Partial Vector Derivative: Is This the Correct Derivative of B?

[Philosophaie]

Is this the correct partial derivative of B?

$\vec{B} = \frac{g \vec{r}}{4 \pi r^3}$

$\frac{\partial \vec B}{\partial r}$ = $-3\frac{g \vec{r}}{4 \pi r^4} + \frac{g}{4 \pi r^3 }(\frac{\partial r_r \hat r}{\partial r})$

 

[Orodruin]

What is $r_r$?

Apart from that it seems as if you are just applying the product rule for derivatives.

 

[Philosophaie]

$r_r$ is the radial part of a vector from the origin to an arbitrary point to be examined:

$\vec{r} = r_r \hat r +r_\theta \hat \theta +r_\phi \hat \phi$

 

[Orodruin]

Then yes and no. Yes because it is technically correct due to $r_r = r$ and $r_\theta = r_\phi = 0$. No since you generally cannot assume that $\partial_r \vec w = \partial_r w_r \hat r$, the general expression is $\partial_r \vec w = \partial_r (w_r\hat r + w_\theta \hat \theta + w_\phi \hat \phi)$ and doing so generally will get you the incorrect result.