# Del operator in spherical coordinates

## Homework Statement

Write the del operator in spherical coordinates?

## Homework Equations

I wrote the spherical unit vectors:
$$\hat{r}$$=sin$$\theta$$.cos$$\phi$$.$$\hat{x}$$+sin$$\theta$$.sin$$\phi$$.$$\hat{y}$$+cos$$\theta$$.$$\hat{z}$$
$$\hat{\phi}$$=-sin$$\phi$$.$$\hat{x}$$+cos$$\phi$$.$$\hat{y}$$
$$\hat{\theta}$$=cos$$\phi$$.cos$$\theta$$.$$\hat{x}$$+sin$$\phi$$+cos$$\theta$$.$$\hat{y}$$-sin$$\theta$$.$$\hat{z}$$

## The Attempt at a Solution

vela
Staff Emeritus
Homework Helper
In cartesian coordinates, you have

$$\nabla = \hat{x} \frac{\partial}{\partial x}+\hat{y} \frac{\partial}{\partial y}+\hat{z} \frac{\partial}{\partial z}$$

Use the chain rule to change variables to $r, \varphi, \vartheta$. For example, you can write

$$\frac{\partial}{\partial x} = \frac{\partial r}{\partial x}\frac{\partial}{\partial r}+\frac{\partial \varphi}{\partial x}\frac{\partial}{\partial \varphi}+\frac{\partial \vartheta}{\partial x}\frac{\partial}{\partial \vartheta}$$

Then express the cartesian unit vectors in terms of the spherical unit vectors. Plug everything in and simplify.

vela
Staff Emeritus