So messing about with spherical coordinates, I wanted at some point to get the expressions for $\frac{\partial r}{\partial x}$ and $\frac{\partial x }{\partial r}$; using the convention:
$$ \left\{ \begin{array}{c} x = r \text{ cos} \theta \text{ sin} \phi \\ y = r \text{ sin} \theta \text{ sin} \phi \\ z = r \text{ cos} \phi. \\ \end{array} \right. $$
For $\frac{\partial x }{\partial r}$ I do:
$$ \frac{\partial{} }{\partial r} [r \text{ cos} \theta \text{ sin} \phi] = \text{ cos} \theta \text{ sin} \phi. $$
For $\frac{\partial r }{\partial x}$, I use:
$$ r^2 = x^2 + y^2 + z^2 $$ $$ \rightarrow \frac{\partial}{\partial x} [r^2] = 2 r \frac{\partial r }{\partial x} = 2 x $$ $$ \rightarrow \frac{\partial r }{\partial x} = \frac{x}{r} = \frac{r \text{ cos} \theta \text{ sin} \phi}{r} = \text{ cos} \theta \text{ sin} \phi. $$
How come I get the same for both?