23. The Exterior Differential

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 a $k$-form is an $(k+1)$-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, and introduce a third operator $i_X$ known as the interior product.

We conclude by proving a result known as Cartan’s Magic Formula^[1], which relates the three operators together: for any vector field $X$ on a smooth manifold $M$, one has $$ \mathcal{L}_X=d \circ i_X + i_X \circ d.$$