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If we make a change of variable in higher dimensional integrals how do we decide the limits of integration. In two dimensions the problem is fairly simple using a geometric interpretation but in 3 and higher dimensions is there some way of doing this?

For eg. In the integral: $$\int_{0}^{a}dx\int_{x}^{a}dy\int_{y}^{a}dzf(x)g(z-y)$$ After making the change of variable $$x=\alpha;\,y+z=\beta;\, z-y=\gamma$$ how does the integration transform for the new variables $\alpha,\beta$ and $\gamma$? I'm more interested in the transformation of the limits of integration and any help/references will be appreciated?

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    This is a math question ideally suited for math.stackexchange.2012-06-04

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