Can somebody help me understand this. Let $\omega$ be a closed two-form on $\mathbb{R}^3$ and $\eta$ a one-form such that $\omega=d\eta$. $M$ is an orientable manifold with boundary $\partial M$. $i:M\to\mathbb{R}^3$ and $j:\partial M\to M$ are inclusion maps.
What I don't understand is, why is it that you can claim that
$$\int_M i^*\omega =\int_M d(j^*(i^*\eta)) $$
I just forgot all the algebra rules involved.
I believe you can exchange the operators $d$ and $i^*$ like this $$i^*(d\eta) = d(i^*\eta)$$ right?
But how does the $j^*$ factor into all of this? As I said, I would love to know the algebraic rules on operations involving pullbacks because I used to have a list but I lost it and I cannot find much info online.
Maybe you have to use Stoke's Theorem:
$\int_Μd{(j^*(i^*(η))}=\int_{\partial M}j^*(i^*(η))=\int_{\partial M}(ij)^*(η))$
which makes sense since $ij$ is a function from $\partial M$ to $\mathbb{R^3}$.
The "algebraic" rule is that the pullback is a contravariant functor, meaning $(fg)^*=g^*f^*$