We begin this lecture by introducing the notion of orientability. We define what it means for a vector space to be orientable, then define what it means for a vector bundle to be orientable.

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.

We then (finally) get round to defining manifolds with boundary, starting with a topological manifold with boundary and then building from this a smooth manifold with boundary, just as we did in Lecture 1 in the boundary-less case.

We conclude by explaining how an orientation of a manifold with boundary $M$ induces an orientation on $\partial M$. This will be crucial in our treatment of Stokes' Theorem over the next couple of lectures.