Mathematics / core
Multivariable and vector calculus
Build the local and integral calculus of several variables through Jacobians, Hessians, coordinate changes, curves, surfaces, and conservation laws.
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Inside this module
Module 01 core
Multivariable and vector calculus
Build the local and integral calculus of several variables through Jacobians, Hessians, coordinate changes, curves, surfaces, and conservation laws.
- 01
Total differentiability and linear approximation
Replace separate coordinate slopes with one linear map that controls errors in every direction.
26 min - 02
The chain rule, gradients, and Jacobians
Compose local linear maps in the correct order and distinguish gradients from general derivatives.
27 min - 03
Multivariable Taylor expansions and Hessians
Use second-order structure to classify stationary points and certify local approximation errors.
28 min - 04
Inverse maps and implicit level sets
Identify when a nonlinear map has a local inverse and when an equation defines a smooth graph.
29 min - 05
Multiple integrals and iterated integration
Translate a region into correct bounds and identify when changing integration order is justified.
28 min - 06
Change of variables and Jacobian determinants
Transform integration regions while accounting for local area or volume scale and multiplicity.
29 min - 07
Curve parametrizations and line integrals
Distinguish accumulated scalar density from oriented work and test when path independence is justified.
29 min - 08
Surface parametrizations and flux integrals
Build surface area and oriented flux from tangent vectors, cross products, and a consistent normal.
30 min - 09
Divergence, curl, and local conservation
Connect derivative combinations with local outflow, circulation, and the balance of a conserved density.
31 min - 10
Green's, Stokes', and divergence theorems
Relate interior derivatives to oriented boundary integrals while checking domains, regularity, and signs.
32 min