What are transformations in mathematics? I'm learning about transformation in mathematics in general. There are many of them like for example Jacobian (which I'm trying to understand now). The "how?" part is very easy. There are formulas and we use them but the "why?" part is what I'm looking for. I fail to understand the intuition behind it. Why are we using transformations ? Any links, suggestions, are all welcome. 
My apologies for ambiguity. Here is an example question. 
Let $X_1$, $X_2$, ..., $X_n$ be independent identically distributed $U (0, \theta)$ random variables. What is the distribution of $(X_1-X_2)$ ?
I can solved this question directly but it is very simple to use Jacobian to solve this. I've the solution so I know how to use it. But why are we using Jacobian and how do I know when to use it ?
 A: Here is an extremely general answer to go with your extremely general question. If you have trouble seeing something because it's hidden behind a tree, generally it's a good idea to move to a different position so that the tree isn't blocking your view. All you've done is apply an affine change of coordinates to your view of the universe. Mathematics is no different: when we want to see something clearly in a problem, we do it in a view that makes it as easy to see as possible.
A: As PEV hinted, changing coordinates can simplify a problem.  In $\mathbb{R}^2$, if all the action is along a couple axes, it helps to have those aligned with $x$ and $y$.  If you want to find the area of a square by integration, if it is $[0,1]\times [0,1]$ it is much easier than if it is $[(-\sqrt{2}/2,-\sqrt{2}/2) \text{to} (-\sqrt{2}/2,\sqrt{2}/2)]\times[(-\sqrt{2}/2,-\sqrt{2}/2) \text{to} (\sqrt{2}/2,-\sqrt{2}/2)]$  Point charge electrostatics is much easier in spherical coordinates.  There are many more examples like this.
