What is the intuition behind this divergence example

Question

So, I am taking a course on electromagnetic theory and I would like to have a firm grasp on the basics. Now there is an example in the book that asks, Find the divergence of a position vector to an arbitrary point. The solution is OP = x i + y j + z k

and the divergence by taking the partial derivative of the component is simply 3

Now, the mechanics is easy to compute, but what does that actually mean? I know it means that at any given point of time there is 3 more vectors entering than leaving a certin point. But I am having a difficulty visualizing it. In all the intution videos I found on youtube they show a sea of vectors flowing in space. But we only have one vector here. I am confused. I guess what I need to get it is to see this flow of vectors for this example.

Answer 1

For dealing with flux I find the heat inside a volume to be the easiest example to build my intuition. In this case, with constant divergence everywhere it means every point is a heat source putting out exactly 3 units per area of heat.

Also, it's not accurate to say that there are three more vectors entering than leaving a point as an interpretation of divergence. This can be seen by calculating a similar example given by $OP = \frac{1}{4}x\boldsymbol{i} + \frac{1}{4}y\boldsymbol{j} + \frac{1}{4}z\boldsymbol{k}$ where the divergence is $\frac{3}{4}$ so you wouldn't even have a whole vector. Sources and sinks tend to be a better interpretation with the size of the divergence being a measure of how quickly the flow is through that point per unit volume.

Answer 2

This question concerns a certain flow field in ${\mathbb R}^3$, namely the field ${\bf v}({\bf x}):={\bf x}$. Such a field could not be realized by flowing water, because ${\rm div}\,{\bf v}({\bf x})=3\ne0$ for all ${\bf x}\in {\mathbb R}^3$. What does the fact ${\rm div}\,{\bf v}({\bf x})\equiv3$ indicate geometrically, or physically? It means that in this flowing process new material is generated at a rate of $3$ units of mass per second and unit of volume. This is difficult to visualize. But imagine a flow of hot gas containing some special ingredient, in which new molecules of interest are chemically generated during the process.