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Seperation Vector

  • Thread starter Yeldar
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  • #1
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Separation Vector

Let [itex]\vec{r}[/itex] be the seperation vector from a fixed point [itex](\acute{x},\acute{y},\acute{z})[/itex] to the source point [itex](x,y,z)[/itex].

Show that:

[tex]\nabla(\frac{1}{||\vec{r}||}) = \frac {-\hat{r}} {||\vec{r}||^2} [/tex]

Now, I've attempted this comeing from the approach that [itex]||\vec{r}|| = (\vec{r} \cdot \vec{r})^\frac {1} {2} [/itex] but it dosent seem to get me anywhere, am I missing something blatently obvious?

Thanks.
 
Last edited:

Answers and Replies

  • #2
MalleusScientiarum
Go back to the definition of the gradient in spherical coordinates.
 
  • #3
6
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Wouldnt that just complicate things further?


In Spherical Coordinates:

[tex]\displaystyle{ \nabla = \hat{r} \frac {\partial{}{}} {\partial{}{r}} + \frac {1}{r} \hat{\phi}\frac {\partial{}{}} {\partial{}{\phi}} + \frac {1}{r sin \phi} \hat{\theta}\frac {\partial{}{}} {\partial{}{\theta}} }[/tex]



I just don't see how that could simplify things?
 
  • #4
6
0
Okay, nevermind on this...

Went with a totally different appraoch and things worked out nicely without having to go into spherical coordinates.


Thanks again.
 

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