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Find $$\int_0^{\infty} \int_0^{\infty} e^{-2xy} \, \mathrm d x \mathrm dy$$ using $u = x^2 - y^2$ and $v=2xy$.

I have tried using the Jacobian matrix to obtain the Jacobian of the transformation. However, confusion arises since I do not know what should be kept constant. Do I directly differentiate $u$ with respect to $x$ while keeping $y$ constant, or do I substitute $y$ from $v=2xy$, and then differentiate $u$ with respect to $x$ while keeping $v$ constant?

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  • $\begingroup$ When finding $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial y}$, just differentiate $u$ as a function of $(x,y)$, ignore the $v$. Similarly, when differentiating $v$, don't worry about $u$. So for example, $\frac{\partial u}{\partial x} = 2x$. $\endgroup$ – Minus One-Twelfth Mar 10 at 11:55
  • $\begingroup$ However, the Jacobian matrix says that we should derive x with respect to u, x wrt v, y wrt u, and y wrt v. Therefore, it is not deriving u wrt to x and y, but it's the other way around. $\endgroup$ – user205891 Mar 10 at 11:59
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    $\begingroup$ There is a trick that you can use if you want: the Jacobian determinant of the inverse map is the inverse (reciprocal) of the Jacobian determinant of the forward map. Therefore, you can just do it the way I described, get the Jacobian determinant of that, and then take the reciprocal. (This will get you something in terms of $x$ and $y$, which you will need to express in terms of $u$ and $v$ for the integral.) $\endgroup$ – Minus One-Twelfth Mar 10 at 12:01
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It is always the old, original, variables with respect the new ones...but you can do it the other way around and take the inverse matrix's determinant!:

$$J=J\frac{(x,y)}{(u,v)}=\begin{vmatrix}\frac{\partial u}{\partial x}&\frac{\partial u}{\partial y}\\\frac{\partial v}{\partial x}&\frac{\partial v}{\partial y}\end{vmatrix}^{-1}=\begin{vmatrix}2x&-2y\\2y&2x\end{vmatrix}^{-1}=\frac1{4(x^2+y^2)}$$

We took the inverse determinant since we differentiated the new variables wrt the original ones...and there exists that relation.

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