# How to calculate divergence of some special fields

netheril96
$$$\nabla \cdot \frac{{\vec e_r }}{{r^2 }} = 4\pi \delta (\vec r)$$$
This can be seen from$$$\nabla \cdot \frac{{\vec e_r }}{{r^2 }} = \frac{1}{{r^2 }}\frac{\partial }{{\partial r}}(r^2 \cdot \frac{1}{{r^2 }}) = \frac{1}{{r^2 }}\frac{\partial }{{\partial r}}(1) = 0(r \ne 0)$$$
And from Gauss' Theorem$$$\int_V {(\nabla \cdot \frac{{\vec e_r }}{{r^2 }})dV = \oint_S {\frac{{\vec e_r }}{{r^2 }} \cdot d\vec S} } = 4\pi$$$
But if I want to directly using the formula of divergence in spherical coordinates,I can only get$$$\nabla \cdot \frac{{\vec e_r }}{{r^2 }} = \frac{1}{{r^2 }}\frac{\partial }{{\partial r}}(\frac{{r^2 }}{{r^2 }})$$$
And integrating this over a volume cannot give me the result of 4π$$$\int_V {(\nabla \cdot \frac{{\vec e_r }}{{r^2 }})dV = } \int_0^\pi {\sin \theta d\theta \int_0^{2\pi } {d\phi \int_0^R {\frac{\partial }{{\partial r}}(\frac{{r^2 }}{{r^2 }})} } } dr = 4\pi \int_0^R {\frac{\partial }{{\partial r}}(\frac{{r^2 }}{{r^2 }})} dr$$$
(Here V is a sphere with radius of R)
So how can I connect it with Dirac Delta?
By the way,I post this here because this problem arises in the electrostatic field of a point charge and I found nothing about such thing in any book concerning δ(x).

Homework Helper
Gold Member
The problem is that $\frac{r^2}{r^2}=\infty$ at $r=0$.

netheril96
The problem is that $\frac{r^2}{r^2}=\infty$ at $r=0$.

So how can I get$$$\int_0^R {\frac{\partial }{{\partial r}}(\frac{{r^2 }}{{r^2 }})} dr = 1$$$
Without integration,you cannot conclude some function with a singularity is δ(x)

Homework Helper
Gold Member
Other than just using Gauss' Law, I suppose an appropriate limiting procedure can be used. I'd start with your expression for $\mathbf{\nabla}\cdot\left(\frac{\textbf{e}_r}{r^2}\right)$ and calculate the limit of it as $r\to 0$

As you have seen $$\delta({\vec r})$$ is not easily treated in spherical coordinates.