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
$$\frac{\partial \bold{r}}{\partial u_p}\cdot\triangledown u_q=\delta_{pq}$$