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Laplacian of a Vector

  • #1

Homework Statement


Show that [itex]\nabla^{2}\left(\frac{1}{\overrightarrow{r}}\right)=0[/itex]


Homework Equations



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?
 

Answers and Replies

  • #2
TSny
Homework Helper
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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?
 
  • #3
##\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?

It looks like a vector
image024.gif

The problem is from Mathematical Methods for Physicists by Tai L. Chow.
 
  • #4
TSny
Homework Helper
Gold Member
12,405
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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:

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