In this lecture we prove Stokes' Theorem (twice).

The local version states for a that for a singular $k$-chain $\mathsf{q}$ and a differential $(k-1)$-form $\omega$, one has
$$ \int_{  \mathsf{q}} d \omega = \int_{ \partial \mathsf{q}}  \omega.$$
The case $k = 1$ is just the Fundamental Theorem of Calulus in disguise.

The global version only works for oriented manifolds and for differential forms with compact support of degree $m-1$. Modulo this, the statement is visually very similar.

$$ \int_{ M, \mathfrak{o}} d \omega = \int_{\partial M , \partial \mathfrak{o}}  \omega.$$

Next lecture we return to the de Rham cohomology groups of a smooth manifold.

Comments and questions?