### Problem

$X,Y\sim n(0,1)$. $X,Y$ are indpendent. $U = X+Y$, $V=X-Y$. Find $f\_{U,V}(u,v).$ 

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### Work

I've made some progress, but there are a number of steps remaining and I'm not sure how to proceed. I know there's a formula for this sort of thing 

$$f\_{U,V}(u,v) = f\_{X,Y}(h1(u,v), h2(u,v))|J|$$

but I wanted to try working it out as explicitly as possible, using the definitions and integral change of variables in order to see what's going on under the hood.

$$f\_{U,V}(u,v)$$
$$= \frac{\partial}{\partial u \partial v} F\_{U,V}(u,v)$$
by the second FTC.
$$= \frac{\partial}{\partial u \partial v} \int\_{\{(u,v): U < u, V < v\}} f\_{U,V}(u,v)dudv $$ by the definition of $F\_{U,V}$.
So we'll change variables from U-V into X-Y coordinates. First we'll measure the area element
$$dudv = \frac{\partial(u,v)}{\partial(x,y)}dxdy=2dxdy$$ using the Jacobian determinant, so that we can rescale appropriately the sum of the various volume chunks.

Second, we change the integrand. Do we plug in values of $u=x+y$ and $v=x-y$ in the integrand? This seems like it would make sense given that we want to replace $u$ and $v$ with $x$ and $y$, and this is the correct equivalence between the two pairs.

Third, we change the region. 
$$R=\{(u,v):U < u, V < v\}$$
$$=\{(u,v):X+Y < u, X-Y < v\}$$
but I'm not sure what we're getting out of this.

Fourth, we have to do something with the partial derivatives. Perhaps chain rule, or something cool with the FTC -- but I don't know how that would work if we've already changed out our $u$ and $v$ that we're supposed to be differentiating with respect to.

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A big part of learning how to do this is surely in getting used to the notation and the manipulations, so I'd most appreciate as explicit a solution as possible, showing how each manipulation works and is motivated.