# Change of variables, transformations, reversibility

1. May 18, 2010

### Jamin2112

1. The problem statement, all variables and given/known data

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?

2. Relevant equations

The theorem, I suppose.

3. 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.

2. May 18, 2010

### Jamin2112

bump (c'mon, guys)