multivariable-chain-rule

Shows a concrete composition f: R²→R² feeding into g: R²→R², tracks a moving input point x, evaluates the Jacobians Df(x) and Dg(f(x)), then animates the chain rule by highlighting the row/column used in the matrix product to form D(g∘f)(x). Emphasizes “evaluate Dg at f(x)” and “compose linear maps = multiply matrices.”

canvasclick to interact

t=0s

practical uses

01.Backpropagation in neural networks (Jacobian products through layers)
02.Change of variables and sensitivity analysis in physics/engineering models
03.Robotics/kinematics: composing coordinate transforms and their derivatives

technical notes

Pure Canvas2D; responsive scale via min(w,h)/baseSize and 6px-ish grid snapping for a blocky aesthetic. Uses time-phased animation (3 steps over ~3.6s) plus moving packets on arrows and per-entry row/column highlighting to illustrate Jacobian matrix multiplication.

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