I second the recommendation to at least flip through Gravitation. It has an intimidating size, but easygoing manner. I had a lot of difficulty with Spivak's Calculus on Manifolds (which has essentially no physical intuition outside the Archimedes exercise at the end), but I think I was uncomfortable with the abstract notions of tensor product and dual vector space at the time I was learning from it. At some point I caught on that df was supposed to eat vector fields and produce functions, and things got a little better.

You might try Sternberg's Advanced Calculus (available on line), especially chapters 11 and 13.

Edit: I like to think of abstract forms as "things to integrate" and Stokes's theorem as some kind of adjunction between boundaries and the derivative. This becomes a bit more meaningful when homology and cohomology are introduced. I don't have much advice for connecting with physical intuition, but I have found it useful to:

1. Decompose div, grad and curl in terms of d and the metric.
2. Work through some E&M starting from a 1-form (strictly speaking a U(1)-connection) A on $\mathbb{R}^{1,3}$ (see Wikipedia).

I second the recommendation to at least flip through Gravitation. It has an intimidating size, but easygoing manner. I had a lot of difficulty with Spivak's Calculus on Manifolds (which has essentially no physical intuition outside the Archimedes exercise at the end), but I think I was uncomfortable with the abstract notions of tensor product and dual vector space at the time I was learning from it. At some point I caught on that df was supposed to eat vector fields and produce functions, and things got a little better.

You might try Sternberg's Advanced Calculus (available on line), especially chapters 11 and 13.

Edit: I like to think of abstract forms as "things to integrate" and Stokes's theorem as some kind of adjunction between boundaries and the derivative. This becomes a bit more meaningful when homology and cohomology are introduced. I don't have much advice for connecting with physical intuition, but I have found it useful to:

1. Decompose div, grad and curl in terms of d and the metric.
2. Work through some E&M starting from a 1-form (strictly speaking a U(1)-connection) A on $\mathbb{R}^{1,3}$ (see Wikipedia).

I second the recommendation to at least flip through Gravitation. It has an intimidating size, but easygoing manner. I had a lot of difficulty with Spivak's Calculus on Manifolds (which has essentially no physical intuition outside the Archimedes exercise at the end), but I think I was uncomfortable with the abstract notions of tensor product and dual vector space at the time I was learning from it.

You might try Sternberg's Advanced Calculus (available on line), especially chapters 11 and 13.

I second the recommendation to at least flip through Gravitation. It has an intimidating size, but easygoing manner. I had a lot of difficulty with Spivak's Calculus on Manifolds (which has essentially no physical intuition outside the Archimedes exercise at the end), but I think I was uncomfortable with the abstract notions of tensor product and dual vector space at the time I was learning from it. At some point I caught on that df was supposed to eat vector fields and produce functions, and things got a little better.

You might try Sternberg's Advanced Calculus (available on line), especially chapters 11 and 13.

Edit: I like to think of abstract forms as "things to integrate" and Stokes's theorem as some kind of adjunction between boundaries and the derivative. This becomes a bit more meaningful when homology and cohomology are introduced. I don't have much advice for connecting with physical intuition, but I have found it useful to:

1. Decompose div, grad and curl in terms of d and the metric.
2. Work through some E&M starting from a 1-form (strictly speaking a U(1)-connection) A on $\mathbb{R}^{1,3}$ (see Wikipedia).