Calculus and analysis

Understands derivatives as rates of change; can compute simple derivatives by hand.

Applies chain rule, partial derivatives, basic integration; computes gradients of simple loss functions.

Uses Jacobians, Hessians; derives backprop step from first principles; applies numerical differentiation.

Analyses convergence with smoothness/Lipschitz conditions; applies functional analysis to variational methods.

Works with measure theory, optimal transport, or differential geometry in ML.