# Spherical co-ordinates grad and div identity.

• Mar 24th 2012, 12:14 PM
breitling
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.
• Mar 27th 2012, 01:50 PM
thelostchild
Re: Spherical co-ordinates grad and div identity.
The first identity comes out because spherical coordinates are orthonormal so have the property
$\mathbf{e}_i \cdot \mathbf{e}_j = \delta_{ij}$
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