Differential Forms: A Complement to Vector Calculus by Weintraub
From Intelligent Perception
Differential Forms: A Complement to Vector Calculus by Steven H. Weintraub.
The book was used for Introduction to differential forms: course.
The book reads well, mostly.
The proofs of some theorems are too long if weighted against their importance.
The discussion of orientation of manifolds is too informal. The last, "advanced" chapter makes up for that somewhat.
No non-trivial exercises (proofs etc). Straightforward computations only.
No discrete differential forms, unfortunately.
II Oriented manifolds
The notion of a manifold (with boundary)
III Differential forms revisited
IV Integration of differential forms over oriented manifolds
The integral of a 0-form over a point (evaluation)
The integral of a 1-form over a curve (line integrals)
The integral of a 2-form over a surface (flux integrals)
The integral of a 3-form over a solid body (volume integrals)
Integration via pull-backs
V The generalized Stokes's Theorem
Statement of the theorem
The fundamental theorem of calculus and its analog for line integrals
Green's and Stokes's theorems
Proof of the GST
VI For the advanced reader
Differential forms in $R^n$ and Poincare's lemma
Manifolds, tangent vectors, and orientations
The basics of De Rham cohomology
Appendix: why does a mirror reverse left and right?