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I have the following problem:

Considerate the following system:

$$x=\cos(u)+\sin(v) $$

$$y=\sin(u)+\cos(v)$$

Show that in a neighborhood of $(u_0, v_0, x_0, y_0)=(0,0,1,1), (u,v)$ can be written as differentiable functions of $x$ and $y$. Furthermore,

$$\left(\frac {\partial u}{\partial x}\right)^2 +\left(\frac {\partial u}{\partial y}\right)^2=\left(\frac {\partial v}{\partial x}\right)^2+\left(\frac {\partial v}{\partial y}\right)^2$$

I made some attempts however I did not succeed in the second part. But related to that , I will try again. I would appreciate any hints in the first part, which I believe is an application of implicit function theorem.

Thanks in advance.

3 Answers 3