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Math Help - Divergence

  1. #1
    Junior Member Pinsky's Avatar
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    Divergence

    Prove that div \vec{e}=0 if e=\frac{\vec{r}}{r^3}

    1. div\vec{e}=div\frac{\vec{r}}{r^3}
    2. =(grad\frac{1}{r^3})\vec{r} + \frac{1}{r^3}div\vec{r}
    3. =-3\vec{r}\frac{1}{r^4}grad{\vec{r}}+\frac{1}{r^3}3

    Here are the first three steps of the solution i have in the book, i know how to go after step three, here is what i can't figure out:

    What is done in step 1 that leads to step 2?
    Why is div\vec{r}=3?
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  2. #2
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    What is r?
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  3. #3
    Junior Member Pinsky's Avatar
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    it's a radivector. Wasn't able to find a right translation of the word. If you don't understand what it is i'll try to explain.
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  4. #4
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    Here is one way to rewrite the notation to avoid confusion.
    R = xi + yj + zk\,\& \,r = \left\| R \right\|.

    Now the definition e = \frac{R}{{r^3 }} is a bit easier to understand.
    Clearly \nabla \cdot R = \frac{\partial }{{\partial x}}(x) + \frac{\partial }{{\partial y}}(y) + \frac{\partial }{{\partial z}}(z) = 3.
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