Separation Vector: Showing $\nabla(\frac{1}{||\vec{r}||})$

[Yeldar]

Separation Vector

Let \vec{r} be the separation vector from a fixed point (\acute{x},\acute{y},\acute{z}) to the source point (x,y,z).

Show that:

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

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

Thanks.

 

[MalleusScientiarum]

Go back to the definition of the gradient in spherical coordinates.

 

[Yeldar]

Wouldnt that just complicate things further?


In Spherical Coordinates:

\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}} }



I just don't see how that could simplify things?

 

[Yeldar]

Okay, nevermind on this...

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


Thanks again.