve image size on sensor or screenOptical Linear Magnification: solve corresponding object sizeAngular Image Scale per PixelAngular Image Scale per Pixel: solve angular field spanAngular Image Scale per Pixel: solve pixel count across fieldPixel Active-Area Fill FactorPixel Active-Area Fill Factor: solve photosensitive pixel areaPixel Active-Area Fill Factor: solve total pixel-cell areaAircraft GroundspeedAircraft Groundspeed: solve ground distance travelledAircraft Groundspeed: solve elapsed flight timeAircraft Climb Gradient PercentageAircraft Climb Gradient Percentage: solve net height gainedAircraft Climb Gradient Percentage: solve horizontal ground distanceAircraft Glide RatioAircraft Glide Ratio: solve horizontal distance in still airAircraft Glide Ratio: solve height lostRunway Longitudinal Slope PercentageRunway Longitudinal Slope Percentage: solve runway endpoint elevation changeRunway Longitudinal Slope Percentage: solve horizontal runway lengthAircraft Rate of ClimbAircraft Rate of Climb: solve altitude gainedAircraft Rate of Climb: solve elapsed climb timeIrrigation Application DepthIrrigation Application Depth: solve water volume appliedIrrigation Application Depth: solve irrigated land areaPackaging Volume UtilizationPackaging Volume Utilization: solve contained product volumePackaging Volume Utilization: solve carton internal volumePallet Footprint UtilizationPallet Footprint Utilization: solve load footprint areaPallet Footprint Utilization: solve available pallet deck areaWatershed Relief RatioWatershed Relief Ratio: solve watershed elevation reliefWatershed Relief Ratio: solve horizontal watershed lengthRainfall Runoff VolumeRainfall Runoff Volume: solve effective runoff depthRainfall Runoff Volume: solve contributing catchment areaGroundwater Hydraulic GradientGroundwater Hydraulic Gradient: solve hydraulic-head differenceGroundwater Hydraulic Gradient: solve horizontal flow-path distanceStride Cadence SpeedStride Cadence Speed: solve distance per strideStride Cadence Speed: solve stride cadenceVertical Jump Height from ReachVertical Jump Height from Reach: solve maximum jump reachVertical Jump Height from Reach: solve standing reachMarine Anchor Scope RatioMarine Anchor Scope Ratio: solve deployed rode lengthMarine Anchor Scope Ratio: solve vertical anchor-to-bow distanceVessel Displacement VolumeVessel Displacement Volume: solve displaced water massVessel Displacement Volume: solve water densityVessel Under-Keel ClearanceVessel Under-Keel Clearance: solve available water depthVessel Under-Keel Clearance: solve vessel operating draftMarine Propeller Pitch RatioMarine Propeller Pitch Ratio: solve nominal propeller pitchMarine Propeller Pitch Ratio: solve propeller diameterTelescope Central-Obstruction PercentageTelescope Central-Obstruction Percentage: solve secondary obstruction diameterTelescope Central-Obstruction Percentage: solve primary aperture diameterTelescope MagnificationTelescope Magnification: solve telescope focal lengthTelescope Magnification: solve eyepiece focal lengthTelescope Exit PupilTelescope Exit Pupil: solve clear telescope apertureTelescope Exit Pupil: solve angular magnificationTelescope True Field of ViewTelescope True Field of View: solve eyepiece apparent fieldTelescope True Field of View: solve telescope magnificationTelescope Light-Gathering RatioTelescope Light-Gathering Ratio: solve first telescope apertureTelescope Light-Gathering Ratio: solve reference telescope apertureTelescope Focal-Plane Plate ScaleTelescope Focal-Plane Plate Scale: solve represented angular fieldTelescope Focal-Plane Plate Scale: solve focal-plane physical spanTelescope Sensor Angular FieldTelescope Sensor Angular Field: solve sensor physical dimensionTelescope Sensor Angular Field: solve effective telescope focal lengthRobot Wheel Linear SpeedRobot Wheel Linear Speed: solve wheel travel per revolutionRobot Wheel Linear Speed: solve wheel revolutions per timeLuminaire Spacing-to-Height RatioLuminaire Spacing-to-Height Ratio: solve centre-to-centre luminaire spacingLuminaire Spacing-to-Height Ratio: solve mounting height above workplaneWarehouse Usable-Cube UtilizationWarehouse Usable-Cube Utilization: solve stored load volumeWarehouse Usable-Cube Utilization: solve usable storage-zone volumeWarehouse Aisle Area ShareWarehouse Aisle Area Share: solve aisle and maneuvering floor areaWarehouse Aisle Area Share: solve total warehouse operating floor areaEarthquake Hypocentral DistanceEarthquake Hypocentral Distance: solve horizontal epicentral distanceEarthquake Hypocentral Distance: solve earthquake focal depthFault Rupture Aspect RatioFault Rupture Aspect Ratio: solve rupture length along strikeFault Rupture Aspect Ratio: solve rupture width down dipRectangular Thin-Film ResistanceRectangular Thin-Film Resistance: solve thin-film sheet resistanceRectangular Thin-Film Resistance: solve effective number of geometric squaresHexagonal Crystal Axial RatioHexagonal Crystal Axial Ratio: solve hexagonal c-axis lattice parameterHexagonal Crystal Axial Ratio: solve basal a-axis lattice parameterCT Helical PitchCT Helical Pitch: solve table travel per gantry rotationCT Helical Pitch: solve total nominal beam collimation widthMRI Voxel VolumeMRI Voxel Volume: solve in-plane voxel face areaMRI Voxel Volume: solve reconstructed slice thicknessRadiographic Geometric MagnificationRadiographic Geometric Magnification: solve source-to-image distanceRadiographic Geometric Magnification: solve source-to-object distanceMedical Image Pixel SpacingMedical Image Pixel Spacing: solve physical field-of-view lengthMedical Image Pixel Spacing: solve reconstructed pixel count along same axisMedical Image Slice CoverageMedical Image Slice Coverage: solve effective centre-to-centre slice spacingMedical Image Slice Coverage: solve represented slice countGeodetic Map Ground DistanceGeodetic Map Ground Distance: solve measured map distanceGeodetic Map Ground Distance: solve representative scale denominatorSurvey Coordinate MisclosureSurvey Coordinate Misclosure: solve east-west coordinate misclosureSurvey Coordinate Misclosure: solve north-south coordinate misclosureGeodetic Spherical Arc DistanceGeodetic Spherical Arc Distance: solve adopted spherical Earth radiusGeodetic Spherical Arc Distance: solve central angle in degreesGeodetic Earth Chord DistanceGeodetic Earth Chord Distance: solve adopted spherical Earth radiusGeodetic Earth Chord Distance: solve central angle in degreesAcoustic Room Mean Free PathAcoustic Room Mean Free Path: solve enclosed room volumeAcoustic Room Mean Free Path: solve total enclosing surface areaCubic Lattice Plane SpacingCubic Lattice Plane Spacing: solve cubic lattice parameterCubic Lattice Plane Spacing: solve root-sum-square Miller-index magnitudeAircraft Wing Aspect RatioAircraft Wing Aspect Ratio: solve wing spanAircraft Wing Aspect Ratio: solve planform-area square-root scale
Trigonometry formula relationships102 optional lessons
Sine ComponentSine Component: solve vector magnitudeSine Component: solve angle in degreesCosine ComponentCosine Component: solve vector magnitudeCosine Component: solve angle in degreesTangent RiseTangent Rise: solve horizontal runTangent Rise: solve incline angle in degreesAngular to Linear SpeedAngular to Linear Speed: solve radiusAngular to Linear Speed: solve angular speedCycle PeriodCycle Period: solve elapsed timeCycle Period: solve completed cyclesCycle FrequencyCycle Frequency: solve completed cyclesCycle Frequency: solve elapsed timePhase from CyclesPhase from Cycles: solve completed cyclesPhase from Cycles: solve elapsed timeAngle-of-Elevation HeightAngle-of-Elevation Height: solve horizontal distanceAngle-of-Elevation Height: solve elevation angle in degreesSloped-Length Vertical ComponentSloped-Length Vertical Component: solve sloped lengthSloped-Length Vertical Component: solve incline angle in degreesInclined-Length Horizontal ComponentInclined-Length Horizontal Component: solve inclined lengthInclined-Length Horizontal Component: solve angle above horizontal in degreesRolling Wheel Travel DistanceRolling Wheel Travel Distance: solve wheel radiusRolling Wheel Travel Distance: solve complete wheel turnsOrthogonal Sinusoid AmplitudeOrthogonal Sinusoid Amplitude: solve cosine-component amplitudeOrthogonal Sinusoid Amplitude: solve sine-component amplitudeSlope Grade PercentageSlope Grade Percentage: solve vertical riseSlope Grade Percentage: solve horizontal runAngle from Turn FractionAngle from Turn Fraction: solve fraction of a full turnAngle from Turn Fraction: solve degrees per full turnUniform Angular DisplacementUniform Angular Displacement: solve constant angular speedUniform Angular Displacement: solve elapsed timePhase Accumulation from Cycle CountPhase Accumulation from Cycle Count: solve completed cycle countPhase Accumulation from Cycle Count: solve phase angle per cycleRamp Rise from Run and AngleRamp Rise from Run and Angle: solve horizontal runRamp Rise from Run and Angle: solve inclination angle in degreesCrosswind ComponentCrosswind Component: solve wind speedCrosswind Component: solve angle from runway heading in degreesHeadwind ComponentHeadwind Component: solve wind speedHeadwind Component: solve angle from runway heading in degreesCircle Boundary-to-Area RatioCircle Boundary-to-Area Ratio: solve circle circumferenceCircle Boundary-to-Area Ratio: solve circle areaDegree–Radian Scale ConversionDegree–Radian Scale Conversion: solve angle in radiansDegree–Radian Scale Conversion: solve degrees per radianHarmonic Time PhaseHarmonic Time Phase: solve angular frequencyHarmonic Time Phase: solve elapsed timeSpatial Phase AdvanceSpatial Phase Advance: solve angular wavenumberSpatial Phase Advance: solve travel distanceVector Dot Product from AngleVector Dot Product from Angle: solve product of vector magnitudesVector Dot Product from Angle: solve included angle in degreesVector Cross-Product Magnitude from AngleVector Cross-Product Magnitude from Angle: solve product of vector magnitudesVector Cross-Product Magnitude from Angle: solve included angle in degreesPeriodic Sample PhasePeriodic Sample Phase: solve sample indexPeriodic Sample Phase: solve phase increment per sampleBragg Diffraction Path DifferenceBragg Diffraction Path Difference: solve twice the interplanar spacingBragg Diffraction Path Difference: solve glancing angle in degreesSurvey Horizontal Distance ProjectionSurvey Horizontal Distance Projection: solve measured slope distanceSurvey Horizontal Distance Projection: solve vertical angle magnitude in degreesSurvey Vertical Elevation ChangeSurvey Vertical Elevation Change: solve measured slope distanceSurvey Vertical Elevation Change: solve signed vertical angle in degreesGeodetic Meridian Convergence ApproximationGeodetic Meridian Convergence Approximation: solve longitude separation from central meridian in degreesGeodetic Meridian Convergence Approximation: solve geodetic latitude in degreesGeodetic Parallel Arc DistanceGeodetic Parallel Arc Distance: solve equatorial arc distance for the same longitude intervalGeodetic Parallel Arc Distance: solve geocentric latitude magnitude in degreesOptical Numerical ApertureOptical Numerical Aperture: solve refractive index of object spaceOptical Numerical Aperture: solve acceptance half-angle in degreesDiffraction-Grating WavelengthDiffraction-Grating Wavelength: solve grating spacing per diffraction orderDiffraction-Grating Wavelength: solve diffraction angle in degreesCrystal Bragg WavelengthCrystal Bragg Wavelength: solve twice lattice-plane spacing divided by diffraction orderCrystal Bragg Wavelength: solve Bragg angle in degrees
Mathematical Physics formula relationships319 optional lessons
Wave Frequency–WavelengthWave Frequency–Wavelength: solve frequencyWave Frequency–Wavelength: solve wavelengthKinetic Energy RelationshipKinetic Energy Relationship: solve massKinetic Energy Relationship: solve speedInverse-Square Field StrengthInverse-Square Field Strength: solve source-strength constantInverse-Square Field Strength: solve distanceTwo-Particle Reduced MassTwo-Particle Reduced Mass: solve first particle massTwo-Particle Reduced Mass: solve second particle massImpedance Magnitude RelationshipImpedance Magnitude Relationship: solve voltage magnitudeImpedance Magnitude Relationship: solve current magnitudePressure–Force–AreaPressure–Force–Area: solve normal forcePressure–Force–Area: solve loaded areaMass–Density–VolumeMass–Density–Volume: solve material densityMass–Density–Volume: solve occupied volumeElectric Charge from Current and TimeElectric Charge from Current and Time: solve steady currentElectric Charge from Current and Time: solve elapsed timeElectric Power from Voltage and CurrentElectric Power from Voltage and Current: solve voltage magnitudeElectric Power from Voltage and Current: solve current magnitudeResistance from Voltage and CurrentResistance from Voltage and Current: solve voltage differenceResistance from Voltage and Current: solve currentImpulse from Force and TimeImpulse from Force and Time: solve average net forceImpulse from Force and Time: solve interaction durationMomentum from Mass and SpeedMomentum from Mass and Speed: solve massMomentum from Mass and Speed: solve signed velocityWave Intensity–Amplitude SquareWave Intensity–Amplitude Square: solve medium proportionality factorWave Intensity–Amplitude Square: solve wave amplitudeTwo Parallel Resistances EquivalentTwo Parallel Resistances Equivalent: solve first resistanceTwo Parallel Resistances Equivalent: solve second resistanceTwo Series Springs EquivalentTwo Series Springs Equivalent: solve first spring stiffnessTwo Series Springs Equivalent: solve second spring stiffnessPerpendicular Torque MagnitudePerpendicular Torque Magnitude: solve lever-arm lengthPerpendicular Torque Magnitude: solve perpendicular forceAngular Impulse from Torque and TimeAngular Impulse from Torque and Time: solve average net torqueAngular Impulse from Torque and Time: solve interaction durationMach NumberMach Number: solve object speedMach Number: solve local speed of soundReynolds Inertial-to-Viscous RatioReynolds Inertial-to-Viscous Ratio: solve inertial transport scaleReynolds Inertial-to-Viscous Ratio: solve viscous transport scaleLinear MomentumLinear Momentum: solve massLinear Momentum: solve velocity componentIdeal Transformer Voltage ScalingIdeal Transformer Voltage Scaling: solve primary voltageIdeal Transformer Voltage Scaling: solve secondary-to-primary turns ratioIdeal Spring Potential EnergyIdeal Spring Potential Energy: solve spring stiffnessIdeal Spring Potential Energy: solve displacement magnitudeCapacitor Stored EnergyCapacitor Stored Energy: solve capacitanceCapacitor Stored Energy: solve voltage magnitudeMass Flow from Volumetric FlowMass Flow from Volumetric Flow: solve fluid densityMass Flow from Volumetric Flow: solve volumetric flow rateThermal ResistanceThermal Resistance: solve temperature differenceThermal Resistance: solve heat-transfer rateResonator Quality Factor from Energy LossResonator Quality Factor from Energy Loss: solve stored energyResonator Quality Factor from Energy Loss: solve energy lost per cycleLossless-Line Characteristic ImpedanceLossless-Line Characteristic Impedance: solve positive inductance per lengthLossless-Line Characteristic Impedance: solve positive capacitance per lengthAtmospheric Wind Dynamic PressureAtmospheric Wind Dynamic Pressure: solve air mass densityAtmospheric Wind Dynamic Pressure: solve wind speedTelescope Diffraction Angular ResolutionTelescope Diffraction Angular Resolution: solve diffraction wavelength coefficientTelescope Diffraction Angular Resolution: solve clear aperture diameterRobot Motor Torque ConstantRobot Motor Torque Constant: solve developed motor torqueRobot Motor Torque Constant: solve motor phase currentRobot Linear-Actuator Mechanical PowerRobot Linear-Actuator Mechanical Power: solve axial actuator forceRobot Linear-Actuator Mechanical Power: solve linear actuator speedRobot Joint Mechanical PowerRobot Joint Mechanical Power: solve joint torqueRobot Joint Mechanical Power: solve joint angular speedWorkplane IlluminanceWorkplane Illuminance: solve luminous flux incident on workplaneWorkplane Illuminance: solve illuminated workplane areaLight-Source Luminous EfficacyLight-Source Luminous Efficacy: solve useful luminous fluxLight-Source Luminous Efficacy: solve electrical input powerPoint-Source IlluminancePoint-Source Illuminance: solve source luminous intensity toward targetPoint-Source Illuminance: solve source-to-target distanceLuminous ExitanceLuminous Exitance: solve luminous flux leaving a surfaceLuminous Exitance: solve emitting surface areaSolar Energy from Equivalent Sun HoursSolar Energy from Equivalent Sun Hours: solve effective photovoltaic output powerSolar Energy from Equivalent Sun Hours: solve equivalent peak-sun hoursBattery Nominal Energy CapacityBattery Nominal Energy Capacity: solve nominal battery voltageBattery Nominal Energy Capacity: solve rated charge capacityAcoustic Sound-Pressure LevelAcoustic Sound-Pressure Level: solve root-mean-square sound pressureAcoustic Sound-Pressure Level: solve reference sound pressureAcoustic Sound-Intensity LevelAcoustic Sound-Intensity Level: solve sound intensity magnitudeAcoustic Sound-Intensity Level: solve reference sound intensityFree-Field Acoustic Distance AttenuationFree-Field Acoustic Distance Attenuation: solve farther source distanceFree-Field Acoustic Distance Attenuation: solve reference source distanceSabine Room Reverberation TimeSabine Room Reverberation Time: solve Sabine volume coefficientSabine Room Reverberation Time: solve total equivalent absorption areaSpecific Acoustic Impedance MagnitudeSpecific Acoustic Impedance Magnitude: solve sound-pressure magnitudeSpecific Acoustic Impedance Magnitude: solve particle-velocity magnitudeAcoustic Sound ExposureAcoustic Sound Exposure: solve time-averaged squared sound pressureAcoustic Sound Exposure: solve exposure durationAcoustic Power per Measurement Surface AreaAcoustic Power per Measurement Surface Area: solve sound power crossing measurement surfaceAcoustic Power per Measurement Surface Area: solve measurement surface areaSeismic Wave Travel TimeSeismic Wave Travel Time: solve seismic propagation path lengthSeismic Wave Travel Time: solve average seismic wave speedSeismic Characteristic ImpedanceSeismic Characteristic Impedance: solve bulk material densitySeismic Characteristic Impedance: solve seismic wave speedFault Seismic MomentFault Seismic Moment: solve rigidity-times-rupture-area coefficientFault Seismic Moment: solve average fault slipBuilding Heating Energy from Degree HoursBuilding Heating Energy from Degree Hours: solve building heat-loss coefficientBuilding Heating Energy from Degree Hours: solve accumulated heating degree-hoursBuilding Envelope U-ValueBuilding Envelope U-Value: solve steady envelope heat-transfer rateBuilding Envelope U-Value: solve area-times-temperature-difference productHVAC Duct Average Air VelocityHVAC Duct Average Air Velocity: solve duct air volume flow rateHVAC Duct Average Air Velocity: solve internal duct flow areaHVAC Sensible Heat-Transfer RateHVAC Sensible Heat-Transfer Rate: solve air heat-capacity flow coefficientHVAC Sensible Heat-Transfer Rate: solve air temperature changeSemiconductor Device Current DensitySemiconductor Device Current Density: solve device current through active regionSemiconductor Device Current Density: solve active current-carrying areaThin-Film Sheet ResistanceThin-Film Sheet Resistance: solve thin-film electrical resistivityThin-Film Sheet Resistance: solve conductive film thicknessMOSFET On-State Conduction LossMOSFET On-State Conduction Loss: solve effective on-state resistanceMOSFET On-State Conduction Loss: solve root-mean-square drain currentSemiconductor Junction Temperature RiseSemiconductor Junction Temperature Rise: solve dissipated device powerSemiconductor Junction Temperature Rise: solve junction-to-reference thermal resistanceCrystallographic Unit-Cell DensityCrystallographic Unit-Cell Density: solve unit-cell massCrystallographic Unit-Cell Density: solve unit-cell volumeReciprocal-Lattice Vector MagnitudeReciprocal-Lattice Vector Magnitude: solve diffraction order or harmonic multiplierReciprocal-Lattice Vector Magnitude: solve real-space plane spacingScherrer Crystallite SizeScherrer Crystallite Size: solve shape-factor-times-wavelength numeratorScherrer Crystallite Size: solve instrument-corrected peak-width cosine productElectrochemical Faradaic ChargeElectrochemical Faradaic Charge: solve average Faradaic currentElectrochemical Faradaic Charge: solve charge-transfer durationElectrolyte Ionic ConductivityElectrolyte Ionic Conductivity: solve ionic current densityElectrolyte Ionic Conductivity: solve electric-field magnitudeRadio Effective Isotropic Radiated PowerRadio Effective Isotropic Radiated Power: solve transmitter power delivered toward antennaRadio Effective Isotropic Radiated Power: solve antenna gain relative to isotropic in linear unitsRadio Noise-Factor Ratio in DecibelsRadio Noise-Factor Ratio in Decibels: solve device noise factorRadio Noise-Factor Ratio in Decibels: solve reference noise factorRadio Voltage Reflection Return LossRadio Voltage Reflection Return Loss: solve voltage-ratio decibel conversion coefficientRadio Voltage Reflection Return Loss: solve voltage reflection-coefficient magnitudeIonizing-Radiation Absorbed DoseIonizing-Radiation Absorbed Dose: solve ionizing energy impartedIonizing-Radiation Absorbed Dose: solve irradiated material massRadiation-Weighted Equivalent DoseRadiation-Weighted Equivalent Dose: solve absorbed dose to tissueRadiation-Weighted Equivalent Dose: solve radiation weighting factorPoint-Source Radiation Dose RatePoint-Source Radiation Dose Rate: solve source dose-rate distance-squared coefficientPoint-Source Radiation Dose Rate: solve source-to-point distanceCT Dose-Length ProductCT Dose-Length Product: solve volume CT dose indexCT Dose-Length Product: solve scanned lengthUltrasound Pulse-Echo DepthUltrasound Pulse-Echo Depth: solve half the assumed tissue sound speedUltrasound Pulse-Echo Depth: solve round-trip echo timeStructural Force-to-Displacement StiffnessStructural Force-to-Displacement Stiffness: solve applied incremental forceStructural Force-to-Displacement Stiffness: solve resulting displacement incrementHarmonic Vibration Velocity AmplitudeHarmonic Vibration Velocity Amplitude: solve displacement amplitudeHarmonic Vibration Velocity Amplitude: solve angular frequencyHarmonic Vibration Acceleration AmplitudeHarmonic Vibration Acceleration Amplitude: solve displacement amplitudeHarmonic Vibration Acceleration Amplitude: solve angular frequencyRotating Unbalance Force AmplitudeRotating Unbalance Force Amplitude: solve rotating mass-eccentricity productRotating Unbalance Force Amplitude: solve angular speedAcoustic Pressure Specific ImpedanceAcoustic Pressure Specific Impedance: solve complex-amplitude pressure magnitudeAcoustic Pressure Specific Impedance: solve particle-velocity magnitudeAcoustic Characteristic ImpedanceAcoustic Characteristic Impedance: solve medium mass densityAcoustic Characteristic Impedance: solve small-signal sound speedAcoustic Sabine Reverberation TimeAcoustic Sabine Reverberation Time: solve Sabine-scaled room volumeAcoustic Sabine Reverberation Time: solve total equivalent absorption areaAcoustic Plane-Wave Energy DensityAcoustic Plane-Wave Energy Density: solve time-averaged acoustic intensityAcoustic Plane-Wave Energy Density: solve medium sound speedAcoustic Power Transmission LossAcoustic Power Transmission Loss: solve natural-logarithm decibel coefficientAcoustic Power Transmission Loss: solve power transmission coefficientAcoustic Directivity IndexAcoustic Directivity Index: solve sound-source directivity factorAcoustic Directivity Index: solve isotropic reference factorHydrologic Runoff VolumeHydrologic Runoff Volume: solve catchment areaHydrologic Runoff Volume: solve effective runoff depthAquifer Drainable Storage VolumeAquifer Drainable Storage Volume: solve saturated aquifer volume representedAquifer Drainable Storage Volume: solve specific yieldGroundwater Seepage VelocityGroundwater Seepage Velocity: solve Darcy flux magnitudeGroundwater Seepage Velocity: solve effective mobile porosityPhotonic Optical IrradiancePhotonic Optical Irradiance: solve incident radiant powerPhotonic Optical Irradiance: solve illuminated projected areaPhotonic Radiant ExposurePhotonic Radiant Exposure: solve time-averaged irradiancePhotonic Radiant Exposure: solve exposure durationOptical Density from TransmittanceOptical Density from Transmittance: solve natural-logarithm base-conversion coefficientOptical Density from Transmittance: solve fractional optical transmittanceOptical Beer-Lambert AttenuationOptical Beer-Lambert Attenuation: solve spectral attenuation coefficientOptical Beer-Lambert Attenuation: solve optical path lengthOptical EtendueOptical Etendue: solve projected optical aperture areaOptical Etendue: solve accepted solid angleLaser Pulse FluenceLaser Pulse Fluence: solve laser pulse energyLaser Pulse Fluence: solve effective illuminated spot areaOptical Photon FluxOptical Photon Flux: solve monochromatic radiant powerOptical Photon Flux: solve energy per photonOptical Propagation Group DelayOptical Propagation Group Delay: solve optical path lengthOptical Propagation Group Delay: solve group slownessElectrophoretic MobilityElectrophoretic Mobility: solve measured migration velocityElectrophoretic Mobility: solve applied electric-field magnitudeCentrifuge Relative ForceCentrifuge Relative Force: solve radius-and-unit conversion coefficientCentrifuge Relative Force: solve rotational speedCycling Gear-Development SpeedCycling Gear-Development Speed: solve distance advanced per crank revolutionCycling Gear-Development Speed: solve crank revolutions per unit timeJump Ballistic Takeoff HeightJump Ballistic Takeoff Height: solve vertical takeoff speedJump Ballistic Takeoff Height: solve twice local gravitational accelerationBiomechanical Joint Angular ImpulseBiomechanical Joint Angular Impulse: solve representative net joint momentBiomechanical Joint Angular Impulse: solve moment application durationSeismic Acoustic ImpedanceSeismic Acoustic Impedance: solve formation bulk densitySeismic Acoustic Impedance: solve compressional-wave speedSeismic WavelengthSeismic Wavelength: solve seismic phase velocitySeismic Wavelength: solve wave frequencySeismic Geometric-Spreading IntensitySeismic Geometric-Spreading Intensity: solve source intensity-distance-squared coefficientSeismic Geometric-Spreading Intensity: solve source distanceSeismic Moment Slip ProductSeismic Moment Slip Product: solve rupture rigidity-area productSeismic Moment Slip Product: solve average fault slipCrystal Unit-Cell DensityCrystal Unit-Cell Density: solve mass represented in one unit cellCrystal Unit-Cell Density: solve unit-cell volumeMaterial Linear Thermal StrainMaterial Linear Thermal Strain: solve linear thermal expansion coefficientMaterial Linear Thermal Strain: solve temperature changeVessel Displacement MassVessel Displacement Mass: solve ambient water densityVessel Displacement Mass: solve immersed displacement volumeShallow-Water Wave CelerityShallow-Water Wave Celerity: solve local gravitational accelerationShallow-Water Wave Celerity: solve uniform water depthMarine Buoyancy ForceMarine Buoyancy Force: solve ambient fluid specific weightMarine Buoyancy Force: solve displaced fluid volumeSoil Profile Water StorageSoil Profile Water Storage: solve profile-average volumetric water fractionSoil Profile Water Storage: solve represented soil-layer thicknessOre Contained Metal MassOre Contained Metal Mass: solve dry ore massOre Contained Metal Mass: solve metal grade as a mass fractionAircraft Coordinated-Turn Radius
Calculus formula relationships360 optional lessons
Monomial Derivative CoefficientMonomial Derivative Coefficient: solve original coefficient aMonomial Derivative Coefficient: solve power nMonomial Antiderivative CoefficientMonomial Antiderivative Coefficient: solve original coefficient aMonomial Antiderivative Coefficient: solve power nExponential Derivative at ZeroExponential Derivative at Zero: solve function coefficient aExponential Derivative at Zero: solve exponential base bSinusoid Second-Derivative AmplitudeSinusoid Second-Derivative Amplitude: solve original amplitudeSinusoid Second-Derivative Amplitude: solve angular frequencyConstant Definite IntegralConstant Definite Integral: solve constant function valueConstant Definite Integral: solve interval widthLinear Ramp IntegralLinear Ramp Integral: solve ending height from zeroLinear Ramp Integral: solve interval widthForward-Difference DerivativeForward-Difference Derivative: solve function-value changeForward-Difference Derivative: solve step widthCentral-Difference DerivativeCentral-Difference Derivative: solve function-value change across stencilCentral-Difference Derivative: solve total stencil widthRectangle-Rule QuadratureRectangle-Rule Quadrature: solve representative function heightRectangle-Rule Quadrature: solve interval widthTrapezoid Average-Height QuadratureTrapezoid Average-Height Quadrature: solve average endpoint heightTrapezoid Average-Height Quadrature: solve interval widthSimpson Weighted-Height QuadratureSimpson Weighted-Height Quadrature: solve Simpson weighted mean heightSimpson Weighted-Height Quadrature: solve interval widthNewton Method CorrectionNewton Method Correction: solve function value f(x)Newton Method Correction: solve derivative f′(x)Relative Error RatioRelative Error Ratio: solve absolute errorRelative Error Ratio: solve reference magnitudeSigned Approximation ErrorSigned Approximation Error: solve approximate valueSigned Approximation Error: solve reference valueConvergence Error ReductionConvergence Error Reduction: solve step-size ratioConvergence Error Reduction: solve method orderDifferential Linearization ChangeDifferential Linearization Change: solve local derivativeDifferential Linearization Change: solve small input changeLocal Truncation Power LawLocal Truncation Power Law: solve step-size ratioLocal Truncation Power Law: solve local method orderCondition-Based Forward Error BoundCondition-Based Forward Error Bound: solve condition numberCondition-Based Forward Error Bound: solve relative input-error boundPower Antiderivative CoefficientPower Antiderivative Coefficient: solve integrand coefficientPower Antiderivative Coefficient: solve original exponentRiemann Rectangle ContributionRiemann Rectangle Contribution: solve sampled function heightRiemann Rectangle Contribution: solve subinterval widthUniform Mesh SpacingUniform Mesh Spacing: solve interval lengthUniform Mesh Spacing: solve subinterval countJacobian Area ScalingJacobian Area Scaling: solve source-region areaJacobian Area Scaling: solve absolute Jacobian determinantLogarithmic Relative ChangeLogarithmic Relative Change: solve positive final valueLogarithmic Relative Change: solve positive initial valueLogarithmic Elasticity RatioLogarithmic Elasticity Ratio: solve input-times-local-derivativeLogarithmic Elasticity Ratio: solve nonzero function valueLogarithmic Derivative ScaleLogarithmic Derivative Scale: solve logarithm coefficientLogarithmic Derivative Scale: solve positive evaluation pointExponential Antiderivative CoefficientExponential Antiderivative Coefficient: solve exponential-term coefficientExponential Antiderivative Coefficient: solve nonzero exponential rateImplicit-Function Sensitivity MagnitudeImplicit-Function Sensitivity Magnitude: solve partial derivative magnitude in input directionImplicit-Function Sensitivity Magnitude: solve nonzero partial derivative magnitude in output directionTrapezoidal Mean OrdinateTrapezoidal Mean Ordinate: solve left endpoint valueTrapezoidal Mean Ordinate: solve right endpoint valuePolar Sector Area from Radian AnglePolar Sector Area from Radian Angle: solve angle in radiansPolar Sector Area from Radian Angle: solve radiusFinite-Volume Face Flux ContributionFinite-Volume Face Flux Contribution: solve normal flux densityFinite-Volume Face Flux Contribution: solve face measureForward Euler State UpdateForward Euler State Update: solve current state valueForward Euler State Update: solve computed Euler incrementGradient Descent Parameter UpdateGradient Descent Parameter Update: solve current parameterGradient Descent Parameter Update: solve learning-rate gradient stepTotal-Derivative Chain ContributionTotal-Derivative Chain Contribution: solve partial derivativeTotal-Derivative Chain Contribution: solve input rate of changeTwo-Direction DivergenceTwo-Direction Divergence: solve first directional partial derivativeTwo-Direction Divergence: solve second directional partial derivativeTwo-Direction LaplacianTwo-Direction Laplacian: solve second derivative in first directionTwo-Direction Laplacian: solve second derivative in second directionCentered Second-Difference CurvatureCentered Second-Difference Curvature: solve second-difference numeratorCentered Second-Difference Curvature: solve uniform mesh spacingTrapezoid Panel IntegralTrapezoid Panel Integral: solve mean endpoint ordinateTrapezoid Panel Integral: solve panel widthSimpson Weighted Panel AverageSimpson Weighted Panel Average: solve weighted ordinate sumSimpson Weighted Panel Average: solve Simpson divisorCourant Travel-to-Cell RatioCourant Travel-to-Cell Ratio: solve distance travelled in one time stepCourant Travel-to-Cell Ratio: solve cell widthDiffusion Mesh NumberDiffusion Mesh Number: solve diffusivity-times-step productDiffusion Mesh Number: solve mesh spacingObserved Numerical Convergence OrderObserved Numerical Convergence Order: solve positive error ratioObserved Numerical Convergence Order: solve positive mesh-size ratioReference-Scaled Numerical Error PercentageReference-Scaled Numerical Error Percentage: solve absolute numerical errorReference-Scaled Numerical Error Percentage: solve reference solution magnitudeIterative Residual Reduction FactorIterative Residual Reduction Factor: solve new residual normIterative Residual Reduction Factor: solve previous residual normTwo-Component Gradient Norm SquaredTwo-Component Gradient Norm Squared: solve first gradient componentTwo-Component Gradient Norm Squared: solve second gradient componentTwo-Direction Hessian TraceTwo-Direction Hessian Trace: solve first diagonal Hessian entryTwo-Direction Hessian Trace: solve second diagonal Hessian entryTwo-by-Two Jacobian DeterminantTwo-by-Two Jacobian Determinant: solve product of main-diagonal derivativesTwo-by-Two Jacobian Determinant: solve product of off-diagonal derivativesLocal Volume Jacobian ScalingLocal Volume Jacobian Scaling: solve source differential volumeLocal Volume Jacobian Scaling: solve absolute Jacobian determinantDefinite-Integral Average Function ValueDefinite-Integral Average Function Value: solve signed definite integralDefinite-Integral Average Function Value: solve interval widthInverse-Function Derivative from Forward DerivativeInverse-Function Derivative from Forward Derivative: solve unit reciprocal scaleInverse-Function Derivative from Forward Derivative: solve nonzero forward derivativeTaylor Approximation Remainder GapTaylor Approximation Remainder Gap: solve exact function valueTaylor Approximation Remainder Gap: solve Taylor polynomial valueGraph Arc-Length Local Stretch FactorGraph Arc-Length Local Stretch Factor: solve unit horizontal differentialGraph Arc-Length Local Stretch Factor: solve derivative magnitudeOptimization Primal–Dual GapOptimization Primal–Dual Gap: solve primal objective boundOptimization Primal–Dual Gap: solve dual objective boundTrust-Region Radius Utilization PercentageTrust-Region Radius Utilization Percentage: solve accepted step normTrust-Region Radius Utilization Percentage: solve trust-region radiusRegularized Objective TotalRegularized Objective Total: solve data-fit lossRegularized Objective Total: solve regularization penaltyQuadratic Constraint Penalty ContributionQuadratic Constraint Penalty Contribution: solve penalty coefficientQuadratic Constraint Penalty Contribution: solve constraint violation magnitudeTrust-Region Model Agreement RatioTrust-Region Model Agreement Ratio: solve actual objective reductionTrust-Region Model Agreement Ratio: solve predicted model reductionArmijo Sufficient-Decrease ThresholdArmijo Sufficient-Decrease Threshold: solve step lengthArmijo Sufficient-Decrease Threshold: solve scaled descent-slope magnitudeL1 Regularization Penalty ContributionL1 Regularization Penalty Contribution: solve regularization coefficientL1 Regularization Penalty Contribution: solve parameter L1 normIterative Method Operation BudgetIterative Method Operation Budget: solve iteration countIterative Method Operation Budget: solve operations per iterationNumerical Mesh Refinement FactorNumerical Mesh Refinement Factor: solve coarse mesh spacingNumerical Mesh Refinement Factor: solve fine mesh spacingPer-Iteration Convergence FactorPer-Iteration Convergence Factor: solve positive total error ratioPer-Iteration Convergence Factor: solve iteration interval countNewton Quadratic Error ConstantNewton Quadratic Error Constant: solve new iteration error magnitudeNewton Quadratic Error Constant: solve previous error magnitudeFixed-Point Iteration Contraction FactorFixed-Point Iteration Contraction Factor: solve new iterate-difference normFixed-Point Iteration Contraction Factor: solve previous iterate-difference normRichardson Extrapolation Error CorrectionRichardson Extrapolation Error Correction: solve coarse-fine approximation differenceRichardson Extrapolation Error Correction: solve refinement power denominatorAdaptive Mesh Error per CellAdaptive Mesh Error per Cell: solve aggregate error indicatorAdaptive Mesh Error per Cell: solve active mesh cell countObserved Local Relative Condition RatioObserved Local Relative Condition Ratio: solve relative output changeObserved Local Relative Condition Ratio: solve relative input perturbationDirectional Derivative from Gradient ProjectionDirectional Derivative from Gradient Projection: solve gradient normDirectional Derivative from Gradient Projection: solve direction cosine with gradientRadius of Curvature from CurvatureRadius of Curvature from Curvature: solve curvature magnitudeRadius of Curvature from Curvature: solve unit reciprocal scaleLine-Integral Average DensityLine-Integral Average Density: solve scalar line integralLine-Integral Average Density: solve curve lengthSurface Flux Average DensitySurface Flux Average Density: solve total oriented surface fluxSurface Flux Average Density: solve surface areaNewton Decrement from Quadratic FormNewton Decrement from Quadratic Form: solve positive Newton quadratic formNewton Decrement from Quadratic Form: solve unit normalization scaleGradient Clipping Scale FactorGradient Clipping Scale Factor: solve clipping thresholdGradient Clipping Scale Factor: solve unclipped gradient normOptimization Momentum History ContributionOptimization Momentum History Contribution: solve momentum retention coefficientOptimization Momentum History Contribution: solve previous momentum stateMini-Batch Count per EpochMini-Batch Count per Epoch: solve training example countMini-Batch Count per Epoch: solve mini-batch sizeTraining Example ExposuresTraining Example Exposures: solve dataset example countTraining Example Exposures: solve completed epoch countLearning-Rate Exponential DecayLearning-Rate Exponential Decay: solve initial learning rateLearning-Rate Exponential Decay: solve integrated positive decay exponentRegularization-Path Coefficient RatioRegularization-Path Coefficient Ratio: solve coefficient magnitude at first penaltyRegularization-Path Coefficient Ratio: solve coefficient magnitude at second penaltyOptimization Capacity-Constraint SlackOptimization Capacity-Constraint Slack: solve constraint capacityOptimization Capacity-Constraint Slack: solve allocated resource amountLinear-Program Reduced Cost from Dual PricingLinear-Program Reduced Cost from Dual Pricing: solve variable objective coefficientLinear-Program Reduced Cost from Dual Pricing: solve dual-price resource contributionSecant Derivative EstimateSecant Derivative Estimate: solve function-value differenceSecant Derivative Estimate: solve input-coordinate differenceNewton Updated IterateNewton Updated Iterate: solve current iterateNewton Updated Iterate: solve Newton correction f(x)/f'(x)Lagrange Interpolation Basis WeightLagrange Interpolation Basis Weight: solve product of evaluation-minus-other-node factorsLagrange Interpolation Basis Weight: solve product of selected-node-minus-other-node factorsInterpolation Error above Best ApproximationInterpolation Error above Best Approximation: solve observed interpolation errorInterpolation Error above Best Approximation: solve best-approximation errorRichardson Extrapolated EstimateRichardson Extrapolated Estimate: solve fine-grid approximationRichardson Extrapolated Estimate: solve estimated asymptotic correctionNewton Linearized Residual ChangeNewton Linearized Residual Change: solve local derivative or Jacobian actionNewton Linearized Residual Change: solve proposed iterate correctionComposite Midpoint-Rule IntegralComposite Midpoint-Rule Integral: solve uniform panel widthComposite Midpoint-Rule Integral: solve sum of midpoint function valuesGaussian-Quadrature Normalized Weighted MeanGaussian-Quadrature Normalized Weighted Mean: solve sum of weighted node valuesGaussian-Quadrature Normalized Weighted Mean: solve sum of quadrature weightsLebesgue Interpolation Error Amplification BoundLebesgue Interpolation Error Amplification Bound: solve one plus Lebesgue constantLebesgue Interpolation Error Amplification Bound: solve best uniform approximation error boundQuadrature Accuracy Gain per EvaluationQuadrature Accuracy Gain per Evaluation: solve validated accuracy digits gainedQuadrature Accuracy Gain per Evaluation: solve additional function evaluationsAlgorithm Runtime Speedup FactorAlgorithm Runtime Speedup Factor: solve baseline execution timeAlgorithm Runtime Speedup Factor: solve improved execution timeParallel Computing Efficiency PercentageParallel Computing Efficiency Percentage: solve measured speedup factorParallel Computing Efficiency Percentage: solve parallel processor countAmdahl Maximum SpeedupAmdahl Maximum Speedup: solve unit workload scaleAmdahl Maximum Speedup: solve serial workload fractionGustafson Scaled SpeedupGustafson Scaled Speedup: solve parallel processor countGustafson Scaled Speedup: solve serial-fraction scaling penaltyAlgorithm Useful-Work Efficiency PercentageAlgorithm Useful-Work Efficiency Percentage: solve useful mathematical operationsAlgorithm Useful-Work Efficiency Percentage: solve total executed operation countAlgorithm Arithmetic IntensityAlgorithm Arithmetic Intensity: solve floating-point operation countAlgorithm Arithmetic Intensity: solve bytes transferred from target memory levelSustained Memory BandwidthSustained Memory Bandwidth: solve bytes transferredSustained Memory Bandwidth: solve elapsed transfer or kernel timeParallel Algorithm Work OverheadParallel Algorithm Work Overhead: solve aggregate parallel workParallel Algorithm Work Overhead: solve best serial workParallel Computing CostParallel Computing Cost: solve parallel processor countParallel Computing Cost: solve parallel elapsed timeSoil Infiltration RateSoil Infiltration Rate: solve water depth infiltratedSoil Infiltration Rate: solve elapsed infiltration timeOpen-Water Evaporation RateOpen-Water Evaporation Rate: solve water-depth lossOpen-Water Evaporation Rate: solve elapsed observation timeTree Height Increment RateTree Height Increment Rate: solve tree height increaseTree Height Increment Rate: solve growth intervalTree Diameter Increment RateTree Diameter Increment Rate: solve stem diameter increaseTree Diameter Increment Rate: solve growth intervalPrecipitation IntensityPrecipitation Intensity: solve accumulated precipitation depthPrecipitation Intensity: solve accumulation durationEnvironmental Temperature Lapse RateEnvironmental Temperature Lapse Rate: solve observed temperature decreaseEnvironmental Temperature Lapse Rate: solve elevation increaseAtmospheric Pressure TendencyAtmospheric Pressure Tendency: solve barometric pressure changeAtmospheric Pressure Tendency: solve elapsed observation timeFood Fermentation Substrate Consumption RateFood Fermentation Substrate Consumption Rate: solve fermentable substrate consumedFood Fermentation Substrate Consumption Rate: solve active fermentation durationLivestock Average Daily GainLivestock Average Daily Gain: solve live-weight gain over intervalLivestock Average Daily Gain: solve growth interval in daysFault Average Slip RateFault Average Slip Rate: solve accumulated fault displacementFault Average Slip Rate: solve elapsed geological intervalSeismic Log-Amplitude Attenuation CoefficientSeismic Log-Amplitude Attenuation Coefficient: solve initial-to-final amplitude ratioSeismic Log-Amplitude Attenuation Coefficient: solve propagation distanceField-Effect Transistor TransconductanceField-Effect Transistor Transconductance: solve small drain-current changeField-Effect Transistor Transconductance: solve small gate-voltage changeElectroplating Mass Deposition RateElectroplating Mass Deposition Rate: solve deposited coating massElectroplating Mass Deposition Rate: solve active plating durationCorrosion Penetration RateCorrosion Penetration Rate: solve equivalent material thickness lostCorrosion Penetration Rate: solve exposure durationIonizing-Radiation Average Dose RateIonizing-Radiation Average Dose Rate: solve absorbed or equivalent dose accumulatedIonizing-Radiation Average Dose Rate: solve exposure durationAcoustic Level Decay RateAcoustic Level Decay Rate: solve measured sound-level reductionAcoustic Level Decay Rate: solve elapsed decay timeStream Longitudinal GradientStream Longitudinal Gradient: solve channel elevation lossStream Longitudinal Gradient: solve channel-path lengthReservoir Hydraulic Retention TimeReservoir Hydraulic Retention Time: solve active reservoir water volumeReservoir Hydraulic Retention Time: solve representative throughflow rateAnalytical Calibration SensitivityAnalytical Calibration Sensitivity: solve calibration response changeAnalytical Calibration Sensitivity: solve analyte concentration changeSeismic Wave Travel SpeedSeismic Wave Travel Speed: solve seismic ray-path distanceSeismic Wave Travel Speed: solve measured travel timeVertical Seismic Profile Interval VelocityVertical Seismic Profile Interval Velocity: solve depth interval between receiversVertical Seismic Profile Interval Velocity: solve one-way travel-time intervalLoom Production RateLoom Production Rate: solve woven fabric length producedLoom Production Rate: solve active loom running time
Differential Equations formula relationships108 optional lessons
Exponential ODE AdvanceExponential ODE Advance: solve initial stateExponential ODE Advance: solve integrated rate ktHalf-Life Period CountHalf-Life Period Count: solve initial quantityHalf-Life Period Count: solve elapsed half-livesEuler State IncrementEuler State Increment: solve derivative at current stateEuler State Increment: solve time stepOscillator Angular-Frequency SquaredOscillator Angular-Frequency Squared: solve spring stiffnessOscillator Angular-Frequency Squared: solve oscillating massIntegrating-Factor ExponentialIntegrating-Factor Exponential: solve initial multiplierIntegrating-Factor Exponential: solve integrated coefficientRelaxation Decay FactorRelaxation Decay Factor: solve initial displacement from equilibriumRelaxation Decay Factor: solve dimensionless elapsed relaxationTime Constant from Reciprocal RateTime Constant from Reciprocal Rate: solve positive rate constantTime Constant from Reciprocal Rate: solve unit scaleExponential Mode AmplificationExponential Mode Amplification: solve initial mode amplitudeExponential Mode Amplification: solve integrated growth exponentDamping Logarithmic DecrementDamping Logarithmic Decrement: solve positive earlier peak amplitudeDamping Logarithmic Decrement: solve positive later peak amplitudeReaction Rate–Timestep NumberReaction Rate–Timestep Number: solve reaction rate magnitudeReaction Rate–Timestep Number: solve time-step widthWave Grid Points per WavelengthWave Grid Points per Wavelength: solve represented wavelengthWave Grid Points per Wavelength: solve spatial grid spacingOne-Dimensional Diffusion RMS LengthOne-Dimensional Diffusion RMS Length: solve twice the positive diffusivityOne-Dimensional Diffusion RMS Length: solve positive elapsed timeLinear Forced Steady-State ResponseLinear Forced Steady-State Response: solve constant forcing magnitudeLinear Forced Steady-State Response: solve positive restoring rateODE Step Amplification FactorODE Step Amplification Factor: solve new perturbation normODE Step Amplification Factor: solve previous perturbation normODE Stiffness Eigenvalue RatioODE Stiffness Eigenvalue Ratio: solve largest decay-rate magnitudeODE Stiffness Eigenvalue Ratio: solve smallest relevant decay-rate magnitudeViscous Damping RatioViscous Damping Ratio: solve actual viscous damping coefficientViscous Damping Ratio: solve critical damping coefficientTime Steps per Oscillation PeriodTime Steps per Oscillation Period: solve oscillation periodTime Steps per Oscillation Period: solve time-step widthLinear Stability Abscissa MarginLinear Stability Abscissa Margin: solve stability-boundary real partLinear Stability Abscissa Margin: solve largest eigenvalue real partForced Oscillation Frequency DetuningForced Oscillation Frequency Detuning: solve forcing angular frequencyForced Oscillation Frequency Detuning: solve natural angular frequencyReaction–Diffusion Characteristic LengthReaction–Diffusion Characteristic Length: solve positive diffusivityReaction–Diffusion Characteristic Length: solve positive reaction-rate magnitudeModal Energy Exponential DecayModal Energy Exponential Decay: solve initial modal energyModal Energy Exponential Decay: solve integrated positive decay exponentBoundary Inflow–Outflow BalanceBoundary Inflow–Outflow Balance: solve total oriented inflowBoundary Inflow–Outflow Balance: solve total oriented outflowRelative Conservation Defect PercentageRelative Conservation Defect Percentage: solve absolute conservation imbalanceRelative Conservation Defect Percentage: solve reference conserved quantity magnitudeEigenmode Timescale from Decay RateEigenmode Timescale from Decay Rate: solve positive decay-rate magnitudeEigenmode Timescale from Decay Rate: solve unit reciprocal scalePDE Discrete State CapacityPDE Discrete State Capacity: solve spatial degree-of-freedom countPDE Discrete State Capacity: solve state variables per spatial degreeAdaptive Time-Step Change FactorAdaptive Time-Step Change Factor: solve proposed new time-step widthAdaptive Time-Step Change Factor: solve current time-step widthOperator-Splitting Local Error ScaleOperator-Splitting Local Error Scale: solve commutator error coefficientOperator-Splitting Local Error Scale: solve time-step widthLargest Lyapunov Exponent from Separation GrowthLargest Lyapunov Exponent from Separation Growth: solve positive perturbation growth factorLargest Lyapunov Exponent from Separation Growth: solve elapsed evolution timePitchfork Normal-Form Equilibrium AmplitudePitchfork Normal-Form Equilibrium Amplitude: solve positive bifurcation parameterPitchfork Normal-Form Equilibrium Amplitude: solve positive cubic saturation coefficientFloquet Cycle Amplification FactorFloquet Cycle Amplification Factor: solve Floquet multiplier magnitudeFloquet Cycle Amplification Factor: solve completed forcing cyclesControl-System Gain Margin FactorControl-System Gain Margin Factor: solve gain at instability thresholdControl-System Gain Margin Factor: solve current open-loop gain scaleControl-System Phase MarginControl-System Phase Margin: solve instability phase boundary in degreesControl-System Phase Margin: solve open-loop phase at gain crossover in degreesExponential Settling Time from Decay RequirementExponential Settling Time from Decay Requirement: solve required positive decay exponentExponential Settling Time from Decay Requirement: solve positive exponential decay rateObserved Poincaré Recurrence RateObserved Poincaré Recurrence Rate: solve detected returns to selected sectionObserved Poincaré Recurrence Rate: solve trajectory observation timeBifurcation Parameter DistanceBifurcation Parameter Distance: solve evaluated control parameterBifurcation Parameter Distance: solve critical bifurcation parameterLogistic Model Positive EquilibriumLogistic Model Positive Equilibrium: solve carrying-capacity scaleLogistic Model Positive Equilibrium: solve net surviving growth fraction
Probability formula relationships231 optional lessons
Binomial Expected SuccessesBinomial Expected Successes: solve success probabilityBinomial Expected Successes: solve trial countIndependent Event UnionIndependent Event Union: solve probability of AIndependent Event Union: solve probability of BRepeated Outcome Tree CountRepeated Outcome Tree Count: solve outcomes per stageRepeated Outcome Tree Count: solve independent stagesOdds from Outcome CountsOdds from Outcome Counts: solve favourable outcome countOdds from Outcome Counts: solve unfavourable outcome countLikelihood RatioLikelihood Ratio: solve likelihood under first hypothesisLikelihood Ratio: solve likelihood under second hypothesisPosterior Odds UpdatePosterior Odds Update: solve prior oddsPosterior Odds Update: solve Bayes factorRelative Frequency RatioRelative Frequency Ratio: solve observed event countRelative Frequency Ratio: solve total observation countIndependent Event IntersectionIndependent Event Intersection: solve probability of event AIndependent Event Intersection: solve probability of event BBayes NumeratorBayes Numerator: solve prior probabilityBayes Numerator: solve evidence likelihoodAt Least One Event from Failure ProbabilitiesAt Least One Event from Failure Probabilities: solve failure probability for first independent eventAt Least One Event from Failure Probabilities: solve failure probability for second independent eventBinary Log-Loss ContributionBinary Log-Loss Contribution: solve outcome weightBinary Log-Loss Contribution: solve predicted probability assigned to outcomeEntropy Category ContributionEntropy Category Contribution: solve category probability weightEntropy Category Contribution: solve same category probabilityPoisson Mean from Rate and ExposurePoisson Mean from Rate and Exposure: solve event rate per exposure unitPoisson Mean from Rate and Exposure: solve total exposureMarkov Transition Probability FlowMarkov Transition Probability Flow: solve source-state probabilityMarkov Transition Probability Flow: solve conditional transition probabilityEffective State Count from EntropyEffective State Count from Entropy: solve state-count scaleEffective State Count from Entropy: solve entropy in natsProbability-Weighted Payoff ContributionProbability-Weighted Payoff Contribution: solve outcome probabilityProbability-Weighted Payoff Contribution: solve outcome payoffInverse Probability WeightInverse Probability Weight: solve selection probabilityInverse Probability Weight: solve weight scaleEffective Outcomes from Entropy BitsEffective Outcomes from Entropy Bits: solve outcome base per bitEffective Outcomes from Entropy Bits: solve entropy in bitsLikelihood Geometric FactorLikelihood Geometric Factor: solve positive likelihood scaleLikelihood Geometric Factor: solve mean log-likelihoodInformation Gain DifferenceInformation Gain Difference: solve prior uncertainty measureInformation Gain Difference: solve posterior uncertainty measureExpected Pair Collision CountExpected Pair Collision Count: solve candidate pair countExpected Pair Collision Count: solve per-pair collision probabilityMutual Information from Entropy ReductionMutual Information from Entropy Reduction: solve marginal entropyMutual Information from Entropy Reduction: solve conditional entropyConditional Entropy from Joint EntropyConditional Entropy from Joint Entropy: solve joint entropy H(X,Y)Conditional Entropy from Joint Entropy: solve conditioning-variable entropy H(Y)Kullback–Leibler Divergence from Cross-Entropy GapKullback–Leibler Divergence from Cross-Entropy Gap: solve cross entropy H(P,Q)Kullback–Leibler Divergence from Cross-Entropy Gap: solve source entropy H(P)Absolute Risk DifferenceAbsolute Risk Difference: solve first event riskAbsolute Risk Difference: solve second event riskRelative Risk Reduction PercentageRelative Risk Reduction Percentage: solve absolute risk reductionRelative Risk Reduction Percentage: solve control-group event riskNumber Needed from Absolute Risk ReductionNumber Needed from Absolute Risk Reduction: solve absolute risk reductionNumber Needed from Absolute Risk Reduction: solve unit probability scaleIndependent Random-Walk RMS DisplacementIndependent Random-Walk RMS Displacement: solve squared step lengthIndependent Random-Walk RMS Displacement: solve independent step countNormalized Mutual Information PercentageNormalized Mutual Information Percentage: solve mutual informationNormalized Mutual Information Percentage: solve selected entropy normalizationVariation of InformationVariation of Information: solve conditional entropy H(X|Y)Variation of Information: solve conditional entropy H(Y|X)Jensen–Shannon Entropy GapJensen–Shannon Entropy Gap: solve mixture-distribution entropyJensen–Shannon Entropy Gap: solve weighted component-entropy meanSource Coding Efficiency PercentageSource Coding Efficiency Percentage: solve source entropy per symbolSource Coding Efficiency Percentage: solve average code length per symbolSource Code Redundancy per SymbolSource Code Redundancy per Symbol: solve average code length per symbolSource Code Redundancy per Symbol: solve source entropy per symbolChannel Capacity Utilization PercentageChannel Capacity Utilization Percentage: solve reliable information rateChannel Capacity Utilization Percentage: solve channel capacityMin-Entropy from Maximum ProbabilityMin-Entropy from Maximum Probability: solve logarithm unit scaleMin-Entropy from Maximum Probability: solve maximum outcome probabilityEffective Outcomes from Collision ProbabilityEffective Outcomes from Collision Probability: solve collision probabilityEffective Outcomes from Collision Probability: solve unit reciprocal scaleBhattacharyya Distance from CoefficientBhattacharyya Distance from Coefficient: solve logarithm unit scaleBhattacharyya Distance from Coefficient: solve positive Bhattacharyya coefficientTotal Correlation Multi-InformationTotal Correlation Multi-Information: solve sum of marginal entropiesTotal Correlation Multi-Information: solve joint entropyLittle's Law Average System PopulationLittle's Law Average System Population: solve effective arrival or throughput rateLittle's Law Average System Population: solve mean time spent in systemQueue Traffic IntensityQueue Traffic Intensity: solve arrival rateQueue Traffic Intensity: solve total service capacity rateWald Random-Sum ExpectationWald Random-Sum Expectation: solve expected number of summed termsWald Random-Sum Expectation: solve expected value of each termCount-Process Dispersion IndexCount-Process Dispersion Index: solve observed count varianceCount-Process Dispersion Index: solve observed count meanTwo-Component Series-System ReliabilityTwo-Component Series-System Reliability: solve first component reliabilityTwo-Component Series-System Reliability: solve second component reliabilityTwo-Component Parallel-System ReliabilityTwo-Component Parallel-System Reliability: solve first component reliabilityTwo-Component Parallel-System Reliability: solve second component reliabilityMarkov Mean Recurrence IntervalMarkov Mean Recurrence Interval: solve stationary state probabilityMarkov Mean Recurrence Interval: solve unit recurrence scaleBranching-Process Expected Next GenerationBranching-Process Expected Next Generation: solve current population countBranching-Process Expected Next Generation: solve mean offspring numberMarkov Detailed-Balance Flow ResidualMarkov Detailed-Balance Flow Residual: solve forward stationary transition flowMarkov Detailed-Balance Flow Residual: solve reverse stationary transition flowAbsorbing-Chain Probability RemainderAbsorbing-Chain Probability Remainder: solve unit total absorption probabilityAbsorbing-Chain Probability Remainder: solve sum of known absorbing-state probabilitiesObserved Renewal Event RateObserved Renewal Event Rate: solve completed renewal countObserved Renewal Event Rate: solve observation durationCumulative Hazard from Survival ProbabilityCumulative Hazard from Survival Probability: solve unit logarithmic scaleCumulative Hazard from Survival Probability: solve survival probabilityNormal Bayesian Posterior PrecisionNormal Bayesian Posterior Precision: solve prior precisionNormal Bayesian Posterior Precision: solve data likelihood precision contributionBayesian Data Precision Weight PercentageBayesian Data Precision Weight Percentage: solve data likelihood precision contributionBayesian Data Precision Weight Percentage: solve posterior total precisionPosterior Probability from OddsPosterior Probability from Odds: solve posterior odds in favourPosterior Probability from Odds: solve unit odds baselineBayesian Posterior Mean ShiftBayesian Posterior Mean Shift: solve posterior parameter meanBayesian Posterior Mean Shift: solve prior parameter meanExponential-Distribution Mean LifetimeExponential-Distribution Mean Lifetime: solve unit time-rate scaleExponential-Distribution Mean Lifetime: solve positive event rateExponential-Distribution Median LifetimeExponential-Distribution Median Lifetime: solve natural logarithm of twoExponential-Distribution Median Lifetime: solve positive event rateGeometric-Distribution Mean Trials to SuccessGeometric-Distribution Mean Trials to Success: solve unit trial scaleGeometric-Distribution Mean Trials to Success: solve success probability per trialWeibull Cumulative HazardWeibull Cumulative Hazard: solve time-to-scale positive ratioWeibull Cumulative Hazard: solve Weibull shape parameterWeibull Survival from Cumulative HazardWeibull Survival from Cumulative Hazard: solve unit initial survivalWeibull Survival from Cumulative Hazard: solve Weibull cumulative hazardQueue Service-Capacity Rate SlackQueue Service-Capacity Rate Slack: solve total service capacity rateQueue Service-Capacity Rate Slack: solve arrival rateM/M/1 Queue Idle ProbabilityM/M/1 Queue Idle Probability: solve unit probability totalM/M/1 Queue Idle Probability: solve traffic intensity rhoM/M/1 Mean System PopulationM/M/1 Mean System Population: solve traffic intensity rhoM/M/1 Mean System Population: solve one minus traffic intensityM/M/1 Mean Time in SystemM/M/1 Mean Time in System: solve unit customer-time scaleM/M/1 Mean Time in System: solve service-rate-minus-arrival-rate slackM/M/1 Mean Queue Waiting TimeM/M/1 Mean Queue Waiting Time: solve traffic intensity rhoM/M/1 Mean Queue Waiting Time: solve service-rate-minus-arrival-rate slackM/M/1 Mean Queue PopulationM/M/1 Mean Queue Population: solve traffic intensity rhoM/M/1 Mean Queue Population: solve one minus traffic intensityErlang Offered Traffic LoadErlang Offered Traffic Load: solve arrival rateErlang Offered Traffic Load: solve mean service or holding timeMulti-Server Offered Load per ServerMulti-Server Offered Load per Server: solve offered traffic in erlangsMulti-Server Offered Load per Server: solve parallel server countQueue Throughput Capacity Utilization PercentageQueue Throughput Capacity Utilization Percentage: solve effective completed-job throughputQueue Throughput Capacity Utilization Percentage: solve nominal service capacityQueue Unserved Arrival RemainderQueue Unserved Arrival Remainder: solve total arrivalsQueue Unserved Arrival Remainder: solve served or completed arrivalsBirthday Approximation Expected Collision PairsBirthday Approximation Expected Collision Pairs: solve reciprocal outcome-space sizeBirthday Approximation Expected Collision Pairs: solve sample countSport Shooting AccuracySport Shooting Accuracy: solve successful target hitsSport Shooting Accuracy: solve attempted shotsGoal Shot Conversion RateGoal Shot Conversion Rate: solve goals scoredGoal Shot Conversion Rate: solve valid shot attemptsAquaculture Stock Survival RateAquaculture Stock Survival Rate: solve surviving accounted animal countAquaculture Stock Survival Rate: solve initial stocked animal countSemiconductor Electrical Die YieldSemiconductor Electrical Die Yield: solve dies passing electrical acceptanceSemiconductor Electrical Die Yield: solve dies electrically testedRadionuclide Activity RateRadionuclide Activity Rate: solve nuclear decays counted or inferredRadionuclide Activity Rate: solve elapsed counting intervalRadionuclide Decay Branching FractionRadionuclide Decay Branching Fraction: solve decays through selected branchRadionuclide Decay Branching Fraction: solve total parent decaysRadiation Detector Counting EfficiencyRadiation Detector Counting Efficiency: solve net recorded detection eventsRadiation Detector Counting Efficiency: solve radiation quanta emitted toward stated basis
Linear Algebra formula relationships240 optional lessons
Diagonal Matrix DeterminantDiagonal Matrix Determinant: solve first diagonal entryDiagonal Matrix Determinant: solve second diagonal entryTwo-Component Vector MagnitudeTwo-Component Vector Magnitude: solve first componentTwo-Component Vector Magnitude: solve second componentRectangular Matrix Entry CountRectangular Matrix Entry Count: solve matrix rowsRectangular Matrix Entry Count: solve matrix columnsTensor Product DimensionTensor Product Dimension: solve first vector-space dimensionTensor Product Dimension: solve second vector-space dimensionTwo-Eigenvalue TraceTwo-Eigenvalue Trace: solve first eigenvalueTwo-Eigenvalue Trace: solve second eigenvalueTwo-Eigenvalue DeterminantTwo-Eigenvalue Determinant: solve first eigenvalueTwo-Eigenvalue Determinant: solve second eigenvalueTwo-Singular-Value Condition NumberTwo-Singular-Value Condition Number: solve largest singular valueTwo-Singular-Value Condition Number: solve smallest singular valueDiagonal Matrix Frobenius NormDiagonal Matrix Frobenius Norm: solve first diagonal entryDiagonal Matrix Frobenius Norm: solve second diagonal entryPlanar Vector Resultant MagnitudePlanar Vector Resultant Magnitude: solve resultant x-componentPlanar Vector Resultant Magnitude: solve resultant y-componentMatrix Dot-Product Operation CountMatrix Dot-Product Operation Count: solve output entriesMatrix Dot-Product Operation Count: solve terms per dot productOuter Product Entry CountOuter Product Entry Count: solve first vector lengthOuter Product Entry Count: solve second vector lengthKronecker Product Row DimensionKronecker Product Row Dimension: solve rows in first matrixKronecker Product Row Dimension: solve rows in second matrixLinear Map Scalar OutputLinear Map Scalar Output: solve input scalarLinear Map Scalar Output: solve linear scale factorDeterminant Area ScalingDeterminant Area Scaling: solve original oriented areaDeterminant Area Scaling: solve matrix determinantRank–Nullity DimensionRank–Nullity Dimension: solve linear-map rankRank–Nullity Dimension: solve nullityEigenvalue SeparationEigenvalue Separation: solve larger real eigenvalueEigenvalue Separation: solve smaller real eigenvalueScalar–Matrix EntryScalar–Matrix Entry: solve original matrix entryScalar–Matrix Entry: solve scalar multiplierLinear-Combination TermLinear-Combination Term: solve basis-coordinate valueLinear-Combination Term: solve basis-component valueSquare Matrix Entry CountSquare Matrix Entry Count: solve matrix-count factorSquare Matrix Entry Count: solve square-matrix orderSparse Matrix Storage PercentageSparse Matrix Storage Percentage: solve stored-entry percentageSparse Matrix Storage Percentage: solve dense entry capacityMatrix Rank DeficiencyMatrix Rank Deficiency: solve maximum possible rankMatrix Rank Deficiency: solve actual matrix rankDirect-Sum DimensionDirect-Sum Dimension: solve first subspace dimensionDirect-Sum Dimension: solve second subspace dimensionTensor Slice Entry CountTensor Slice Entry Count: solve first slice dimensionTensor Slice Entry Count: solve second slice dimensionRelative Residual Norm PercentageRelative Residual Norm Percentage: solve residual normRelative Residual Norm Percentage: solve reference right-hand-side normMatrix Aspect RatioMatrix Aspect Ratio: solve row countMatrix Aspect Ratio: solve column countVector Projection CoefficientVector Projection Coefficient: solve dot product with basis vectorVector Projection Coefficient: solve basis-vector normLeading Spectral GapLeading Spectral Gap: solve largest ordered eigenvalueLeading Spectral Gap: solve second-largest ordered eigenvalueUniform Block-Matrix Row CountUniform Block-Matrix Row Count: solve block-row countUniform Block-Matrix Row Count: solve rows per blockDegrees of Freedom after Independent ConstraintsDegrees of Freedom after Independent Constraints: solve unconstrained variable countDegrees of Freedom after Independent Constraints: solve independent constraint countVector Subspace CodimensionVector Subspace Codimension: solve ambient-space dimensionVector Subspace Codimension: solve subspace dimensionTwo-Singular-Value Squared EnergyTwo-Singular-Value Squared Energy: solve first singular valueTwo-Singular-Value Squared Energy: solve second singular valueAnisotropic Plane Area ScaleAnisotropic Plane Area Scale: solve first principal length scaleAnisotropic Plane Area Scale: solve second principal length scaleOperator Norm Amplification BoundOperator Norm Amplification Bound: solve operator normOperator Norm Amplification Bound: solve input vector normMatrix Left NullityMatrix Left Nullity: solve matrix row countMatrix Left Nullity: solve matrix rankEigenvalue Geometric DefectEigenvalue Geometric Defect: solve algebraic multiplicityEigenvalue Geometric Defect: solve geometric multiplicityTwo-Coordinate Manhattan NormTwo-Coordinate Manhattan Norm: solve absolute first coordinateTwo-Coordinate Manhattan Norm: solve absolute second coordinateRectangular Block Grid CountRectangular Block Grid Count: solve block-row countRectangular Block Grid Count: solve block-column countMatrix Factorization Fill-In PercentageMatrix Factorization Fill-In Percentage: solve new nonzero entries createdMatrix Factorization Fill-In Percentage: solve original nonzero entriesRelative Eigenvalue Gap PercentageRelative Eigenvalue Gap Percentage: solve absolute eigenvalue separationRelative Eigenvalue Gap Percentage: solve reference eigenvalue magnitudeOverlapping Subspaces Combined DimensionOverlapping Subspaces Combined Dimension: solve sum of individual subspace dimensionsOverlapping Subspaces Combined Dimension: solve intersection dimensionMean Eigenvalue from Matrix TraceMean Eigenvalue from Matrix Trace: solve matrix traceMean Eigenvalue from Matrix Trace: solve matrix dimensionDeterminant Geometric Scaling FactorDeterminant Geometric Scaling Factor: solve absolute determinantDeterminant Geometric Scaling Factor: solve matrix dimensionVector Cosine SimilarityVector Cosine Similarity: solve vector dot productVector Cosine Similarity: solve product of vector normsOrthogonal Projection Energy PercentageOrthogonal Projection Energy Percentage: solve squared projected normOrthogonal Projection Energy Percentage: solve squared original-vector normOrthogonal Residual EnergyOrthogonal Residual Energy: solve squared original-vector normOrthogonal Residual Energy: solve squared projected-vector normMatrix Elimination Pivot Growth FactorMatrix Elimination Pivot Growth Factor: solve largest magnitude during eliminationMatrix Elimination Pivot Growth Factor: solve largest original matrix magnitudeMatrix Row Diagonal-Dominance MarginMatrix Row Diagonal-Dominance Margin: solve absolute diagonal entryMatrix Row Diagonal-Dominance Margin: solve sum of off-diagonal magnitudesBlock Krylov Subspace CapacityBlock Krylov Subspace Capacity: solve iteration countBlock Krylov Subspace Capacity: solve starting block widthLow-Rank Matrix Storage Compression RatioLow-Rank Matrix Storage Compression Ratio: solve dense matrix parameter countLow-Rank Matrix Storage Compression Ratio: solve factorized parameter countLeading Singular-Value Energy PercentageLeading Singular-Value Energy Percentage: solve squared leading singular valueLeading Singular-Value Energy Percentage: solve squared Frobenius normMatrix Stable RankMatrix Stable Rank: solve squared Frobenius normMatrix Stable Rank: solve squared spectral normMean Singular ValueMean Singular Value: solve nuclear norm or singular-value sumMean Singular Value: solve singular-value countRayleigh QuotientRayleigh Quotient: solve quadratic form x-transpose-A-xRayleigh Quotient: solve squared vector normDeterminant Product LawDeterminant Product Law: solve determinant of first square matrixDeterminant Product Law: solve determinant of second square matrixScalar Schur ComplementScalar Schur Complement: solve retained pivot block valueScalar Schur Complement: solve eliminated coupling correctionPreconditioner Condition Improvement FactorPreconditioner Condition Improvement Factor: solve original condition numberPreconditioner Condition Improvement Factor: solve preconditioned condition numberMahalanobis Distance from Quadratic FormMahalanobis Distance from Quadratic Form: solve positive covariance-scaled quadratic formMahalanobis Distance from Quadratic Form: solve unit normalization scaleBasis Orthogonality DefectBasis Orthogonality Defect: solve product of basis-vector normsBasis Orthogonality Defect: solve absolute basis determinantEffective Participation CountEffective Participation Count: solve sum of squared normalized weightsEffective Participation Count: solve unit reciprocal scaleCondition-Number Decimal Digit LossCondition-Number Decimal Digit Loss: solve positive condition numberCondition-Number Decimal Digit Loss: solve decimal logarithm baseKrylov Dimension Utilization PercentageKrylov Dimension Utilization Percentage: solve realized Krylov subspace dimensionKrylov Dimension Utilization Percentage: solve maximum generated-vector capacityIterative Solver Vector Storage FootprintIterative Solver Vector Storage Footprint: solve stored work-vector countIterative Solver Vector Storage Footprint: solve bytes per work vectorGershgorin Relative Row Radius PercentageGershgorin Relative Row Radius Percentage: solve off-diagonal absolute row sumGershgorin Relative Row Radius Percentage: solve absolute diagonal centerRank-k Retained Spectral Energy PercentageRank-k Retained Spectral Energy Percentage: solve sum of retained squared singular valuesRank-k Retained Spectral Energy Percentage: solve full squared Frobenius normTruncated SVD Discarded EnergyTruncated SVD Discarded Energy: solve full squared Frobenius normTruncated SVD Discarded Energy: solve retained squared singular-value sumMatrix Solve Operation ThroughputMatrix Solve Operation Throughput: solve estimated arithmetic operation countMatrix Solve Operation Throughput: solve elapsed solve timeModel Parameter Sparsity PercentageModel Parameter Sparsity Percentage: solve zero parameter countModel Parameter Sparsity Percentage: solve total parameter countModel Parameter Compression RatioModel Parameter Compression Ratio: solve original parameter countModel Parameter Compression Ratio: solve compressed parameter countNormalized Algebraic Connectivity PercentageNormalized Algebraic Connectivity Percentage: solve second-smallest Laplacian eigenvalueNormalized Algebraic Connectivity Percentage: solve chosen degree or scale boundTensor Total Entry Count from Mode ProductsTensor Total Entry Count from Mode Products: solve selected mode dimensionTensor Total Entry Count from Mode Products: solve product of all remaining mode dimensionsTucker Core Entry CountTucker Core Entry Count: solve selected multilinear rankTucker Core Entry Count: solve product of remaining multilinear ranksCP Tensor Factor Parameter CountCP Tensor Factor Parameter Count: solve CP rankCP Tensor Factor Parameter Count: solve sum of tensor mode dimensionsTucker Tensor Storage Compression RatioTucker Tensor Storage Compression Ratio: solve dense tensor entry countTucker Tensor Storage Compression Ratio: solve Tucker core-plus-factor parameter countTensor Contraction Output Entry CountTensor Contraction Output Entry Count: solve product of free dimensions from first tensorTensor Contraction Output Entry Count: solve product of free dimensions from second tensorTensor Contraction Multiplication CountTensor Contraction Multiplication Count: solve output entry countTensor Contraction Multiplication Count: solve product of contracted dimensionsTensor Mode-Unfolding Aspect RatioTensor Mode-Unfolding Aspect Ratio: solve selected mode dimensionTensor Mode-Unfolding Aspect Ratio: solve product of remaining mode dimensionsTensor Average Mode DimensionTensor Average Mode Dimension: solve sum of tensor mode dimensionsTensor Average Mode Dimension: solve tensor orderTensor Rank–Border-Rank GapTensor Rank–Border-Rank Gap: solve tensor rankTensor Rank–Border-Rank Gap: solve tensor border rankTensor Nuclear-to-Frobenius Norm RatioTensor Nuclear-to-Frobenius Norm Ratio: solve selected tensor nuclear normTensor Nuclear-to-Frobenius Norm Ratio: solve tensor Frobenius normDiscrete Spectral-Radius Stability MarginDiscrete Spectral-Radius Stability Margin: solve unit-circle stability boundaryDiscrete Spectral-Radius Stability Margin: solve iteration or map spectral radius
Quantum Mathematics formula relationships63 optional lessons
Scaled De Broglie RelationshipScaled De Broglie Relationship: solve scaled momentumScaled De Broglie Relationship: solve unit-conversion factorScaled Photon Energy–FrequencyScaled Photon Energy–Frequency: solve scaled Planck constantScaled Photon Energy–Frequency: solve photon frequencyAngular Momentum Quantum ScaleAngular Momentum Quantum Scale: solve quantum indexAngular Momentum Quantum Scale: solve scaled reduced Planck constantUncertainty ProductUncertainty Product: solve position uncertaintyUncertainty Product: solve momentum uncertaintyEnergy–Time Uncertainty ProductEnergy–Time Uncertainty Product: solve energy uncertaintyEnergy–Time Uncertainty Product: solve characteristic time intervalAngular-Momentum Uncertainty ProductAngular-Momentum Uncertainty Product: solve first component uncertaintyAngular-Momentum Uncertainty Product: solve second component uncertaintyPhoton Momentum from EnergyPhoton Momentum from Energy: solve photon energyPhoton Momentum from Energy: solve propagation speedScaled Probability Amplitude SquareScaled Probability Amplitude Square: solve normalization scaleScaled Probability Amplitude Square: solve real amplitude magnitudeQuantum Overlap Probability ScaleQuantum Overlap Probability Scale: solve normalization factorQuantum Overlap Probability Scale: solve state-overlap magnitudeQuantum Energy-Level GapQuantum Energy-Level Gap: solve upper energy levelQuantum Energy-Level Gap: solve lower energy levelQuantum Dynamical Phase MagnitudeQuantum Dynamical Phase Magnitude: solve energy-time actionQuantum Dynamical Phase Magnitude: solve reduced Planck constantQuantum State Normalization RemainderQuantum State Normalization Remainder: solve target total probabilityQuantum State Normalization Remainder: solve sum of known basis probabilitiesEffective Quantum Dimension from PurityEffective Quantum Dimension from Purity: solve density-matrix purityEffective Quantum Dimension from Purity: solve unit reciprocal scaleQuantum State InfidelityQuantum State Infidelity: solve unit fidelity scaleQuantum State Infidelity: solve state fidelityQuantum Trace Distance from Trace NormQuantum Trace Distance from Trace Norm: solve density-operator difference trace normQuantum Trace Distance from Trace Norm: solve unit convention scaleNormalized von Neumann Entropy PercentageNormalized von Neumann Entropy Percentage: solve von Neumann entropyNormalized von Neumann Entropy Percentage: solve maximum entropy log dimensionQuantum Relative Phase DifferenceQuantum Relative Phase Difference: solve second basis-component phaseQuantum Relative Phase Difference: solve first basis-component phaseRabi Pulse AreaRabi Pulse Area: solve Rabi angular-frequency magnitudeRabi Pulse Area: solve pulse durationCoherent Oscillation Cycle CapacityCoherent Oscillation Cycle Capacity: solve coherence timeCoherent Oscillation Cycle Capacity: solve ordinary oscillation frequencyQuantum Survival Exponential DecayQuantum Survival Exponential Decay: solve initial survival probabilityQuantum Survival Exponential Decay: solve integrated positive loss exponentQuantum Purity LossQuantum Purity Loss: solve initial density-matrix purityQuantum Purity Loss: solve final density-matrix purity
Complex and Fourier formula relationships126 optional lessons
Complex Product ModulusComplex Product Modulus: solve first complex modulusComplex Product Modulus: solve second complex modulusComplex Sum Real PartComplex Sum Real Part: solve first real componentComplex Sum Real Part: solve second real componentComplex Difference Real PartComplex Difference Real Part: solve minuend real componentComplex Difference Real Part: solve subtrahend real componentComplex Component ModulusComplex Component Modulus: solve real componentComplex Component Modulus: solve imaginary componentFourier Bin ResolutionFourier Bin Resolution: solve sampling rateFourier Bin Resolution: solve sample countUniform Sample PeriodUniform Sample Period: solve record durationUniform Sample Period: solve sample intervalsAngular Frequency from CyclesAngular Frequency from Cycles: solve completed cyclesAngular Frequency from Cycles: solve elapsed timeWave Number from Phase CyclesWave Number from Phase Cycles: solve phase cycles across distanceWave Number from Phase Cycles: solve represented distanceNormalized Fourier CoefficientNormalized Fourier Coefficient: solve raw coefficient component sumNormalized Fourier Coefficient: solve sample countFourier Bin FrequencyFourier Bin Frequency: solve frequency-bin indexFourier Bin Frequency: solve frequency-bin widthComplex-Power ModulusComplex-Power Modulus: solve base complex modulusComplex-Power Modulus: solve real exponentPhase VelocityPhase Velocity: solve angular frequencyPhase Velocity: solve wave numberComplex Modulus SquaredComplex Modulus Squared: solve real componentComplex Modulus Squared: solve imaginary componentFourier Quadrature Mode EnergyFourier Quadrature Mode Energy: solve cosine coefficientFourier Quadrature Mode Energy: solve sine coefficientComplex Quadrature RMSComplex Quadrature RMS: solve in-phase componentComplex Quadrature RMS: solve quadrature componentSpatial Wavelength from Angular WavenumberSpatial Wavelength from Angular Wavenumber: solve cycle-scale factorSpatial Wavelength from Angular Wavenumber: solve angular wavenumberFFT Angular Bin SpacingFFT Angular Bin Spacing: solve cycle-scale factorFFT Angular Bin Spacing: solve transform lengthSampled Observation WindowSampled Observation Window: solve sample countSampled Observation Window: solve sampling intervalSpectral Resolution from Window DurationSpectral Resolution from Window Duration: solve window durationSpectral Resolution from Window Duration: solve cycle-scale factorHarmonic FrequencyHarmonic Frequency: solve harmonic numberHarmonic Frequency: solve fundamental frequencyBeat Frequency DifferenceBeat Frequency Difference: solve higher component frequencyBeat Frequency Difference: solve lower component frequencyOctave Frequency FactorOctave Frequency Factor: solve frequency ratio per octaveOctave Frequency Factor: solve octave intervalReal Power Factor RatioReal Power Factor Ratio: solve real powerReal Power Factor Ratio: solve apparent powerReactive-to-Apparent Power RatioReactive-to-Apparent Power Ratio: solve reactive power magnitudeReactive-to-Apparent Power Ratio: solve apparent powerSinusoidal RMS-to-Peak ScaleSinusoidal RMS-to-Peak Scale: solve RMS amplitudeSinusoidal RMS-to-Peak Scale: solve peak-to-RMS factorPower Level Ratio in DecibelsPower Level Ratio in Decibels: solve positive measured powerPower Level Ratio in Decibels: solve positive reference powerAmplitude Level Ratio in DecibelsAmplitude Level Ratio in Decibels: solve positive measured amplitudeAmplitude Level Ratio in Decibels: solve positive reference amplitudeAmplitude Ratio in NepersAmplitude Ratio in Nepers: solve positive measured amplitudeAmplitude Ratio in Nepers: solve positive reference amplitudeSpectral Density from Bin PowerSpectral Density from Bin Power: solve power contained in frequency binSpectral Density from Bin Power: solve bin bandwidthGroup Velocity from Dispersion SlopeGroup Velocity from Dispersion Slope: solve angular-frequency changeGroup Velocity from Dispersion Slope: solve angular-wavenumber changeGroup Delay from Phase Slope MagnitudeGroup Delay from Phase Slope Magnitude: solve phase change magnitudeGroup Delay from Phase Slope Magnitude: solve angular-frequency changeResonance Quality from BandwidthResonance Quality from Bandwidth: solve resonant centre frequencyResonance Quality from Bandwidth: solve half-power bandwidthSignal-to-Noise Power Ratio in DecibelsSignal-to-Noise Power Ratio in Decibels: solve positive signal powerSignal-to-Noise Power Ratio in Decibels: solve positive noise powerSpectral Amplitude Dynamic Range in DecibelsSpectral Amplitude Dynamic Range in Decibels: solve positive largest spectral amplitudeSpectral Amplitude Dynamic Range in Decibels: solve positive smallest resolved amplitudeSampling Oversampling FactorSampling Oversampling Factor: solve actual sample rateSampling Oversampling Factor: solve minimum Nyquist sample rateFFT Zero-Padding FactorFFT Zero-Padding Factor: solve padded transform lengthFFT Zero-Padding Factor: solve original sample lengthTime–Bandwidth ProductTime–Bandwidth Product: solve effective signal durationTime–Bandwidth Product: solve effective bandwidthWindow Coherent GainWindow Coherent Gain: solve sum of window coefficientsWindow Coherent Gain: solve window sample countSpectral Flatness PercentageSpectral Flatness Percentage: solve geometric mean spectral powerSpectral Flatness Percentage: solve arithmetic mean spectral powerParseval Mean Spectral EnergyParseval Mean Spectral Energy: solve total transform-domain squared magnitudeParseval Mean Spectral Energy: solve transform normalization countTotal Harmonic Distortion PercentageTotal Harmonic Distortion Percentage: solve root-sum-square harmonic amplitudeTotal Harmonic Distortion Percentage: solve fundamental amplitudeImage-Rejection Power Ratio in DecibelsImage-Rejection Power Ratio in Decibels: solve positive desired-sideband powerImage-Rejection Power Ratio in Decibels: solve positive image power
Discrete Mathematics formula relationships384 optional lessons
Cartesian Product CardinalityCartesian Product Cardinality: solve elements in set ACartesian Product Cardinality: solve elements in set BBinary Relation Pair CapacityBinary Relation Pair Capacity: solve source-set elementsBinary Relation Pair Capacity: solve target-set elementsFinite Function CountFinite Function Count: solve codomain choices per inputFinite Function Count: solve domain elementsFixed-Length String CountFixed-Length String Count: solve alphabet sizeFixed-Length String Count: solve string lengthComplete Bipartite Edge CountComplete Bipartite Edge Count: solve vertices in first partComplete Bipartite Edge Count: solve vertices in second partUndirected Graph Average DegreeUndirected Graph Average Degree: solve edge countUndirected Graph Average Degree: solve vertex countSet Symmetric-Difference SizeSet Symmetric-Difference Size: solve elements in unionSet Symmetric-Difference Size: solve elements in intersectionGrid State CountGrid State Count: solve states available per cellGrid State Count: solve independent grid cellsRadix Codeword CountRadix Codeword Count: solve symbols in the radixRadix Codeword Count: solve codeword lengthBoolean Assignment CountBoolean Assignment Count: solve truth values per variableBoolean Assignment Count: solve Boolean variable countFull Binary Tree Leaf CapacityFull Binary Tree Leaf Capacity: solve branching choices per levelFull Binary Tree Leaf Capacity: solve tree depthUniform K-Ary Tree Leaf CapacityUniform K-Ary Tree Leaf Capacity: solve children per internal nodeUniform K-Ary Tree Leaf Capacity: solve tree depthComplete Bipartite Graph Edge CapacityComplete Bipartite Graph Edge Capacity: solve first partition sizeComplete Bipartite Graph Edge Capacity: solve second partition sizeSimple Graph Edge Density PercentageSimple Graph Edge Density Percentage: solve present edgesSimple Graph Edge Density Percentage: solve possible edgesForest Edge CountForest Edge Count: solve vertex countForest Edge Count: solve connected-component countBoolean Truth-Table Row CountBoolean Truth-Table Row Count: solve truth values per variableBoolean Truth-Table Row Count: solve variable countFixed-Length Codeword CapacityFixed-Length Codeword Capacity: solve alphabet sizeFixed-Length Codeword Capacity: solve codeword lengthGraph Complement Edge CountGraph Complement Edge Count: solve possible simple edgesGraph Complement Edge Count: solve present graph edgesComplete Directed Graph Arc CountComplete Directed Graph Arc Count: solve vertex countComplete Directed Graph Arc Count: solve other-vertex choices per sourceUnordered Pair CapacityUnordered Pair Capacity: solve item countUnordered Pair Capacity: solve other-item choicesFinite Group Direct-Product OrderFinite Group Direct-Product Order: solve first group orderFinite Group Direct-Product Order: solve second group orderBinary Relation CountBinary Relation Count: solve truth choices per ordered pairBinary Relation Count: solve ordered-pair countFull M-Ary Tree Leaf CountFull M-Ary Tree Leaf Count: solve children-minus-one branching factorFull M-Ary Tree Leaf Count: solve internal node countGraph Cycle RankGraph Cycle Rank: solve edge-minus-vertex excessGraph Cycle Rank: solve connected-component countFinite Set Complement CardinalityFinite Set Complement Cardinality: solve universal-set cardinalityFinite Set Complement Cardinality: solve subset cardinalityFinite Quotient Group OrderFinite Quotient Group Order: solve finite group orderFinite Quotient Group Order: solve normal subgroup orderHomomorphism Kernel–Image OrderHomomorphism Kernel–Image Order: solve kernel orderHomomorphism Kernel–Image Order: solve image orderCoprime Residue Density PercentageCoprime Residue Density Percentage: solve coprime residue countCoprime Residue Density Percentage: solve complete residue countUniform Left-Degree Bipartite Edge CountUniform Left-Degree Bipartite Edge Count: solve left-partition vertex countUniform Left-Degree Bipartite Edge Count: solve degree of each left vertexCell-Complex Euler CharacteristicCell-Complex Euler Characteristic: solve total even-dimensional cell countCell-Complex Euler Characteristic: solve total odd-dimensional cell countUniform Hypergraph Incidence CountUniform Hypergraph Incidence Count: solve hyperedge countUniform Hypergraph Incidence Count: solve vertices per hyperedgeSimplicial Facet Incidence CountSimplicial Facet Incidence Count: solve simplex countSimplicial Facet Incidence Count: solve facets per simplexGraph Component ReductionGraph Component Reduction: solve initial connected-component countGraph Component Reduction: solve final connected-component countBoolean Formula Clause DensityBoolean Formula Clause Density: solve clause countBoolean Formula Clause Density: solve variable countLinear Code Rate PercentageLinear Code Rate Percentage: solve information-symbol countLinear Code Rate Percentage: solve codeword-symbol countCode Redundancy PercentageCode Redundancy Percentage: solve redundant-symbol countCode Redundancy Percentage: solve codeword-symbol countGCD–LCM Product TheoremGCD–LCM Product Theorem: solve greatest common divisorGCD–LCM Product Theorem: solve least common multiplePrime-Counting Density PercentagePrime-Counting Density Percentage: solve prime count in intervalPrime-Counting Density Percentage: solve integer count in intervalMean Consecutive-Prime GapMean Consecutive-Prime Gap: solve span from first to last primeMean Consecutive-Prime Gap: solve number of consecutive gapsModular Orbit Period RepetitionsModular Orbit Period Repetitions: solve total orbit stepsModular Orbit Period Repetitions: solve fundamental period lengthEuler Totient Residue DensityEuler Totient Residue Density: solve coprime residue count phi(n)Euler Totient Residue Density: solve modulus nCoprime Chinese-Remainder Combined ModulusCoprime Chinese-Remainder Combined Modulus: solve first coprime modulusCoprime Chinese-Remainder Combined Modulus: solve second coprime modulusDirichlet Convolution Divisor TermDirichlet Convolution Divisor Term: solve first arithmetic-function value f(d)Dirichlet Convolution Divisor Term: solve paired value g(n/d)Denominator-Squared Rational Approximation ErrorDenominator-Squared Rational Approximation Error: solve absolute rational approximation errorDenominator-Squared Rational Approximation Error: solve positive denominatorSquare-Free Integer Density PercentageSquare-Free Integer Density Percentage: solve square-free integer countSquare-Free Integer Density Percentage: solve integers examinedComplementary Divisor FactorComplementary Divisor Factor: solve positive integerComplementary Divisor Factor: solve known positive divisorUniform Occupancy LoadUniform Occupancy Load: solve object countUniform Occupancy Load: solve container countPigeonhole Capacity ExcessPigeonhole Capacity Excess: solve object countPigeonhole Capacity Excess: solve one-per-container capacityTwo-Set Inclusion–Exclusion OvercountTwo-Set Inclusion–Exclusion Overcount: solve sum of individual set cardinalitiesTwo-Set Inclusion–Exclusion Overcount: solve union cardinalityDerangement Share PercentageDerangement Share Percentage: solve derangement countDerangement Share Percentage: solve total permutation countGraph Global Transitivity PercentageGraph Global Transitivity Percentage: solve closed connected triplet countGraph Global Transitivity Percentage: solve all connected triplet countGraph Cut ConductanceGraph Cut Conductance: solve edge boundary sizeGraph Cut Conductance: solve smaller side volumeMatching Vertex Saturation PercentageMatching Vertex Saturation Percentage: solve vertices covered by matchingMatching Vertex Saturation Percentage: solve total graph verticesGraph Community Modularity ExcessGraph Community Modularity Excess: solve observed within-community edge fractionGraph Community Modularity Excess: solve null-model expected fractionRelative Hamming Distance PercentageRelative Hamming Distance Percentage: solve differing coordinate countRelative Hamming Distance Percentage: solve codeword lengthFinite Covering-Space Euler CharacteristicFinite Covering-Space Euler Characteristic: solve covering degreeFinite Covering-Space Euler Characteristic: solve base-space Euler characteristicProduct-Space Euler CharacteristicProduct-Space Euler Characteristic: solve first-space Euler characteristicProduct-Space Euler Characteristic: solve second-space Euler characteristicConnected-Sum Euler CorrectionConnected-Sum Euler Correction: solve sum of component Euler characteristicsConnected-Sum Euler Correction: solve gluing correctionSet Jaccard Similarity PercentageSet Jaccard Similarity Percentage: solve intersection cardinalitySet Jaccard Similarity Percentage: solve union cardinalitySet Dice Similarity CoefficientSet Dice Similarity Coefficient: solve intersection cardinalitySet Dice Similarity Coefficient: solve sum of individual set cardinalitiesSet Overlap CoefficientSet Overlap Coefficient: solve intersection cardinalitySet Overlap Coefficient: solve smaller set cardinalityGraph Degree Centrality PercentageGraph Degree Centrality Percentage: solve vertex degreeGraph Degree Centrality Percentage: solve maximum possible degreeGraph Closeness CentralityGraph Closeness Centrality: solve reachable other-vertex countGraph Closeness Centrality: solve sum of shortest-path distancesGraph Edge ExpansionGraph Edge Expansion: solve cut boundary-edge countGraph Edge Expansion: solve vertices in smaller subsetGraph Matching Vertex DeficiencyGraph Matching Vertex Deficiency: solve total graph verticesGraph Matching Vertex Deficiency: solve vertices covered by matchingVertex-Cover Matching Lower-Bound GapVertex-Cover Matching Lower-Bound Gap: solve selected vertex-cover sizeVertex-Cover Matching Lower-Bound Gap: solve matching lower-bound sizeAverage Local Clustering CoefficientAverage Local Clustering Coefficient: solve sum of eligible local clustering coefficientsAverage Local Clustering Coefficient: solve eligible vertex countGraph Diameter-to-Radius RatioGraph Diameter-to-Radius Ratio: solve graph diameterGraph Diameter-to-Radius Ratio: solve graph radiusMax-Flow Min-Cut Certificate GapMax-Flow Min-Cut Certificate Gap: solve selected cut capacityMax-Flow Min-Cut Certificate Gap: solve feasible flow valueMinimization Linear-Program Integrality GapMinimization Linear-Program Integrality Gap: solve integer-feasible optimumMinimization Linear-Program Integrality Gap: solve linear-relaxation lower boundMinimization Approximation-Algorithm RatioMinimization Approximation-Algorithm Ratio: solve algorithm objective valueMinimization Approximation-Algorithm Ratio: solve optimal objective valueGame-Theory Price of AnarchyGame-Theory Price of Anarchy: solve worst equilibrium social costGame-Theory Price of Anarchy: solve globally optimal social costGame-Theory Price of StabilityGame-Theory Price of Stability: solve best equilibrium social costGame-Theory Price of Stability: solve globally optimal social costAverage Online-Learning RegretAverage Online-Learning Regret: solve cumulative regretAverage Online-Learning Regret: solve completed decision roundsDirected Graph Arc Reciprocity PercentageDirected Graph Arc Reciprocity Percentage: solve directed arcs having a reverse counterpartDirected Graph Arc Reciprocity Percentage: solve total directed arc countClosed Orientable-Surface Euler CharacteristicClosed Orientable-Surface Euler Characteristic: solve sphere Euler baseline twoClosed Orientable-Surface Euler Characteristic: solve twice the orientable genusClosed Nonorientable-Surface Euler CharacteristicClosed Nonorientable-Surface Euler Characteristic: solve sphere Euler baseline twoClosed Nonorientable-Surface Euler Characteristic: solve nonorientable crosscap numberRegular Graph Edge CountRegular Graph Edge Count: solve vertex countRegular Graph Edge Count: solve common vertex degreeGallai Independence–Vertex-Cover IdentityGallai Independence–Vertex-Cover Identity: solve maximum independent-set sizeGallai Independence–Vertex-Cover Identity: solve minimum vertex-cover sizeGraph Vertex-Degree VarianceGraph Vertex-Degree Variance: solve mean squared vertex degreeGraph Vertex-Degree Variance: solve mean vertex degreeChromatic Number–Clique Lower-Bound GapChromatic Number–Clique Lower-Bound Gap: solve graph chromatic numberChromatic Number–Clique Lower-Bound Gap: solve maximum clique sizeEdge-Chromatic Maximum-Degree GapEdge-Chromatic Maximum-Degree Gap: solve edge chromatic numberEdge-Chromatic Maximum-Degree Gap: solve maximum vertex degreeGraph Toughness Cut RatioGraph Toughness Cut Ratio: solve removed vertex countGraph Toughness Cut Ratio: solve components created after removalRSA Semiprime ModulusRSA Semiprime Modulus: solve first prime factorRSA Semiprime Modulus: solve second prime factorRSA Semiprime Euler TotientRSA Semiprime Euler Totient: solve first prime minus oneRSA Semiprime Euler Totient: solve second prime minus oneHash-Table Load FactorHash-Table Load Factor: solve stored item countHash-Table Load Factor: solve available bucket countBloom-Filter Bits per ElementBloom-Filter Bits per Element: solve allocated Bloom-filter bit countBloom-Filter Bits per Element: solve inserted element countBrute-Force Keyspace Exhaustion TimeBrute-Force Keyspace Exhaustion Time: solve candidate keyspace sizeBrute-Force Keyspace Exhaustion Time: solve tested candidates per unit timeBlock-Code Redundancy Symbol CountBlock-Code Redundancy Symbol Count: solve codeword symbol countBlock-Code Redundancy Symbol Count: solve information symbol countBlock-Code Error-Correction CapacityBlock-Code Error-Correction Capacity: solve minimum distance minus oneBlock-Code Error-Correction Capacity: solve two-error correction divisorBlock-Code Error-Detection CapacityBlock-Code Error-Detection Capacity: solve minimum code distanceBlock-Code Error-Detection Capacity: solve unit distance offsetSingleton-Bound Distance SlackSingleton-Bound Distance Slack: solve codeword-minus-information redundancy plus oneSingleton-Bound Distance Slack: solve minimum code distanceCritical-Path Activity Total FloatCritical-Path Activity Total Float: solve late activity start timeCritical-Path Activity Total Float: solve early activity start timeCritical-Path Activity Free FloatCritical-Path Activity Free Float: solve earliest successor start timeCritical-Path Activity Free Float: solve activity early finish timeSchedule Activity LatenessSchedule Activity Lateness: solve actual or planned completion timeSchedule Activity Lateness: solve due dateSchedule Activity EarlinessSchedule Activity Earliness: solve due dateSchedule Activity Earliness: solve actual or planned completion timeSchedule Critical RatioSchedule Critical Ratio: solve time remaining until due dateSchedule Critical Ratio: solve remaining processing timeParallel-Machine Average Assigned LoadParallel-Machine Average Assigned Load: solve total assigned processing timeParallel-Machine Average Assigned Load: solve parallel machine countWeighted Completion-Time ContributionWeighted Completion-Time Contribution: solve job priority weightWeighted Completion-Time Contribution: solve job completion timeImage Bit-Depth Level CountImage Bit-Depth Level Count: solve binary states per bitImage Bit-Depth Level Count: solve bits per channelUncompressed Raster Storage BitsUncompressed Raster Storage Bits: solve total raster pixel countUncompressed Raster Storage Bits: solve stored bits per pixelProduction Takt TimeProduction Takt Time: solve net available production timeProduction Takt Time: solve required customer unit countAssembly-Line Balance EfficiencyAssembly-Line Balance Efficiency: solve total assigned task timeAssembly-Line Balance Efficiency: solve stations times cycle-time capacityAssembly-Line Balance DelayAssembly-Line Balance Delay: solve complete capacity percentageAssembly-Line Balance Delay: solve line-balance efficiency percentageProduction Bottleneck UtilizationProduction Bottleneck Utilization: solve bottleneck demanded processing timeProduction Bottleneck Utilization: solve bottleneck available timeProduction Throughput RateProduction Throughput Rate: solve completed conforming unit countProduction Throughput Rate: solve elapsed production timeProduction Flow Time from WIPProduction Flow Time from WIP: solve average work-in-process unitsProduction Flow Time from WIP: solve effective throughput rateProduction Setup-Time ShareProduction Setup-Time Share: solve total setup timeProduction Setup-Time Share: solve total scheduled production timeProduction Queue-Time ShareProduction Queue-Time Share: solve total waiting timeProduction Queue-Time Share: solve total end-to-end flow timeProduction Capacity CushionProduction Capacity Cushion: solve complete capacity percentageProduction Capacity Cushion: solve planned utilization percentageProduction Value-Added Time ShareProduction Value-Added Time Share: solve value-adding processing timeProduction Value-Added Time Share: solve total end-to-end flow timeRoad Traffic Flow RelationshipRoad Traffic Flow Relationship: solve traffic densityRoad Traffic Flow Relationship: solve space-mean speedTraffic Queue Clearance TimeTraffic Queue Clearance Time: solve queued vehicle countTraffic Queue Clearance Time: solve net queue discharge rateDigital Audio Sample DurationDigital Audio Sample Duration: solve audio sample-frame countDigital Audio Sample Duration: solve sample ratePCM Channel Bit RatePCM Channel Bit Rate: solve sample ratePCM Channel Bit Rate: solve stored bits per sampleDigital Audio Buffer LatencyDigital Audio Buffer Latency: solve buffer frame countDigital Audio Buffer Latency: solve sample rateUncompressed Audio Storage BitsUncompressed Audio Storage Bits: solve aggregate audio bit rateUncompressed Audio Storage Bits: solve recording durationAstrophotography Total Integration TimeAstrophotography Total Integration Time: solve accepted sub-exposure durationAstrophotography Total Integration Time: solve accepted exposure countAstrophotography Raw Storage RequirementAstrophotography Raw Storage Requirement: solve raw frame file sizeAstrophotography Raw Storage Requirement: solve stored frame countRobot Encoder Distance per CountRobot Encoder Distance per Count: solve wheel travel per revolutionRobot Encoder Distance per Count: solve effective encoder counts per revolutionRobot PWM Duty CycleRobot PWM Duty Cycle: solve pulse on-timeRobot PWM Duty Cycle: solve pulse periodPhotovoltaic Module Array Nameplate PowerPhotovoltaic Module Array Nameplate Power: solve module nameplate powerPhotovoltaic Module Array Nameplate Power: solve installed module countPhotovoltaic Series-String VoltagePhotovoltaic Series-String Voltage: solve module operating voltagePhotovoltaic Series-String Voltage: solve modules connected in seriesPhotovoltaic Parallel-String CurrentPhotovoltaic Parallel-String Current: solve current from one matched stringPhotovoltaic Parallel-String Current: solve parallel string countBattery Series-Bank Nominal VoltageBattery Series-Bank Nominal Voltage: solve nominal voltage per series unitBattery Series-Bank Nominal Voltage: solve units connected in seriesBattery Parallel-Bank Charge CapacityBattery Parallel-Bank Charge Capacity: solve rated charge capacity per matched stringBattery Parallel-Bank Charge Capacity: solve parallel string countCrystal Unit-Cell Formula MassCrystal Unit-Cell Formula Mass: solve mass per crystallographic formula unitCrystal Unit-Cell Formula Mass: solve formula units per unit cellDigital Modulation Bits per SymbolDigital Modulation Bits per Symbol: solve uncoded or coded bit rateDigital Modulation Bits per Symbol: solve symbol rate

Clear answers

Learning mathematics with calculators

Where should I start?

Start at Step 1 unless its concepts already feel familiar. This path includes the prerequisites that directly support quantum mechanics; other mathematics is available separately below.

How do I learn with a calculator?

Predict first, calculate second, inspect the working, then change one input and explain why the result changed.

Can this pathway lead toward quantum mechanics?

Yes, by building the required mathematics in order: algebra, trigonometry, probability, calculus, linear algebra, differential equations, complex numbers and Fourier methods before quantum models.

MW SysArc Certified