(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Theorem.The change of variables is reversible near (u_{0},v_{0}) (with continuous partial derivatives for the reverse functions) if and only if the Jacobian of the transformation is nonzero at (u_{0},v_{0}).

1. Consider the change of variables x=x(u,v)=uv and y=y(u,v)=u^{2}-v^{2}.

(a). Find the coordinates (u,v) that go to a common value (x_{0},y_{0}) under than change of variables. (That is, find the points of intersection of the level curves x_{0}=x(u,v) and y_{0}=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)=(x_{0},y_{0}).

x_{0}=uv

y_{0}=u^{2}-v^{2}.

Graphing this in the u-v plane would look like a hyperbola, v=x_{0}/u, that intersects a square root thingy, v= +/- √(u^{2}-y_{0}). 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 x_{0}^{2}=u^{2}(u^{2}-y_{0}), 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=u^{3}+y^{3}. 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.

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# Homework Help: Change of variables, transformations, reversibility

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