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Math Help - divergence of unit vector

  1. #1
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    divergence of unit vector

    show that the divergence of the unit vecotor r is 2/r

    I have done this using cartesian coordinates which is quite tedious,i was hoping someone could show me a way to do it with tensors
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  2. #2
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    Quote Originally Posted by ulysses123 View Post
    show that the divergence of the unit vecotor r is 2/r

    I have done this using cartesian coordinates which is quite tedious,i was hoping someone could show me a way to do it with tensors
    First we need to know that

    \hat{r}=dr \quad \hat{\theta}=rd\theta \quad \hat{\phi}=r\sin(\theta)d\phi

    is a basis and

    *1=r^2\sin(\theta)dr \wedge d\theta \wedge d\phi

    we have the 1-form

    dr

    The divergence is given by

    *d*f=\nabla \cdot \mathbf{f}

    where * is the Hodge Dual

    *dr=*\hat{r}=r^2\sin(\theta)d\theta \wedge d\phi

    This gives a two form, now taking the derivative

    d*dr=2r\sin(\theta)dr \wedge d\theta \wedge d\phi=\frac{2}{r}(dr) \wedge (rd\theta) \wedge (r\sin(\theta)d\phi)

    This gives a three form and now taking the dual gives a 0-form

    *d*dr=\frac{2}{r}

    So this is the divergence in spherical coordinates.
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