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Suppose I have a p-dimensional integral:

$$\int_{0}^{\infty}\int_{0}^{\infty}\dots \int_{0}^{\infty}f(x_1,x_2,\dots,x_p)dx_1dx_2\dots dx_p$$

And then I make a rotation + translation transform:

$$W=A^{T}(X-b)$$

Question: How will the region of integration $X>0$ change in the $W$ space?

Can assume $A$ is a matrix of eigenvectors of a real symmetric positive definite matrix if this makes the answer easier.

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    I'm afraid the answer is that the region of integration will change into a horrible mess that will be hard to deal with systematically. For the inner integrations, the limits will depend on the outer variables. Think of transforming the first quadrant of the plane to see what happens.2011-04-03
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    @joriki - I had a feeling it would be something disgusting. Oh well, just have to try another route :(. Just would have been nice, as my function $f(.)$ "decouples" under a rotation transform into independent products. Still wouldn't mind seeing the ugly solution, if someone has the energy to work it out.2011-04-03

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