Stokes theorem.

Recall the Fundamental Theorem of Calculus: If $f$ is a differentiable function then

$$f(b) - f(a) = \int_b^a \frac{df }{dt}(x) dx.$$

In the language we have developed in the previous section this can be written as

$$f(b) - f(a) = \int_{[a, b]} df$$

where we orient the $1$-dimensional manifold $[a, b]$ in the positive direction. We want to prove a more general result that will include Stokes theorem, Green's theorem, Gauss' theorem and the Divergence theorem. If $M$ is an oriented manifold of dimension $n$ with boundary $\partial M$ and $\omega$ is an $n-1$ form then Stoke's theorem says that

$$\int_M d\omega = \int_{\partial M} \omega.$$

Before we prove this result we need to make sense of the idea of a manifold with boundary.