# Multivariable Inverse Function problem

Consider the system of equations

\left\{\begin{align*} &x^5 v^2 + 2y^3 u = 3\\ &3yu - xu v^3 = 2\;. \end{align*}\right.

Show that near the point $(x,y,u,v) = (1,1,1,1)$, this system defines $u$ and $v$ implicitly as functions of $x$ and $y$. For such local functions $u$ and $v$, define the local function $f$ by $f(x,y) = \big(u(x,y),v(x,y)\big)$. Find $Df(1,1)$.

I found that we can express $u$ and $v$ in terms of $x$ and $y$ because I found the Jacobian of $u$ and $v$ near $(1,1,1,1)$ is nonzero. However I am stuck when trying to define the local function of $u$ and $v$ in terms of $x$ and $y$.

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Are you trying to explicitly solve the system? That could be impossible. The point is to compute the derivative without actually computing $u$ and $v$, and there are formulas for that (check, e.g., en.wikipedia.org/wiki/Implicit_function_theorem). – student Nov 29 '12 at 6:22
I am not trying to solve the system, I am trying to find a way to write the system as u and v in terms of x and y – tamefoxes Nov 29 '12 at 6:57
The way you say it, I understand that you are trying to find $u$ and $v$ in terms of $x$ and $y$. This is what I mean by solving the system, and this could be impossible. You don't need to do this to compute $Df(1, 1)$. – student Nov 29 '12 at 6:59