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-