# 22. The Lie Derivative Revisited

Today we revisit the Lie Derivative $\mathcal{L}_X$ from Lecture 10 and show how it can be intepreted as a *tensor derivation* on the tensor algebra of a manifold, i.e. a local operator $\mathcal{L}_X \colon \mathscr{T}(M) \to \mathscr{T}(M)$ which commutes with contractions and acts like a derivation with respect to the tensor product.

We explain why Lie derivative of an alternating tensor of type $(0,k)$ — i.e. a differential $k$-form — is again alternating. Thus the Lie derivative can also be thought of as a local operator $\Omega(M) \to \Omega(M)$. Such an operator is called a *graded derivation.*

Next lecture we will study a particularly famous graded derivation: the *exterior differential*.

Comments and questions?