Solve for $x$ and $y$ in terms of $u$ and $v$. Then compute the Jacobian.

Question

solve for x and y in terms of u and v. Then compute the Jacobian

let: $u = x+y$ and $v = x-y$

since:

$$\frac{\partial (x,y)}{\partial(u,v)}* \frac{\partial (u,v)}{\partial(x,y)} = 1$$

but

$$\frac{\partial (u,v)}{\partial(x,y)} = -2$$

hence the jacobian is $$\frac{\partial (x,y)}{\partial(u,v)} = -\frac{1}{2}$$

but I don't know how to solve for x and y in terms of u and v

Answer 1

Adding and respectively, subtracting the two equations $u=x+y$ and $v=x-y$, we get $u+v=2x$, $u-v=2y$. Thus $x=\frac{u+v}{2}$ and $y=\frac{u-v}{2}$.