Let $U=\{(x,y)\in\mathbb{R}^2\colon x\neq 0\}$. For each $(x,y)\in U$, let $$u(x,y)=\frac{x^4+y^4}{x}\quad\text{ and }\quad v(x,y)=\sin(x)+\cos(y).$$ Let $f=(u,v)$ on $U$ and $c=(\pi/2, \pi/2).$ Can we solve for $x,y$ in terms of $u,v$ near $c$? Find $\partial x/\partial u$ near $f(c).$
I'm trying to solve this problem. I don't know what the problem means.
First, I can show that a given function has the inverse at a given point c. Is it not enough? Or I have to solve the system of equations with two variables?