# Exterior derivative of 1-form and derivatives of sections of $T^*M$

Let $$M$$ be a compact manifold and consider a differential form $$\alpha\in\Omega^1(M)$$, which we can think of as a map $$\alpha:M\to T^*M$$. Since $$T^*M$$ is a smooth manifold we can compute the differential of this map $$D\alpha:TM\to T(T^*M)$$ How does this differential differ from the exterior derivative on 1-forms? I have a feeling that $$D\alpha(X) = d(\alpha(X))$$, but I'm not sure how to show this.

• What is $X$?${}{}$ – Camilo Arosemena-Serrato Jun 2 at 19:13
• Presumably a smooth vector field on M. – yousuf soliman Jun 3 at 2:15
• Your feeling can't be right: These objects live in different places. Moreover, $D\alpha(X)$ depends pointwise just on the value of $X$ at the point, whereas $d(\alpha(X))$ involves differentiating the vector field $X$. – Ted Shifrin Jun 3 at 17:56