Applied Differential Geometry
William L. Burke
Cambridge University Press
A unique book. Changes the way one thinks about geometry. The concepts and tools become second nature. I strongly recommend it for engineers who need differential geometry in their research (they do, whether they know it or not).

To give an example from page 134: "Vector fields that do not commute are called anholonomic. If two transformations commute, then the system would never leave a 2-surface. This obvious results is called the Frobenius Theorem."

Now after reading about the Frobenius Theorem elsewhere, few people would call in "obvious." Nonetheless, when you read Burke, you will agree. (Granted, it will not happen at first reading unless you are already familiar with the material. So you will read the book several times, which only adds to the pleasure.) Afterwards, you will be happy to consult the proof elsewhere.

Caveat: this book is not the place to go for a formal presentation. It may cause conniptions in the more ideological bourbakistes. Nothing should prevent one from also reading some of the excellent texts that present the material in a precise way, for instance those by Manfredo Perdigão do Carmo, Spivak, or Lang. Nonetheless, Burke is the one to go for the intuition.