Early undergraduateAnalysisPlanned

Multivariable and vector calculus

Calculus III: partial derivatives, the derivative as a linear map, optimisation with Lagrange multipliers, multiple integrals and change of variables, line and surface integrals, and the theorems of Green, Stokes and Gauss.

math.analysis.multivariable-calculus

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Builds on

Calculus of one variable Linear algebra

Required prerequisites, all the way down: 36

You will learn to

Compute partial and directional derivatives, gradients and Jacobian matrices.
Solve constrained optimisation problems with Lagrange multipliers.
Evaluate multiple, line and surface integrals and apply Green's theorem, Stokes's theorem and the divergence theorem.

Definition

The derivative of f : ℝⁿ → ℝᵐ at a is the linear map Df(a) such that f(a + h) = f(a) + Df(a)h + o(|h|) as h → 0.
The gradient ∇f of a real-valued function collects its partial derivatives.

Theorem

Chain rule: D(g ∘ f)(a) = Dg(f(a)) ∘ Df(a).
Change of variables in multiple integrals uses the absolute value of the Jacobian determinant.
Divergence theorem: the flux of F out of a bounded region V with smooth boundary equals ∭_V div F dV.
If all partial derivatives of f exist and are continuous near a, then f is differentiable at a.

Common misconceptions

Thinking the existence of partial derivatives implies differentiability.
Forgetting the Jacobian factor when changing variables.

Leads to

Classical mechanics Differential geometry of curves and surfaces Complex analysis Partial differential equations and Fourier series Optimisation Smooth manifolds

Optional discoveries

Functors The chain rule says that taking derivatives respects composition; in categorical language, differentiation at a point is functorial.