Dirac delta and divergence

[shaun_chou]

I know that this question was posted before but I just couldn't get it using another way around. So your comments are highly appreciated. In the textbook, $$\nabla\bullet\left(\widehat{r}/r^{2}\right)=4\pi\delta^{3}\left(r\right)$$. But when I want to calculate the divergence using Catesian coordinates then it comes to calculate $${\frac{\partial}{\partial\,x} x/(x^{2}+y^{2}+z^{2})^{3/2}}$$ but I can't get the results of "zero" as it claimed. Where did I go wrong?

 

[Bob_for_short]

You have to add three derivatives to get zero, not only x-component. You will obtain 1/r³ - 1/r³.

 

[charng]

The Dirac delta function is a mathematical tool used to represent a point source in space. It is often used in physics and engineering to model the behavior of a point charge or a point mass. In this context, the Dirac delta function can be thought of as a distribution that is concentrated at a single point, with an infinite magnitude at that point and zero magnitude everywhere else.

The divergence of a vector field is a measure of the outward flux of the field from a given point. In the case of the vector field \widehat{r}/r^{2}, which represents the electric field of a point charge, the divergence is given by \nabla\bullet\left(\widehat{r}/r^{2}\right). This is equal to 4\pi\delta^{3}\left(r\right) because the field is radially symmetric and the flux is concentrated at the origin, where r=0.

When calculating the divergence using Cartesian coordinates, it is important to remember that the vector field \widehat{r}/r^{2} is not defined at the origin. This means that the partial derivatives with respect to x, y, and z are also not defined at the origin. Therefore, the expression {\frac{\partial}{\partial\,x} x/(x^{2}+y^{2}+z^{2})^{3/2}} is also not defined at the origin.

To properly calculate the divergence using Cartesian coordinates, we need to use the definition of the divergence in terms of partial derivatives:

\nabla\bullet\left(\widehat{r}/r^{2}\right)=\frac{\partial}{\partial\,x} \left(\frac{x}{(x^{2}+y^{2}+z^{2})^{3/2}}\right) + \frac{\partial}{\partial\,y} \left(\frac{y}{(x^{2}+y^{2}+z^{2})^{3/2}}\right) + \frac{\partial}{\partial\,z} \left(\frac{z}{(x^{2}+y^{2}+z^{2})^{3/2}}\right)

Using this definition, we can see that the divergence of \widehat{r}/r^{2} is indeed zero everywhere except at the origin, where it is infinite. This is consistent with the fact that the