Integration along fibers The following statements are  from the book heat kernel and dirac operator chapter 1.
" Let $\pi : M \rightarrow B $ be a fiber bundle with n-dimensional fiber, such that both M and B are oriented. If $\alpha \in {A}_c^k(M)$ is a compactly-supported differential form on M, its integral over the fibers of $ M \rightarrow B $ is the differential form $\int\limits_{M/B} \alpha \in A^{k-n}(B)$ such that
\begin{equation} 
\int\limits_B (\int\limits_{M/B} \alpha) \wedge \beta = \int\limits_M \alpha \wedge \pi^* \beta \quad  ... (1.15)
\end{equation}
for all differential forms $\beta$ on the base B. We sometimes write $ \pi_* \alpha $ instead of $ \int\limits_{M/B} $. It follows easily from the (1.15) that
\begin{equation} 
\pi_*(\alpha \wedge \pi^* \beta ) = \pi_* \alpha \wedge \beta \quad ...(1.16)
\end{equation}
for all $\alpha \in A_c(M) $ and $\beta \in A(\beta).$ "
My questions are the following:

*

*what is the intuition behind defining the integral of $\alpha$ over the fibers by the equation (1.15), and why the notion of integration along fiber  is important ?


*how to prove that equation 1.15 implies equation 1.16 ?
 A: For your first question about why integration along the fiber is important, note that it gives a cochain map of degree $-n$,
$$\pi_*: A^k(M) \to A^{k-n}(B).$$
If $M$ was an oriented rank $n$ vector bundle over $B$ then
it can be shown that this map induces isomorphism at the level of cohomology, $H^k(M) \to H^{k-n}(B)$. This is called the Thom isomorphism. You can find a proof in Bott & Tu's classic.
Now for your second part, notice that (1.15) is the defining property for $\pi_*\alpha$. That is if there were any other form $\omega$, which satisfied (1.15): $\int_B \omega\wedge\beta = \int_M \alpha\wedge\pi^*\beta$ for all $\beta$ (of appropriate degree) on the base, then we would have $\omega = \pi_*\alpha$. So, (1.15) uniquely characterizes $\pi_*\alpha$.
T show that $\pi_*(\alpha\wedge\pi^*\beta) = \pi_*\alpha\wedge\beta$, it suffices to show that $\pi_*\alpha \wedge \beta$ indeed satisfies the defining property (1.15) for $\alpha\wedge\pi^*\beta$. That is we need to show that, for any $\phi \in A^*(B)$.
$$
\int_B (\pi_*\alpha\wedge\beta)\wedge\phi= \int_M (\alpha\wedge\pi^*\beta)\wedge\pi^*\phi
$$
but this is obvious because of the defining property of $\pi_*\alpha$ we have,
$$
\int_B (\pi_*\alpha\wedge\beta)\wedge\phi = \int_B \pi_*\alpha\wedge(\beta\wedge\phi) = \int_M \alpha \wedge \pi^*(\beta\wedge\phi) = \int_M (\alpha \wedge \pi^*\beta)\wedge\pi^*\phi.
$$
Lastly, regarding your comment to @Ted about references, I would suggest Bott & Tu's book. They define integration along the fiber more explicitly using local charts (similar to what Ted was suggesting you to do). Then both (1.15) and (1.16) follow easily from that definition. FWIW, these formulas together are called the Projection Formulas (see Proposition 6.15 in Bott & Tu).
