Chapter 1 Vector Analysis: Problem 1.16

Problem 1.16 Sketch the vector function

$$\vec{v} = \frac{\hat{r}}{r^2}$$

and compute its divergence. The answer may surprise you.. can you explain it?

Solution:

 Let's first sketch the given vector function.

NOTE: Use this site to sketch vector function and play with it: https://c3d.libretexts.org/CalcPlot3D/index.html 

Since it seems too complicated, let's see it from the top and see the field on the $x-y$ axis

Now as we can see there is a net outward flux at the origin since the field is generated from the origin. So divergence at origin should be some positive value.

We can't say anything about other coordinates as the field is expanding out but at the same time the magnitude is decreasing.

Now compute the divergence:

 $$\vec{v} = \frac{\hat{r}}{r^2}$$

$$\vec{\nabla}\cdot \vec{v} = \frac{1}{r^2\sin\theta}\big[\frac{\partial}{\partial r}(r^2\sin\theta v_r)+\frac{\partial}{\partial \theta}(r\sin\theta v_{\theta})+\frac{\partial}{\partial \phi}(rv_{\phi})\big]$$

$$\vec{\nabla}\cdot \vec{v} = \frac{1}{r^2\sin\theta}\big[\frac{\partial}{\partial r}(r^2\sin\theta \frac{1}{r^2})\big] = 0$$

So according to the result we computed, divergence should be zero at every point but from the sketch divergence is positive at the origin.

So what is wrong here?

The fact that the vector field is not well defined at the origin proves it all. The field blew up at the origin. So to deal with such ill- defined vector field we'll use dirac delta function which will be coming in the sections ahead.

If you have any doubt regarding the solution or you want solution of some problem which is not posted please let me know by commenting. This encourages me to answer more question because sometime it feels like all I am doing is just a waste. If it helps someone I will be happy to do it.