A more general definition of a source and a sink for a vector field

Question

As far as I can tell the definition of a source and a sink respectively are given in terms of the divergence operator.

That is, given a vector field $\vec{D}$, it has a source in point $P$ if its divergence $\text{div}\vec{D}$ is pozitive in $P$ or a sink if it's negative. For example, in electromagnetism, one says $\text{div}\vec{D} = \rho_v$ where $\rho_v$ is the volume charge density and $\vec{D}$ is the electric flux density.

But let's say $\vec{D}$ is given by a positive point charge $q$ located at $(0,0,0)$ which creates the field

$$\vec{D} = \text{const} \frac{\vec{R}}{|\vec{R}|^3}$$

where $\vec{R}=x\vec{i}+y\vec{j}+z\vec{k}$.

In this case, $\text{div}\vec{D}=0$ everywhere, however the origin is a sort of a source as the field "emerges" from there and the net flux over each surface enclosing the charge is positive.

My question is: are there any other definitions of a source and sink? Possibly some that are a bit more general and encompass more particular cases such as the one I've last mentioned?

Answer 1

I think one intuitive generalization comes from the divergence theorem! Namely, if we know that a vector field has positive divergence in some region, then the integral over the surface of any ball around that region will be positive. That encompasses your example, because that way, we never need to look at the singularity at $x = 0$, we just look at balls around that singularity!

Denote by $B_r(p)$ the open ball of radius $r > 0$ around $p$, and denote by $\partial B_r(p)$ its boundary surface.

Let $U \subset \mathbb{R}^n$ be an open set, and $p \in \mathbb{R}^n$ a point so that there is an $\epsilon > 0$ so that the spheres $\partial B_r(p)$ are contained in $U$ for all $r < \epsilon$.

Given a continuous vector field $X : U \to \mathbb{R^n}$, we say that a point $p \in U$ is...

• ...a source for $X$ if there is an $\epsilon > 0$ so that $$ \oint_{\partial B_r(p)} X(y) \, dy > 0 \quad \forall r < \epsilon.$$
• ...a sink for $X$ is if there is an $\epsilon > 0$ so that $$ \oint_{\partial B_r(p)} X(y) \, dy < 0 \quad \forall r < \epsilon$$

If your vector field can be extended to be smooth in the whole interior $B_r(p)$ of the spheres $S_r(p)$, then the divergence theorem tells us

$$\oint_{\partial B_r(p)} X(y) \, dy = \int_{B_r(p)} \text{div} X(y) \, dy,$$

and then your definition implies this one, because if $\text{div} X(p) > 0$ in a single point, then by continuity arguments there is going to be a whole ball $B_r(p)$ on which $\text{div} X > 0$.

You'll find that your example fits in perfectly with this definition and you can very easily calculate the integrals on balls around zero, and they are all going to be positive, even though you can never touch the point zero itself.

I'm not citing from any textbooks or so, so beware, this is just my own opinion on a reasonable generalization :)

EDIT: An alternative is to change the definition of the divergence, but still using this idea of integrating balls around points, See for example in this question and answer.

Answer 2

In case that the vector field is integrable you can give a much more topological definition.

Let $\vec{D}$ be an integrable vector field and $d$ its flux. Let $p$ such that $\vec{D}(p)=0$.

$p$ is a $\textit{sink}$ iff there exists an open set $U$ containing $p$ such that $\overline{d(U)} \subset U$.

$p$ is a $\textit{source}$ iff there exists an open set $U$ containing $p$ such that $\overline{U} \subset {d(U)} $.