# Tuesday, November 10

:::{.exercise title="?"}
In coordinates, show that the following two definitions of $[X, Y]$ are equivalent:

1. $[X, Y](f) \da X(Y(f)) - Y(X(f))$
2. $[X, Y] \da \mathcal{L}_X Y \da \dd{}{t}\evalfrom_{t=0} D\Phi^X_{-t}\qty{ Y_{\Phi^X_t(p)} }$ where $\Phi_t^X: M\to M$ is flowing along $X$ for time $t$.

:::