I want to evaluate $$\int_0^∞ \int_0^∞ t^{x-1} s^{y-1} e^{-(s+t)} dsdt$$ which I need for some relation of gamma function. The book is suggested change of variables $s = uv$ and $t = u(1 - v)$ and claimed that $0 < s, t <∞$ implies $0 < u < ∞$ and $0 < v < 1$. I want to know how rigorously this change of variables works :
1- Intuitively, $0 < u < ∞$ and $0 < v < 1$ when $0 < s, t <∞$ for example $v$ cannot be less than $0$ because of $s = uv$ and cannot be greater than $1$ because of $t = u(1 - v)$ but why this implies sets of all $v$ is not a proper subset of $[0,1]$? Mathematically, how one proves $0 < s, t <∞$ implies $0 < u < ∞$ and $0 < v < 1$?
2- When I studied advanced calculus, in one variable there was always a bijection when change of variables imposed, but in two variables the book used always intuition for a One-to-one correspondence. How to establish a bijection between two sheets u-v and s-t? A general method of construction of a bijection in two dimension (to include this special) would be even better because I can learn finally the method that the book missed to express.
Solve for $u$ and $v$ in terms of $s$ and $t$. Get $u=s+t$ and $v=\frac{s}{s+t}$.
This immediately tells you what the ranges are for $u$ and $v$.