Demonstration of a basic formula involving differential forms

Question

I'm writing some notes on Lie Groups and I'm not sure if I should demonstrate this formula or not. Assume $\omega$ is a differencial form and $X,Y$ fields con a Manifold M, is there a simple way to demonstrate that $$d\omega(X,Y)=X\left(\omega\left(Y\right)\right)-Y\left(\omega\left(X\right)\right)-\omega\left(\left[X,Y\right]\right)$$

Answer 1

That formula is often used as the definition! Since you're asking this question, I'll assume you're using the other common definition of $d$ for one-forms, which is the coordinate formula $d\omega_{ij} = \partial_i \omega_j - \partial_j \omega_i.$ Contracting with $X^i Y^j$ we get $$d \omega(X,Y) = Y^j X(\omega_j) - X^j Y(\omega_j).$$

From the Leibniz rule we have $ Y^j X(\omega_j) = X(\omega_j Y^j) - \omega_j X(Y^j)$ and similarly for the other term, so we get

$$ d \omega(X,Y) = X(\omega(Y)) - Y(\omega(X)) -\omega_j (X(Y^j) - Y(X^j)).$$

Recognizing the last term as $\omega([X,Y])$ we have the desired formula.