Exploring the Dirac-Delta Function: Divergence at the Origin

[brianparks]

In introducing the dirac-delta function, my electrodynamics discusses the following function:

$$v = 1 / r^2$$r

The text states that "at every location, v is directed radially outward; if there ever was a function that ought to have a large positive divergence, this is it. And yet, when you actually calculate the divergence (using Eq. 1.71), you get precisely zero."

At least on the positive portion of the x-y-z axis, it seems to me that v would have a negative divergence, meaning that as you move away from the origin, the magnitude of the vectors (which all point radially outward from the origin), would get smaller. That is, for every point other than the origin, the amount going in would exceed the amount coming out (the definition of a sink). For example:

Origin---------->-------->------>---->-->

Does this not indicate a negative divergence?

This question has caused me to wonder if my understanding of the divergence function is flawed.

Any help is greatly appreciated,

--Brian

 

[Tide]

The divergence is zero for r > 0 because the field never diverges from the radial direction.

 

[brianparks]

But doesn't the field weaken as you move radially outward? If so, wouldn't that mean that the divergence is negative?

Maybe my understanding of divergence is wrong.

Consider this function:

$$v = 1 / x^2$$x

From 0 to infinity, it would look like this:

---------->--------->-------->------->------>----->---->--->-->

and so on...

Wouldn't it have a negative divergence?

Thanks again for any help,
--Brian

 

[pervect]

But doesn't the field weaken as you move radially outward? If so, wouldn't that mean that the divergence is negative?

Maybe my understanding of divergence is wrong.

Consider this function:

$$v = 1 / x^2$$x

From 0 to infinity, it would look like this:

---------->--------->-------->------->------>----->---->--->-->

and so on...

Wouldn't it have a negative divergence?

Thanks again for any help,
--Brian

See http://hyperphysics.phy-astr.gsu.edu/hbase/diverg.html#c3

for the appropriate formulas for divergence in spherical coordiantes.

 

[ehild]

In introducing the dirac-delta function, my electrodynamics discusses the following function:

$$v = 1 / r^2$$r

The text states that "at every location, v is directed radially outward; if there ever was a function that ought to have a large positive divergence, this is it. And yet, when you actually calculate the divergence (using Eq. 1.71), you get precisely zero."


--Brian

Let us try.

$$\Vec{v}(x,y,z)=\frac{1}{x^2+y^2+z^2}(x\Vec{i}+y\Vec{j}+z\Vec{k})$$

$$\frac{\partial v_x}{\partial x} =-\frac{2x^2}{(x^2+y^2+z^2)^2}+\frac{1}{x^2+y^2+z^2}= \frac{-2x^2}{r^4} + \frac{1}{r^2}$$


In the same way,

$$\frac{\partial v_y}{\partial y} =-\frac{2y^2}{r^4}+\frac{1}{r^2}$$

and

$$\frac{\partial v_z}{\partial z} =-\frac{2z^2}{r^4}+\frac{1}{r^2}$$

The divergence is

$$\mbox{Div}(\Vec{v})=\frac{\partial v_x}{\partial x} +\frac{\partial v_y}{\partial y} +\frac{\partial v_z}{\partial z}=$$

$$=\frac{-2x^2-2y^2-2z^2}{r^4}+\frac{3}{r^2}=\frac{-2r^2}{r^4}+\frac{3}{r^2}=\frac{1}{r^2}$$


It is the function
$$\Vec{v}(\Vec{r}) = \frac{\Vec{r}}{r^3}$$
whose divergence is zero everywhere except at the origin.

ehild