Change of variables, transformations, reversibility

[Jamin2112]

Homework Statement

Theorem. The change of variables is reversible near (u₀,v₀) (with continuous partial derivatives for the reverse functions) if and only if the Jacobian of the transformation is nonzero at (u₀,v₀).

1. Consider the change of variables x=x(u,v)=uv and y=y(u,v)=u²-v².

(a). Find the coordinates (u,v) that go to a common value (x₀,y₀) under than change of variables. (That is, find the points of intersection of the level curves x₀=x(u,v) and y₀=y(u,v) in the u-v plane.) Generally, this transformation transforms how many points in the u-v plane to an image point in the x-y plane?

Homework Equations

The theorem, I suppose.

The Attempt at a Solution

So, I want to see what happens when I fix x and y. Let's say (x,y)=(x₀,y₀).

x₀=uv
y₀=u²-v².

Graphing this in the u-v plane would look like a hyperbola, v=x₀/u, that intersects a square root thingy, v= +/- √(u²-y₀). The two curves should intersect at 2 points, meaning that we don't have a 1-1 transformation. But I can't find the coordinates (u,v) of this intersection. I can simplify it to x₀²=u²(u²-y₀), but can't solve for u from there.

I have this problem on every other problem on this assignment. For example, the next problem has x=uv, y=u³+y³. How do I solve for u,v? Seems like I would need to know the formula for solving a 3rd degree polynomial. Or is there an easier way?

Thanks in advance. Expound as much as possible.

 

[Jamin2112]

bump (c'mon, guys)