Exterior derivative of a one-form evaluated on vector fields
= Exterior derivative of a one-form evaluated on vector fields
For a smooth one-form $\alpha$ and smooth <vector fields> $X,Y$,
$$
d\alpha(X,Y)=X(\alpha(Y))-Y(\alpha(X))-\alpha([X,Y]).
$$
In coordinates the derivatives of the components of $X,Y$ cancel against the <Lie bracket of vector fields>, leaving $(\partial_i\alpha_j-\partial_j\alpha_i)X^iY^j$. This formula extends the coordinate definition of the <exterior derivative> to arbitrary frames.