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.
William L. Burke
A casual introduction with some pretty off the wall applications to cosmology, water waves, etc. Unfortunately out of print.
William L. Burke
This used to be the first quarter of the General Relativity Course. It mutated into this.
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.
David M. Bressoud
Uses differential forms throughout.
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.
Dominic G. B. Edelen
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.
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.
S. Parrott
Math book. QC631.P34 1987