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Laplacian of a Vector

  1. Dec 28, 2014 #1
    1. The problem statement, all variables and given/known data
    Show that [itex]\nabla^{2}\left(\frac{1}{\overrightarrow{r}}\right)=0[/itex]


    2. Relevant equations

    3. The attempt at a solution
    Let [itex]\nabla=\hat{i}\frac{\partial}{\partial x}+\hat{j}\frac{\partial}{\partial y}+\hat{k}\frac{\partial}{\partial z}[/itex]

    and [itex]\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}, \mid r\mid=\sqrt{x^{2}+y^{2}+z^{2}}[/itex],

    [itex]\hat{r}=\hat{i}+\hat{j}+\hat{k}[/itex]

    First calculate [itex]\nabla\left(\frac{1}{\overrightarrow{r}}\right)=\nabla\left(\frac{1}{\mid r\mid\hat{r}}\right)=\nabla\left(\frac{1}{\mid r\mid}\hat{r}\right)=\left[\hat{i}\frac{\partial}{\partial x}+\hat{j}\frac{\partial}{\partial y}+\hat{k}\frac{\partial}{\partial z}\right]\left[\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{x^{2}+y^{2}+z^{2}}}\right][/itex]

    [itex] = \hat{i}\frac{\partial}{\partial x}\left(x^{2}+y^{2}+z^{2}\right)^{\frac{-1}{2}}+\hat{j}\frac{\partial}{\partial y}\left(x^{2}+y^{2}+z^{2}\right)^{\frac{-1}{2}}+\hat{k}\frac{\partial}{\partial z}\left(x^{2}+y^{2}+z^{2}\right)^{\frac{-1}{2}} [/itex]

    Am I doing it the right way?
     
  2. jcsd
  3. Dec 28, 2014 #2

    TSny

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    ##\frac{1}{\vec{r}}## doesn't make sense because it involves dividing by a vector.

    Are you sure you aren't supposed to take the Laplacian of ##\frac{1}{|\vec{r}|}## instead?
     
  4. Dec 28, 2014 #3

    It looks like a vector image024.gif
    The problem is from Mathematical Methods for Physicists by Tai L. Chow.
     
  5. Dec 28, 2014 #4

    TSny

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    You're right, it does look like ##1/\vec{r}##. I think it must be a misprint. It does turn out that ##\nabla^2(1/r)=0## (except at ##r = 0##).

    So, I'm thinking that's what was meant.

    Part (c) also seems to have some misprints, unless the author is purposely trying to test whether you can tell if an expression is meaningless.
     
    Last edited: Dec 28, 2014
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