(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove bijection between the regions

0<x<1, 0<y<1, 0<u, 0<v, u+v<pi/2

2. Relevant equations

x=sinu/cosv y = sinv/cosu

3. The attempt at a solution

We need to show that an inverse function exists to prove the bijection so obviously, (u,v) maps to one and only one (x,y) for the above. But what about the other way around? What is the best approach

Do we need to calculate:

u=arcsin(xcosv), v=arccos(sinu/x), u=arccos(sinv/y), v=arcsin(ycosu) and then look at each individually? Or could we divide one by the other and obtain tan(u)tan(v)=xy so that u=arctan(xy/tan(v) and v=arctan(xy/tan(u))?

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Proving bijection between a region

**Physics Forums | Science Articles, Homework Help, Discussion**