This site is being phased out.
Differential Forms: A Complement to Vector Calculus by Weintraub
I used the book once to teach an early course on differential forms. The book reads well, mostly. On the one hand, I found the book too advanced: the proofs of some theorems are too long if weighed against their importance. On the other hand, it seemed too elementary as there are no exercises with proofs but only ones that require straightforward computations. The discussion of orientation of manifolds is too informal in my view even though the last, "advanced" chapter makes up for that somewhat.
No discrete differential forms, unfortunately.
Contents
Preface
The algebra of differential forms
The fundamental correspondence
II Oriented manifolds
The notion of a manifold (with boundary)
III Differential forms revisited
k-forms
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?