# Study materials to help understand the generalized Stokes' theorem both intuitively and rigorously?

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?

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

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.