[SOLVED] bivariate RVs, joint and marginal densities, and transformations

Hello, I'd like some help on this tough problem. Thanks in advance!

Given the joint density:

$\displaystyle f_{XY}(x,y) = {1 \over 2\pi} e^{-{1 \over 2}((x-\mu)^2 + (y-\mu)^2)}, \text{where } \mu \in\mathbb{R} $

find the joint density function

$\displaystyle f_{UV}(u,v) \text{ of } U = X - Y \text{ and } V = X + Y $

and find out if U, V are independent.

My work for this part of the problem:

I got a joint density function

$\displaystyle f_{UV}(u,v) = {1 \over 4\pi} e^{-{1 \over 2}(({U+V \over 2}-\mu)^2+({V-U \over 2}-\mu)^2)} $

is that correct? and also, how do I integrate that with the crazy exponents? or is there a simpler way to test for independence other that checking

$\displaystyle f_{UV}(u,v) = f_U(u)f_V(v) $

?

The second part of the problem:

Let the transformation from (X,Y) -> (R,θ) be defined as

$\displaystyle R = \sqrt{(X-\mu)^2+(Y-\mu)^2}$

$\displaystyle \theta=$

$\displaystyle arctan \left ({(Y-\mu) \over (X-\mu)} \right ), \text{if }X \ge \mu$

$\displaystyle -\pi + arctan \left ({(Y-\mu) \over (X-\mu)} \right ), \text{if }X < \mu; Y < \mu$

$\displaystyle \pi + arctan \left ({(Y-\mu) \over (X-\mu)} \right ), \text{if }X < \mu; Y \ge \mu$

find the joint density and marginal densities.

So far on this part I have it solved for X and Y:

$\displaystyle X = \sqrt{R^2 - (Y-\mu)^2} + \mu$

$\displaystyle Y = tan(\theta)(X-\mu) + \mu$

I tried plugging in the equations into each other but I don't know how to get rid of the tan(). Any help is appreciated.