In this lecture we study differential forms. These are arguably more important than tensors, since they can (a) be differentiated and (b) be integrated.

We covered differentation today: the exterior differential of an $r$-form is an $(r+1)$-form. This generalises the idea that the differential of a $0$-form (i.e. a function $f$) is a 1-form (namely, $df$). We then prove that the exterior differential commutes with the Lie derivative.

Later we will cover integration of differential forms, and work our way towards a proof of Stokes’ Theorem. This will necessitate finally defining manifolds with boundary, and also what it means for a manifold to be oriented.

Once we have done this, Stokes’ Theorem takes the following form: Let $M$ be an oriented manifold of dimension $n$ with boundary. If $\omega$ is an $(n-1)$-form on $M$ with compact support then $$\int_M d \omega= \int_{ \partial M} \omega.$$