Plane transformations

Question

I need help in understanding how plane transformations work:

for example, let $$A = \{(x,y) \in \mathbb{R}^2: x^2 + y^2 < 1\}$$ Now let's change coordinates like this: $$x = u^2 - v^2$$ $$y = 2uv$$

How to draw the resulting set in the plane $uv$?

I don't want to focus only on this exercise, so the more general the answers are (maybe with a couple of useful links) the better :-)

Answer 1

See if you use these values of $x$ and $y$ in the region $$x^2+y^2<1$$ you find $$-1<u^2+v^2<1$$ that shows region of interior of a circle in $uv$-plane

Have you got your answer? If not please leave a comment, perhaps I can help more.