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.

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