I recommend Yvonne Chocquet-Bruhat, Géométrie différentielle et systèmes extérieurs because it is so short and has exercises and is in the notation you want. I recommend it most highly (even if you can't get it with her inscribed autograph.)

I also recommend very strongly her much longer (but try to get the first edition, which is still plenty long) Analysis, Manifolds and Physics by Yvonne Choquet-Bruhat, Cecile Dewitt-Morette, and Margaret Dillard-Bleick which has many exercises and much more Physics...but is too long. Goodness, it even includes Brownian Motion and path integrals....

That said, Dirac and Schroedinger have good and very short Physics books on the subject, I recommend those too even though they are not quite what you asked for.

Bob Geroch's book is valuable too.