Dear MSE: My goal is to understand the generalized Stokes' theorem both intuitively and rigorously. Could someone give advice or recommend study materials to help understand the generalized Stokes' theorem both intuitively and rigorously?

Baby Rudin seems to contain a terse treatment of the generalized Stokes' theorem in Chapter 10: Integration of Differential Forms. Are there study materials suitable to accompany Baby Rudin's terse treatment?

  • $\begingroup$ I suggest the book "Vector Analysis" by Jänich. $\endgroup$
    – user204299
    Jun 15 '16 at 23:50

The treatment in Baby Rudin is awful. I don't think I've met anyone who wasn't thoroughly confused by it. You are not alone. (And I would caution against trying to read the later chapters, too. In particular, please don't try to learn Lebesgue integration from that book.)

A proper treatment requires learning some basic differential geometry. My favorite introduction is Loring Tu's Introduction to Manifolds. It covers all the material necessary for Stokes's theorem, proves the theorem, and does even more.

As far as intuition goes, you should think of Stokes's theorem as a generalized version of theorems like the fundamental theorem of calculus and Green's theorem. Stokes's theorem says $$\int_{\partial M} \omega = \int_M d\omega,$$ where $M$ is some manifold and $\omega$ is some "differential form" on $M$, and $\partial M$ is the boundary of $M$.

When $M$ is an interval $[a,b]$, we get $$\int_{\{a,b\}} f = \int_{[a,b]} f'(x)\, dx$$ which is just the usual fundamental theorem of calculus. Here the boundary of $M=[a,b]$ is just the set of endpoints $\{a,b\}$, and $\omega=f$.

We can also look the case when $M$ is some two-dimensional domain in the plane. It turns out that in this case Stokes's theorem recovers Green's theorem.


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