What is a pullback in simple calculus context? The definition of a pullback provided by my text is quite accessible

Let $\phi : M \to N$, $f:N \to \mathbb{R}$, then $f\circ \phi: M \to \mathbb{R}$, where $\phi^*f = f\circ\phi$ and $\phi^*$ is the pullback of
  $f$ by $\phi$

But when I look up some examples, things like category theory and sheafs and differential forms pop up! 
I only have a background in calculus, can someone provide a simple examples from calculus to illustrate the idea of a pullback? and why is pullback problematic/pathological/not desired??
 A: The chain rule and its attendant method of substitution in calculus are the most elementary "useful" example that comes to mind. Let $M = [\alpha, \beta]$ and $N = [a, b]$ be compact $1$-manifolds with boundary...er...I mean, closed, bounded intervals.


*

*If $\phi:M \to N$ is a differentiable map (a.k.a. function) and $f:N \to \mathbf{R}$ is differentiable, then the pullback of the $0$-form $f$ on $N$ is the $0$-form $\phi^{*}f = f \circ \phi$ on $M$, and
$$
d(\phi^{*}f) = f'\bigl(\phi(x)\bigr)\phi'(x)\, dx = \phi^{*}(df).
$$

*If $\phi:M \to N$ is an increasing, continuously-differentiable bijection, and $f:N \to \mathbf{R}$ is a continuous function, then the change of variables formula
$$
\int_{a}^{b} f(x)\, dx
  = \int_{\alpha}^{\beta} f\bigl(\phi(t)\bigr)\, \phi'(t)\, dt
$$
asserts the fact that if $\omega = f(x)\, dx$, then
$$
\int_{N} \omega = \int_{M} \phi^{*}\omega.
$$
The change of variables theorem in multivariable calculus is a higher-dimensional, slightly less-elementary example of a pullback, used every time you convert a Cartesian integral to polar (or cylindrical, or spherical...) coordinates, or integrate a function or a vector field over a curve or surface.
