Today we define tensor fields, and show how to pull them back via a smooth map, and push them forward via a diffeomorphism. We prove that the Lie derivative $\mathcal{L}_X$ from Lecture 8 uniquely extends to a tensor derivation.

##### Remark

Historically, a tensor field was defined as a quantity on a manifold that behaved in a certain way with respect to coordinate transformations. This point of view is somewhat messy (albeit still marginally useful), and its treatment is relegated to Problem Sheet J.