# 7. Vector fields

We begin this lecture by introducing the *Einstein Summation Convention*. This may at first seem like more effort than it’s worth, but I guarantee by the end of the course you will be thankful for it!

We then define *vector fields*** **as smooth sections of the tangent bundle, and define the *Lie bracket*** **of two vector fields. We define a *Lie algebra*** **and show that the space of vector fields forms an infinite-dimensional Lie algebra.

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