Multivariable and vector calculus

前置知识

36 项必需的前置知识（一直追溯到底）

你将学会

• 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.

定义

• 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.

定理

• 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.

常见误解

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

通向

可选发现

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

学习路径

• Undergraduate mathematics major
• Applied mathematics
• Geometry and topology
• Analysis, PDE and dynamics
• Mathematical physics

延伸阅读

• John H. Hubbard and Barbara Burke Hubbard, Vector Calculus, Linear Algebra, and Differential Forms