Del operator in spherical coordinates

[tirwit]

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

I have no idea where to start... Please help, I'm going crazy with this...

 

[vela]

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]

It seems I suggested one of the more painful ways of finding the gradient.

Here's a web page that describes several different methods you can try:

http://math.mit.edu/classes/18.013A/HTML/chapter09/section04.html