Math & statistics
Three-dimensional dot product
Three-dimensional dot product: A·B = AxBx + AyBy + AzBz.
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Math & statistics
Three-dimensional cross product
Three-dimensional cross product: A×B = (AyBz−AzBy, AzBx−AxBz, AxBy−AyBx).
Math & statistics
Angle between two 2D vectors
Angle between two 2D vectors: Angle = arccos[(A·B)/(|A||B|)], returned in degrees from 0 to 180.
Math & statistics
Polar and Cartesian coordinate converter
Polar and Cartesian coordinate converter: Polar to Cartesian: x=r cos θ, y=r sin θ.
Math & statistics
Harmonic mean
Harmonic mean: Harmonic mean = count / Σ(1/value).
Math & statistics
Geometric mean
Geometric mean: Compute exp(mean(ln(value))) for positive observations.
Math & statistics
Coefficient of variation
Coefficient of variation: Sample coefficient of variation = sample standard deviation / arithmetic mean × 100.
Math & statistics
Mean absolute deviation about the mean
Mean absolute deviation about the mean: Find the arithmetic mean, then average the absolute distances of observations from that mean.
Math & statistics
Cohen’s d for independent samples
Cohen’s d for independent samples: Cohen’s d = difference of means divided by pooled sample SD; pooled variance weights each sample variance by n−1.
Math & statistics
Probability of at least one success
Probability of at least one success: For independent equal-probability trials, P(at least one success) = 1−(1−p)^n.
Math & statistics
Fractional odds implied probability
Fractional odds implied probability: Implied probability = denominator / (numerator + denominator) × 100.
Math & statistics
Finite population sample correction
Finite population sample correction: Corrected size = ceiling(n₀ / [1+(n₀−1)/N]).
Math & statistics
Polynomial synthetic division
Polynomial synthetic division: Synthetic division divides a polynomial by x−r.
Math & statistics
Modular exponentiation
Modular exponentiation: Compute base^exponent mod modulus using exponentiation by squaring with exact BigInt integer arithmetic.
Physics & engineering
Ideal projectile horizontal range
Ideal projectile horizontal range: For equal launch and landing heights without drag, horizontal range = v² sin(2θ)/g.
Physics & engineering
Free-fall time from rest
Free-fall time from rest: For a body released from rest with constant gravity and no drag, fall time = √(2h/g).
Physics & engineering
Centripetal force
Centripetal force: The inward force needed for uniform circular motion is F = mv²/r.
Physics & engineering
Angular velocity from rotation rate
Angular velocity from rotation rate: Angular velocity in radians/second = RPM × 2π/60.
Physics & engineering
Torque from force and angle
Torque from force and angle: Torque magnitude = radius × force × |sin(angle between radius and force)|, in newton-metres.
Physics & engineering
Impulse and momentum change
Impulse and momentum change: Impulse = average net force × duration.
Physics & engineering
Archimedes buoyant force
Archimedes buoyant force: Buoyant force = fluid density × displaced volume × gravitational acceleration, directed upward in a static fluid.
Physics & engineering
Hydrostatic gauge pressure
Hydrostatic gauge pressure: Gauge pressure increase below a static fluid surface is ρgh.
Physics & engineering
Ideal gas pressure
Ideal gas pressure: Ideal gas pressure P = nRT/V, using R = 8.31446261815324 J/(mol·K).
Physics & engineering
Sensible heat energy
Sensible heat energy: Heat energy Q = mass × specific heat capacity × temperature change.
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