Change of Variable - form $u = x + y$, $v= \frac{y}{x+y}$ into $f(x,y) = \sqrt{x^2 - y^2}$

Question

I need to solve a double integral using change of variables. Solving the integral is not actually where I am confused, it's the simpler part of forming $f(x,y)$ using $u$ and $v$. Here is what I was given:

$f(x,y) = \sqrt{x^2 - y^2}$ , $u = x + y$, $v= \frac{y}{x+y}$

How on EARTH can I form this $u$ and this $v$ into the given $f(x,y)$??

Answer 1

You need to express x and y in terms of u and v. For instance:

\begin{align*} u&=x+y\\ v&=\frac{y}{x+y}=\frac{y}{u}\\ \end{align*}

So $y=uv$ then $x=u-y=u-uv=u(1-v)$. Then just plug it in

$$f(x,y)=f(u(1-v),uv)=\sqrt{u^2-2u^2v+v^2u^2-u^2v^2}=\sqrt{u^2(1-2v}=|u|\sqrt{1-2v}$$

Answer 2

Note that$ y=uv$ and x$=u-uv$.

Thus$$ x^2 - y^2 = (x - y)(x + y)=(u-2uv)u = u^2(1-2v)$$