Spherical co-ordinates grad and div identity.

Struggling with the following:

Prove the identity $$ \nabla = e_{r}(e_{r} \cdot \nabla) + e_{\theta}(e_{\theta} \cdot \nabla) + e_{\phi}(e_{\phi} \cdot \nabla).$$ Given the vector fields $F=F_{r}e_{r}+F_{\theta}e_{\theta}+ F_{\phi}e_{\phi}$ show that$$ \nabla \cdot F=\frac{1}{r^{2}}\frac \partial {{dr}}(r^{2}F_r)+\frac{1}{rsin\theta}\frac \partial {d \theta}(sin\theta F_\theta)+\frac{1}{rsin\theta}\frac \partial {d\phi} $$Any help will be most appreciated, many thanks.

Re: Spherical co-ordinates grad and div identity.

The first identity comes out because spherical coordinates are orthonormal so have the property

$\displaystyle \mathbf{e}_i \cdot \mathbf{e}_j = \delta_{ij}$

Can you see why if that property holds then that identity holds?

The second formula occurs because the unit vectors are also functions of the coordinates themselves, a proof of it can be found here

http://www.csupomona.edu/~ajm/materials/delsph.pdf

(also please put your latex in TEX tags!)

Re: Spherical co-ordinates grad and div identity.

Many thanks for that link, yes still struggling to get my head around the first part