- #1

fatpotato

- Homework Statement
- Find the new region of integration under a given change of variables.

- Relevant Equations
- Jacobian matrix and determinant

Hello,

I have to compute a double integral of the form ## \int_{0}^{\infty} \int_{0}^{\infty} f(u,v) du dv##, where ##f(u,v)## is not relevant. The following change of variable is advised as a hint: ## u = zt ## and ## v = z(1-t)##.

From there, I can reformulate with respect to ##z## and ##t##: ##z = u+v##, ## t = \frac{u}{u+v} ##, but I am absolutely unable to find the limit of integrations of the new region, or to find someone that can clearly explain to me how to find the new bounds.

Sure, I can guess that the new region for ##z## is ##[0, \infty[##, but how would one find the new region for ##t##? Presence of both ##u## and ##v## in the denominator makes it awkward to work with limits.

I suppose that I could try finding ## \lim_{(u,v) \rightarrow (\infty, \infty)} u + v = \infty## for the ##z## variable, but this method does not work for ##t##, as the limit is not defined (or at least, I cannot find it).

Can someone please help me? From several threads I looked at, either here or on Stackexchange, it seems that there is no systematic way of finding the new region, but there has to be at least a few heuristics one can follow to, isn't there?

I have to compute a double integral of the form ## \int_{0}^{\infty} \int_{0}^{\infty} f(u,v) du dv##, where ##f(u,v)## is not relevant. The following change of variable is advised as a hint: ## u = zt ## and ## v = z(1-t)##.

From there, I can reformulate with respect to ##z## and ##t##: ##z = u+v##, ## t = \frac{u}{u+v} ##, but I am absolutely unable to find the limit of integrations of the new region, or to find someone that can clearly explain to me how to find the new bounds.

Sure, I can guess that the new region for ##z## is ##[0, \infty[##, but how would one find the new region for ##t##? Presence of both ##u## and ##v## in the denominator makes it awkward to work with limits.

I suppose that I could try finding ## \lim_{(u,v) \rightarrow (\infty, \infty)} u + v = \infty## for the ##z## variable, but this method does not work for ##t##, as the limit is not defined (or at least, I cannot find it).

Can someone please help me? From several threads I looked at, either here or on Stackexchange, it seems that there is no systematic way of finding the new region, but there has to be at least a few heuristics one can follow to, isn't there?