Problem whit the use of Implicit function theorem

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Problem

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Hi guys!

I want to define for the part $a)$ $$x(u,v); \ \ y(u,v)$$ but in the problem change the order in the components. I calculate the Jacobian matrix and obtain an equation that I not comprhend because it remains in terms of the same function. I think that $a)$ and $b)$ are the same problem.

I appreciate any help and suggestions you can give me.