Let's consider the usual spherical coordinates: $$ \begin{cases} x = r \sin \theta \cos \phi\\ y = r \sin \theta \sin \phi\\ z = r \cos \theta \end{cases} $$ with $r \in \mathbb{R}^+$, $\theta \in [0, \pi[$ and $\phi \in [0, 2\pi[$.
I need to calculate $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ using $\frac{\partial}{\partial r}$, $\frac{\partial}{\partial \theta}$ and $\frac{\partial}{\partial \phi}$. I did that a long time ago but totally forgot the method... What's the best way to find the answer ? Is there any shortcut since I don't need $\frac{\partial}{\partial z}$ ?
You have the relation $$\begin{pmatrix} \frac{\partial}{\partial x}\\ \frac{\partial}{\partial y}\\ \frac{\partial}{\partial z} \end{pmatrix} = \left(\frac{\partial(x,y,z)}{\partial(r,\theta,\phi)}\right)^{T,-1} \begin{pmatrix} \frac{\partial}{\partial r}\\ \frac{\partial}{\partial \theta}\\ \frac{\partial}{\partial \phi} \end{pmatrix} $$