Spherical coordinates and partial derivatives

[batboio]

Hello! My problem is that I want to find (\frac{\partial}{{\partial}x}, \frac{\partial}{{\partial}y}, \frac{\partial}{{\partial}z}) in spherical coordinates. The way I am thinking to do this is:

\frac{\partial}{{\partial}x}{\psi}(r(x,y,z),{\theta}(x,y,z),{\phi}(x,y,z))=\frac{{\partial}{\psi}}{{\partial}r}\frac{{\partial}r}{{\partial}x}+\frac{{\partial}{\psi}}{{\partial}{\theta}}\frac{{\partial}{\theta}}{{\partial}x}+\frac{{\partial}{\psi}}{{\partial}{\phi}}\frac{{\partial}{\phi}}{{\partial}x}=\frac{1}{\frac{{\partial}x}{{\partial}r}}\frac{{\partial}{\psi}}{{\partial}r}+\frac{1}{\frac{{\partial}x}{{\partial}{\theta}}}\frac{{\partial}{\psi}}{{\partial}{\theta}}+\frac{1}{\frac{{\partial}x}{{\partial}{\phi}}}\frac{{\partial}{\psi}}{{\partial}{\phi}},

where \psi is an arbitrary function. And so:

\frac{\partial}{{\partial}x}=\frac{1}{\frac{{\partial}x}{{\partial}r}}\frac{{\partial}}{{\partial}r}+\frac{1}{\frac{{\partial}x}{{\partial}{\theta}}}\frac{{\partial}}{{\partial}{\theta}}+\frac{1}{\frac{{\partial}x}{{\partial}{\phi}}}\frac{{\partial}}{{\partial}{\phi}}

The problem is that I think I've seen somewhere that \frac{{\partial}r}{{\partial}x}=\frac{1}{\frac{{\partial}x}{{\partial}r}} BUT I am really far from being sure that this is true.

And if this is true so far then what happens with \frac{{\partial}{\phi}}{{\partial}z} when \frac{{\partial}z}{{\partial}{\phi}}=0

 

[Some Pig]

In general, let u₁=r, u₂=θ, u₃=φ, then
\frac{\partial \bold{r}}{\partial u_p}\cdot\triangledown u_q=\delta_{pq}