So, I've got a problem understanding the "algorithm" for changing variables in a more-than-one-dimensional integral. For the two-dimensional case, I've got a specific problem that I'm looking at:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\int^{a}_{0}\left(\int^{2a-x}_{x}\frac{y-x}{4a^2+(y+x)^2}dy\right)dx[/tex]

which I assume is an example of the more general case:

[tex]\int^{b}_{a}\left(\int^{g(x)}_{f(x)}\phi(x,y)dy\right)dx[/tex].

For the particular case, I am to make the given substitutions

[tex]u=x+y[/tex]

[tex]v=x-y[/tex]

and evaluate the integral.

I'm trying to figure out how to determine the new boundaries of integration. In the two dimensional example given above, it's easy enough todrawthe region of integration and figure it out, but what if it's not such a simple situation? How does one juggle integration regions? Without drawing it, I'm left with the ugly inequalities

[tex]\{x<y<2a-x\} \leftrightarrow \{u+v<u-v<4a-(u+v)\}[/tex]

and

[tex]\{0<x<a\} \leftrightarrow \{0 < u+v < 2a\}[/tex],

which I can't seem to mentally sort through. In all of the examples I've found online and in textbooks, the region has been drawn. Is this just for illustrative (pun?) purposes, or is it because that's the only straightforward way to determine the regions of integration?

Thanks!

Some background about my understanding:

I've had a basic introduction to the metric G, scale factors, and the Jacobian and the application to transformations.

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# Change of Variables in Multiple Dimensions

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