Exterior derivative of a restricted 1-form

Question

I had a question about the exterior derivative of a restricted 1-form. Let S be a surface in 3-space. Suppose we define a 1-form on an open set containing the surface such that the restricted 1-form is zero. How do you then prove that the restricted exterior derivative is zero? Thanks.

Answer 1

Let the one form be $\alpha$, and let $\iota: S \rightarrow U $ be the inclusion to the open subset $U$. Then the restricted one form is $\alpha\big\lvert_S=\iota^*\alpha$, by assumption we have that $ \iota^*\alpha=0$. Then we have that

$$\iota^*d\alpha = d\iota^*\alpha =0$$

Since the pullback commutes with the exterior derivative.