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Math Help - Spherical co-ordinates grad and div identity.

  1. #1
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    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.
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  2. #2
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    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

    (also please put your latex in TEX tags!)
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  3. #3
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    Re: Spherical co-ordinates grad and div identity.

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