De Rham derivative of a differential 1-form

Definition

Definition by formula

Let $M$ be a differential manifold. Let $\omega$ be a differential 1-form, i.e. a section of the cotangent bundle $T^{*}M$. We define $d\omega$ as the following alternating 2-form:

$(d\omega )(X,Y):=X(\omega Y)-Y(\omega X)-\omega ([X,Y])$