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Differential Forms: A Complement to Vector Calculus by Weintraub

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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

I Differential forms

The algebra of differential forms

Exterior differentiation

The fundamental correspondence

II Oriented manifolds

The notion of a manifold (with boundary)

Orientation

III Differential forms revisited

1-forms

k-forms

Push-forwards and pull-backs

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

Gauss's theorem

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?