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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 X1, X2, . . ., Xn be independent identically distributed U (0, θ) random variables. What is the distribution of (X1-X2) ?

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 ?

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See this for example: mathoverflow.net/questions/2127/… –  PEV Feb 5 '11 at 16:12
    
Solutions to Black Scholes (a PDE) become clearer under a transformation of coordinates. –  PEV Feb 5 '11 at 16:13
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Could you elaborate a little on what you have in mind, maybe give some more examples and add a few details? I find it very difficult to see what you might be asking about. For example, if you could briefly explain one or two of the "how?"s that you've understood, then maybe someone could help you with the "why?"s. –  t.b. Feb 5 '11 at 16:19
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This is an extremely general question, as transformations in mathematics are an extremely general concept; I would appreciate something more specific. –  Qiaochu Yuan Feb 5 '11 at 16:20
    
I made it more specific. Thanks for all of your help. –  Sunil Feb 5 '11 at 18:43
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2 Answers

up vote 4 down vote accepted

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.

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Hi, I gave an example. Can you shed some more light on it? Thanks –  Sunil Feb 5 '11 at 19:46
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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.

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