What is the pdf of g(x,y) were x and y are two random variables from a uniform distribution? 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 $a&lt;x&lt;b$ and $c&lt;y&lt;d$. 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.
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.