TITLE: Intuition about pullbacks in differential geometry
QUESTION [16 upvotes]: I am struggling to understand the role of pullbacks after noticing that they are used when defining an integral of $k$-forms on a manifold.
Let $F:M \to N$ be a map between differentiable manifolds. The pullback of a covector $\omega \in (T_{F(p)}N)^*$ under $F$ at a point $p \in M$ is $F^*:(T_{F(p)}N)^* \to (T_pM)^*$ with
$$(F^*)_p(\omega)X = \omega((F_*)_pX)$$
where $X \in T_p(M)$.
I know that the pushforward map $F_*$ takes a tangent vector at a point in $M$ to a tangent vector at a point in $N$. Am i missing something else?
The pullback takes a covector in $N$ to a covector in $M$, but I can't see why it is important and don't have any intuition in it. I don't get why the pullback is used in integration either. I'd be grateful for any help.

REPLY [5 votes]: I think Jame's answer is really good, I just want to add one thing, regarding intuition, that may be obvious but it took myself some time to realize. 
Pullbacks is really the natural way to integrate given a certain parametrization. When you are in $\mathbb{R^n}$ it is just a fancy mechanism to compute the jacobian a.k.a. to make a correct change of variables a.k.a. to reparametrize the (hyper)surface. 
For example, let's compute $\int_{R}\omega$ where $R$ is the region bounded by the circle $x^2+y^2=4$ and the line $y=x$ and $\omega=(1+\frac{y^2}{x^2})dx\wedge dy$ .
So first we want to find a suitable parametrization. This turns out to be $\phi(r,\theta)=(rcos\theta,rsin\theta)$ with $0\leq r \leq2$ and $0\leq\theta\leq\frac{\pi}{4}$. So we have :
$\int_{R}\omega=\int_{[0,1]\times[0,\frac{\pi}{4}]}φ^*(ω)=\int_0^1 \int_0^\frac{\pi}{4}\frac{1}{cos^2θ}det[J_φ]dθ\wedge dr=\int_0^1 \int_0^\frac{\pi}{4}(-tanθ)'rdθ \wedge dr $ 
and it is easy from now on.
The important thing to note is that through all the abstract formalism needed to define forms in abstract manifolds, what you get is a way to compute the Jacobian. Try writting all the definitions in a map between surfaces in $\mathbb{R^3}$ to gain a better understanding.
The book thath helped me clear the confusion about differential forms is A geometric approach to differential forms  by Bachman and it has many excerscises and examples. When you feel comfortable you can move to Tu's Introduction to Manifolds.