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!
Best Answer
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).