partial derivative
One variable's slope
A partial derivative measures how a function of multiple variables changes when only one variable changes while the others stay fixed.
In calculus notation, ∂f/∂x represents the rate of change of f with respect to x alone.
The concept is fundamental to multivariable calculus, physics, and engineering: The partial derivative with respect to time gives the instantaneous rate of change.
Rate of change across space
- HessianGerman soldier-for-hire, 1770s
- chain ruleComposite function derivative method
- multivariableHaving several independent variables
- differentiationRecognizing meaningful differences
Curve linking equal elevations
- critical pointDecisive juncture in events
- saddle pointStationary point, not extremum
- Taylor expansionFunction as infinite polynomial
- linearizationApproximating nonlinear with linear
continuity · differentiability · domain · surface · vector field · manifold · boundary · limit
To make different
- slopeAngled terrain or steepness
- tangentLine touching curve once
- sensitivityResponsiveness to stimuli
- marginalBarely adequate
increment · response · perturbation · independent variable · dependent variable
Science of heat-energy relationships
- heat capacityEnergy to raise temperature
- fluxContinuous change or flow
- field theoryStudy of spatial phenomena
- mechanicsScience of motion
entropy · enthalpy · isothermal · isobaric
Smallest possible quantity
- maximumGreatest permitted quantity
- extremumFunction's boundary value
- saddle pointStationary point, not extremum
- convexityOutward-curving quality
gradient
Rate of change across space
- HessianGerman soldier-for-hire, 1770s
- chain ruleComposite function derivative method
- multivariableHaving several independent variables
contour line
Curve linking equal elevations
- critical pointDecisive juncture in events
- saddle pointStationary point, not extremum
- Taylor expansionFunction as infinite polynomial
linearization · continuity · differentiability · domain · surface · vector field · manifold · boundary · limit
change
To make different
- slopeAngled terrain or steepness
- tangentLine touching curve once
- sensitivityResponsiveness to stimuli
marginal · increment · response · perturbation · independent variable · dependent variable
thermodynamics
Science of heat-energy relationships
- heat capacityEnergy to raise temperature
- fluxContinuous change or flow
- field theoryStudy of spatial phenomena
mechanics · entropy · enthalpy · isothermal · isobaric
minimum
Smallest possible quantity
- maximumGreatest permitted quantity
- extremumFunction's boundary value
- saddle pointStationary point, not extremum
- neural networksneural network, neural networks, hidden layers, backpropagation, layers
- Autodiffbackpropagation, chain rule, gradients
- differential operatorsdel, gradient, divergence, curl, Hessian
- math symbolsplus, minus, equals, divide, multiply