Here are some useful references to books on differential forms.

A Course in Mathematics for Students of Physics

Bamberg and Sternberg

I wrote a glowing review of this for the American Journal of Physics. The solid reference to start with if you have lots of time, etc.

Spacetime, Geometry, Cosmology

William L. Burke

A casual introduction with some pretty off the wall applications to cosmology, water waves, etc. Unfortunately out of print.

Applied Differential Geometry

William L. Burke

This used to be the first quarter of the General Relativity Course. It mutated into this.

Advanced Calculus: A Differential Forms Approach

Harold M. Edwards

When I first stumbled across this book last year at Computer Literacy Bookstore I thought, about time. Then I discovered that it was a reprint of a 1969 book. He was really ahead of his time.

Second Year Calculus: From Celestial Mechanics to Special Relativity

David M. Bressoud

Uses differential forms throughout.

Differential Forms

Harley Flanders

This is a classic. His attempt to write down to physicists at times makes this feel inelegant, like his definition of differential forms as "those things that you integrate", but it repays study. I neglected this one for decades.

Applied Exterior Calculus

Dominic G. B. Edelen

Div Grad and Curl Are Dead

William L. Burke

In preparation. I intend for this to be the short sweet 100 page introduction to forms that every Junior should read if they are interested in the mathematics of this century.

Tensor Analysis for Physicists

J. A. Schouten

This was the seminal book for me, filled with eggcrate pictures and even photos of plaster models. The antithesis of his book Ricci Calculus. Long out of print. Published in 1951. Not at UCSC.

Relativistic Electrodynamics and Differential Geometry

S. Parrott

Math book. QC631.P34 1987