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

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