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