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.