One way to define the algebra of differential forms $\Omega(M)$ on a smooth manifold $M$ (as explained by John Baez's week287) is as the exterior algebra of the dual of the module of derivations on the algebra $C^{\infty}(M)$ of smooth functions $M \to \mathbb{R}$. Given that derivations are vector fields, 1-forms send vector fields to smooth functions, and some handwaving about area elements suggests that k-forms should be built from 1-forms in an anticommutative fashion, I am **almost** willing to accept this definition as properly motivated.

One can now define the exterior derivative $d : \Omega(M) \to \Omega(M)$ by defining $d(f dg_1\ \dots\ dg_k) = df\ dg_1\ \dots\ dg_k$ and extending by linearity. I am **almost** willing to accept this definition as properly motivated as well.

Now, the exterior derivative (together with the Hodge star and some fiddling) generalizes the three main operators of multivariable calculus: the divergence, the gradient, and the curl. My intuition about the definitions and properties of these operators comes mostly from basic E&M, and when I think about the special cases of Stokes' theorem for div, grad, and curl, I think about the "physicist's proofs." What I'm not sure how to do, though, is to relate this down-to-earth context with the high-concept algebraic context described above.

**Question:** How do I see conceptually that differential forms and the exterior derivative, as defined above, naturally have physical interpretations generalizing the "naive" physical interpretations of the divergence, the gradient, and the curl? (By "conceptually" I mean that it is very unsatisfying just to write down the definitions and compute.) And how do I gain physical intuition for the generalized Stokes' theorem?

(An answer in the form of a textbook that pays special attention to the relationship between the abstract stuff and the physical intuition would be fantastic.)

From Calculus to Cohomologyby Madsden and Tornehave? It's not really about physical intuition (which is why I'm making this a comment), but it might be helpful. $\endgroup$ – Akhil Mathew Jan 3 '10 at 15:284more comments