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I think I understand that the divergence vector operator $\nabla \cdot$ yields a scalar $\nabla \cdot \vec{F}$ that represents how much a given point is a "source" or a "sink". My curiosity has prompted a few questions that I could not find answers to via search engines:

1) The scalar $\nabla \cdot \vec{F}$ is a multivariable function of spatial coordinates; would finding the maximum of this function yield the point in space that is the most abundant "source" present in vector field $\vec{F}$?

2) What do we call the "stuff" that is emanating from the sources and sucked into the sinks? I'm already familiar with a few of the classic examples in physical models, I'm looking for a more general mathematical understanding of what this "stuff" is in the abstract (and how it relates to its vector field).

3) Due to my ignorance, this whole "source" and "sink" quantification seems a bit vague and non-rigorous to me. Would it be more meaningful to normalize the divergence $\nabla \cdot \vec{F}$, perhaps dividing it by its maximum value (or the magnitude of the larger extreme, beit max or min)? Then the normalized divergence would have a range of $[-1,1]$ and could give the percentage that a chosen points acts as a source ($(0,1]$) or a sink ($[-1,0)$), correct?

4) Is a point at which the divergence is zero called a node?

5) Are there any physical examples of the divergence of a vector field being infinite? Does this tend to have useful physical meaning, or do scientists/engineers seek to tame such singularities by "renormalization" or other such tricks?

Thank you for your time!

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1) You are right. You can compute the extrema of this function and see how the sources and sinks are distributed and what are their magnitudes.

2) This "stuff" has a name only when it does represents something. I mean, if you are talking about fluid mechanics, then the divergence of the velocity field means compressibility. Since the divergence of a vector field is defined in a point (this is only a mathematical artifact, "because matter is empty") I should have said mass creation or destruction.

To illustrate this, the net amount $\Delta q$ of some physical quantity that is exchanged through any closed surface $S$ within a vector field $\vec{F}$ with constant divergence $\mathrm{div}{\vec{F}}=c$ is proportional only to the volume $V$ this surface encloses: $$\Delta q =c\,V$$

3) In fact the divergence of a vector field has a definition: $$\mathrm{div}\vec{F}=\lim_{V\to 0}\frac{1}{V}\int_{\partial V}{\vec{F}\cdot\vec{n}\,d\sigma}$$ where it represents the net exchange of some quantity through a surface enclosing a volume $V$ when it shrinks to $0$.

Of course you can do it that way. You can define a function that tells you the normalised divergence you have if you consider this offers more valuable information to you. Regarding your intervals, yes, you are correct.

4)I am not used to this terminology

5) There are many points of view regarding the answer to this question but mine is that the infinity is not a physical quantity. Infinity means big enough wrt. what is considered (this is related to what I have said before: "matter is empty"). Singularities appear basically when mathematical models cannot describe what is going on in there. For example when assuming a potential solution for the flow past a wing, a singularity appears in the trailing edge, because the viscosity has been obviated (which is a physically incorrect assumption).

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Yes, you are absolutely right. For example, imagine that the vector field $\mathbf F(\mathbf x)$ represents the rate (and direction) of heat flow per unit area inside some material. If $V$ is any three-dimensional region within our material and if $\partial V$ is the two-dimensional boundary surface of this region, then the divergence theorem says that $$ \oint_{\partial V} \mathbf F . d \mathbf A = \int_V (\nabla . \mathbf F) dV.$$ The left-hand side represents the total rate at which heat flows out of the region $V$, so the whole equation tells you that the rate at which heat flows out is equal to the integral of $\nabla . \mathbf F$ over $V$. This already establishes a link between $\nabla . \mathbf F$ and your nice idea of "stuff emanating".

Ideally, we would like to express this idea in a more local manner, without integrating $\nabla . \mathbf F$ over a volume. To do this, we will take $V$ to be $B(\mathbf x_0, r)$, the spherical ball of radius $r$ centred at a fixed point $\mathbf x_0$. Assuming $\nabla . \mathbf F$ varies continuously as $\mathbf x$ varies, we can take the limit as $r \to 0$ to obtain $$\lim_{r \to 0} \frac{1}{{\rm Vol}(B(\mathbf x_0, r))} \oint_{\partial V} \mathbf F . d \mathbf A = \nabla . \mathbf F(\mathbf x_0) . \ \ \ \ (\ast)$$ If you stare this is, you will realise that the thing on the left-hand side represents the rate per unit volume at which heat is being generated at the point $\mathbf x_0$. And that is the local interpretation of the divergence $\nabla . \mathbf F(\mathbf x_0)$.

The more positive the value of $\nabla . \mathbf F (\mathbf x_0)$, the faster heat is being pumped into the system at $\mathbf x_0$; the more negative the value, the faster heat is being removed. It's exactly as you said in your point (1). There is no benefit to be gained by normalising $\nabla . \mathbf F$ as suggested in your point (3), because the magnitude of $\nabla . \mathbf F$ is meaningful in determining the rate at which heat is being generated or removed.

Now then, how can we go from this physical example to a purely mathematical way of understanding what $\nabla . \mathbf F$ represents? The answer is simple: Just take ($\ast$) as your "definition" of the divergence! And that answers your point (2).

But before we move on, we should try to visualise this. Below, I've plotted the vector field $\mathbf F = (x,y,z)$, which has constant, positive divergence. (Actually, I only plotted a two-dimensional cross-section, but I hope it's clear enough.) You can see that the field lines point outwards, like the needles on a hedgehog. This means that the surface integral $\oint_{\partial V} \mathbf F . d \mathbf A$ will be positive on any closed surface $\partial V$. A visual way of answering your point (2) is therefore to say that $\nabla . \mathbf F(\mathbf x_0)$ measures the hedgehog-like quality of the field lines around $\mathbf x_0$.

enter image description here

Onto your question (4): what are we to make of points of zero divergence? These are simply points where no net heat is being generated or absorbed, or equivalently, points around the vector field has no net hedgehog-like or anti-hedgehog-like quality. There is no particular reason to think of these points as "nodes", as you suggested. In fact, there are plenty of examples of vector fields with zero divergence everywhere: take for example, $\mathbf F = (-y,x,0)$, which has a rotational look to it. (This vector field has non-zero curl, but that is another story...)

Finally, you asked about vector fields with infinite divergence at certain points. Here is a famous example: $$ \mathbf F = \frac {\mathbf e_r}{4\pi r^2},$$ In this equation, $r$ is the spherical polar radial coordinate and $\mathbf e_r$ is the unit vector pointing radially outwards everywhere. This has zero divergence everywhere, except at the origin, where it has infinite divergence. (The infinity can be made more precise using delta functions.) In physics, one of Maxwell's equations states that the divergence of the electric field at a given point is equal to the charge density at that point. This means that our $\mathbf F$ represents the electric field around a point charge localised at the origin.

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