How does the transformation $u=x+y$, $v=x/y$ transform the first quadrant?

Question

How is the region $(x,y) \in [0,\infty] \times [0,\infty]$ transformed under the change of coordinates given by $$u=x+y$$ $$v=x/y$$ Would appreciate any hints on how to find the image of such transformations.

Answer 1

Let $f(x,y ) =(u,v) =\left( x+y ,\frac{x}{y} \right) .$ Clearly $f([0,\infty )\times [0,\infty))\subset [0,\infty )\times [0,\infty) .$ On the other hand take any $(u,v)\in [0,\infty )\times [0,\infty)$ then $\left(\frac{uv}{v+1} ,\frac{u}{v+1}\right)\in [0,\infty )\times [0,\infty)$ and $f \left(\frac{uv}{v+1} ,\frac{u}{v+1}\right) =(u,v)$ hence $f([0,\infty )\times [0,\infty)) = [0,\infty )\times [0,\infty) .$