# 20. Cartan's Magic Formula and orientability

In this lecture we first state and prove a result known as Cartan’s Magic Formula^{[1]}, which tells us that for any vector field $X$ on a smooth manifold $M$, one has $$ \mathcal{L}_X=d \circ i_X + i_X \circ d.$$

We then move on to the notion of *orientability*. We define what it means for a vector space to be orientable (and throw shade on how linear algebra is usually taught at the same time), then define what it means for a vector bundle to be orientable. Finally, we define a manifold to be orientable if its tangent bundle is orientable as a vector bundle.

##### Remark

There are many different ways to approach orientability. They are all equivalent, but sometimes this is a little tricky to see. For instance, in Algebraic Topology it is common to work with a “homological” definition. I treated this in my course last year for vector bundles here and for manifolds here.

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