# 3D Dirac Delta Function

#### cordyceps

Griffiths' section 1.5.3 states that the divergence of the vector function r/r^2 = 4*Pi*δ^3(r). Can someone show me how this is derived and what it means physically? Thanks in advance.

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#### gabbagabbahey

Homework Helper
Gold Member
Griffiths' section 1.5.3 states that the divergence of the vector function r/r^2 = 4*Pi*δ^3(r). Can someone show me how this is derived and what it means physically? Thanks in advance.
First, use the divergence theorem to show that $$\int_{\mathcal{V}}( \vec{\nabla}\cdot\frac{\hat{r}}{r^2})d\tau=4\pi$$ for any surface enclosing the origin (use a spherical surface centered at the origin for simplicity).

Then, calculate the divergence explicitly using the formula on the inside of the front cover for divergence in spherical coords. You should find that it is zero everywhere except at the origin where it blows up (because the 1/r terms correspond to dividing by zero).

Finally, put the two together by using eq 1.98 with $$f(\vec{r})\delta^3(\vec{r})\equiv \vec{\nabla}\cdot\frac{\hat{r}}{r^2}$$

As for a physical interpretation, there really isn't one (although there are physical consequences as you'll see in chapter 2 and beyond); but there is a geometric interpretation....if you sketch the vector function $$\frac{\hat{r}}{r^2}$$ (figure 1.44 in Griffiths) you'll see why there must be an infinite divergence at the origin. This is all discussed in section 1.5.1

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#### cordyceps

Sorry, I'm kinda slow- why does LaTeX Code: f(\\vec{r})\\delta^3(\\vec{r})\\equiv \\vec{\\nabla}\\cdot\\frac{\\hat{r}}{r^2}?

#### cordyceps

Can't use latex. Why does f(R)δ^3(R) = the divergence of R/r^2?

#### gabbagabbahey

Homework Helper
Gold Member
Can't use latex. Why does f(R)δ^3(R) = the divergence of R/r^2?
The fact that the divergence of $$\frac{\hat{r}}{r^2}$$ is zero everywhere, except at the origin where it is infinite, but when integrated gives a constant, means that it must be some unknown normal function $f(\vec{r})$ times a delta function so, you call it $f(\vec{r})\delta^3(\vec{r})$ and solve for $f$ by setting the integral (eq. 1.98) equal to the value you calculated using the diverergence theorem ($4\pi$)

#### cordyceps

Ok. I think I got it. Thanks a lot!

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