In this lecture we give a more concrete definition of a tangent vector, by making use of a partition of unity. This is a key technical tool that we will often use in the course. It serves as a bridge between local computations and global computations on a manifold—it allows us to “decompose” a globally defined object into a bunch of locally defined objects, and also to “glue together” a bunch of locally defined objects into one global one. The main reason we insisted our manifolds had to be paracompact was to ensure that partitions of unity exist.

Next lecture we will give another alternative definition of tangent vectors as velocity vectors to curves. In Problem Sheet B there is (yet) another equivalent definition of tangent vectors. (The moral of the story is: You can never have enough ways to define a tangent vector.)

##### Remark

The proof that partitions of unity exist is mostly non-examinable—recall that I use a $( \clubsuit)$ to denote non-examinable material.