differential forms

(1.2 hours to learn)

Summary

Differential forms are multilinear functions on vector fields which satisfy certain properties analogous to determinants. They are used to define a notion of integration on manifolds.

Context

This concept has the prerequisites:

determinant and volume (The volume interpretation of determinants is a motivation for the volume form.)
vector fields (Differential forms are functions which take multiple vector fields as input.)
determinant (Differential forms are defined in terms of determinants.)

Goals

Define differential forms in terms of multilinear functions on a vector space

Show that differential forms can be represented in terms of determinants of submatrices

Define the wedge product of differential forms

Be able to manipulate the wedge product algebraically

Core resources (read/watch one of the following)

-Paid-

→ Multivariable Mathematics

A textbook on linear algebra and multivariable calculus with proofs.

Section 8.2.1, "The multilinear setup," pages 335-339
Section 8.2.2, "Differential forms on R^n and the exterior derivative," up to Proposition 2.2, pages 339-340

[external website]