Find ∂x/∂u if I know ∂u/∂x?

Question

I have the following transformation

$\it{u=x^2-y^2}$

$\it{v=2xy}$

I want to calculate $\frac{∂x}{∂u}$ $(0,2)$ and I know that the transformation is bijective around the point $(1,1)$. The answer is $\frac{1}{4}$, but I am unsure how to tackle this problem. Help would be appreciated.

Answer 1

Compute the Jacobian of the map $(x,y) \mapsto (u(x,y),v(x,y))$ at $(x,y)=(1,1)$.

$\frac{\partial x}{\partial u}$ is the entry $1,1$ of the inverse of that Jacobian.

The Jacobian is

$$ J = \begin{pmatrix} \partial{u}/\partial{x} & \partial{u}/\partial{y} \\ \partial{v}/{\partial{x}} & \partial{v}/\partial{y} \end{pmatrix} = \begin{pmatrix} 2& -2 \\ 2 & 2 \end{pmatrix} $$

and its inverse $$\begin{pmatrix} 1/4& 1/4 \\ -1/4& 1/4 \end{pmatrix} $$

Leading to the desired result.

Answer 2

Write $J$ for the Jacobian matrix $$ J = \pmatrix{ \partial{u}/\partial{x} & \partial{u}/\partial{y} \\ \partial{v}/{\partial{x}} & \partial{v}/\partial{y} }. $$ You want to find the upper left entry of the inverse matrix to $J$.