I am woring with curvilinear coordinates and got stuck on an exercise where we have a transformation from cartesian to parabolic coordinates $(x,y,z)\rightarrow (u,v,\varphi)$. My question is how I inverse this, I believe the Jacobian or functional matrix has something to do with it:

$\textbf{F}=\begin{bmatrix} \frac{\partial x}{\partial u_{1}} & \frac{\partial x}{\partial u_{2}} & \frac{\partial x}{\partial u_{3}} \\ \frac{\partial y}{\partial u_{1}} & \frac{\partial y}{\partial u_{2}} & \frac{\partial y}{\partial u_{3}} \\ \frac{\partial z}{\partial u_{3}} & \frac{\partial z}{\partial u_{3}} & \frac{\partial z}{\partial u_{3}} \end{bmatrix}$

The parabolic transformation is expressed as

$\begin{equation} \begin{cases} x=uv\cos\varphi \\ y=uv\sin\varphi \\ z=\frac{1}{2}(u^{2}-v^{2}) \end{cases} \end{equation}$

So how do I calculate the $(u,v,\varphi)\rightarrow (x,y,z)$ transformation?


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