Is there a formula analogous to the single random variable case i.e the pdf of y(x) where x is generated from a pdf=f(x) is given by: $g(y)=f(x(y))|\frac{dx}{dy}|$ ?
Specifically, I have the following problem: X,Y are two uniform distributions with $a<x<b$ and $c<y<d$. Find the pdf of $Z=\frac{1-r^2}{1+\rho^2}$ where $r=\sqrt2\rho$ and $\rho=\sqrt{(x^2+y^2)}$
There is no general formula (that I know of), (unless, you mean the this one, which is the definition): $$P(z)=\int\delta\Big(z-g(x,y)\Big)P(x,y)dx\,dy$$ $\delta$ is the Dirac delta distribution, $P(x,y)$ is the (joint) PDF of $X$ and $Y$ and $P(z)$ is the PDF of $Z$.
However, your problem is relatively simple as you can first calculate the PDF of $\rho$ and then use the formula you cited for $Z$. The PDF of $\rho$, unfortunately, is not very nice. You can get the answer for $a=c=0$ and $b=d=1$ in this question, and you can probably generalize it to other cases, but it will not be nice.