# Using the chain rule to derive a formula for partial derivativs in spherical coordinates

For the spherical coordinates;

$$x=r\sin{ \theta} \cos{\phi}$$

$$y=r \sin{\theta} \sin{\phi}$$

$$z=r\cos{\theta}$$

Where $$\theta$$ is the polar angle and $$\phi$$ is the azimutal angle, Is it possible to uae the chain rule for multivariable functions in order to derive an expression for the partial derivatives $$\frac{\partial}{\partial x}, \frac{\partial}{\partial y}$$ and $$\frac{\partial}{\partial z}$$ in terms of $$\frac{\partial}{\partial r}, \frac{\partial}{\partial \theta}$$ and $$\frac{\partial}{\partial \phi}$$ ?

More specifically, I have an operator in cartesian coordinates that I want to write in the spherical coordinates (without using results from the internet, I want to understand how to derive the expression). The operator is;

$$L_x=-i\hbar \left(y \frac{\partial}{\partial z} - z\frac{\partial}{\partial y}\right)$$

That is the angular momentum operator for the $$x$$ coordinate in physics (up to the fact that Im not sure how to write h_bar properly)

Im not sure what method can I use to replace the partial derivatives in cartesian coordinates in the operator, with the spherical partial derivativea.

• \hbar = $\hbar$ – user170231 Feb 23 at 14:47