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 $$ \mathcal{L}_X=d \circ i_X + i_X \circ d.$$

  1. Somewhat like Santa Claus, this formula is more magical if you are five years old. ↩︎

Comments and questions?