Let's say that $X,Y$ are independent standard normal random variables. I am interested in the distribution $P(X+Y\le 2t)$. Clearly, the domain of integration in this case is $-\infty<x<\infty$ and $-\infty<y\le 2t-x.$
If I were to introduce a change of variables $u=\frac{1}{2}(x+y)$ and $v=\frac{1}{2}(x-y)$, how can I systematically determine the domain of integration for $u,v$?
From the looks of it, clearly $u\le t$. And judging from the values of $x,y$, $-\infty < u \le t$ and $-\infty <v<\infty.$ But this was merely obtained by looking at the values of $x$ and $y$. How would I know that the region of integration on the $uv$-plane would have, for instance, the bounds on $v$ being a function of $u$?
My system is to express the inequalities describing the domain in terms of new variables. The original domain was described by $x+y\le 2t$, with no other constraints. Substitution $x = u+v$ and $y = u-v$ turns this into $$(u+v) + (u-v ) \le 2t$$ which simplifies to $u\le t$; no other constraints.