# 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

## See also

-No Additional Notes-