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