Let $\omega$ be a 1-form, and $X, Y$ vector fields. Then $d\omega(X, Y) = X \omega(Y) - Y \omega(X) - \omega([X, Y])$

Question

These are defined in a smooth manifold, $d$ is the exterior derivative, and $[\cdot, \cdot]$ is the Lie Bracket. I tried opening up these in coordinates, but it got overwhelming pretty quickly. Anyone knows a smart way to prove this?

Answer 1

The "smart" way to prove this (as one finds in the standard textbooks) is to check that the right-hand side defines a tensor; i.e., it is linear (in each slot) over the space of $C^\infty$ functions. After that, by multilinearity and the observed linearity over $C^\infty$ functions, we can just check for $X=\partial/\partial x^i$ and $Y=\partial/\partial x^j$ (for which the Lie bracket term of course disappears). But this is easy.