# What is the physical meaning of divergence?

• I
Hawkingo
I want to visualize the concept of divergence of a vector field.I also have searched the web.Some says it is
1.the amount of flux per unit volume in a region around some point
2.Divergence of vector quantity indicates how much the vector spreads out from the certain point.(is a measure of how much a field comes together or flies apart.).
3.The divergence of a vector field is the rate at which"density"exists in a given region of space.
4.divergence measures the net flow of fluid out of (i.e. diverging from) a given point. If fluid is flowing instead into that point the divergence will be negative.

I am confused with all these definition.Can someone give me a proper visualizable definition which also satisfies in some way all the above definition and descriptions? Thanks for your response.

Gold Member
2022 Award
1, 2, and 4 are valid intuitive ideas about the meaning. I've no clue what 3 should even mean.

The best physically intuitive argument of the definition of the operator "divergence on a vector field" is to consider a fluid flow. Now put any volume ##V## into the flowing fluid. It's boundary surface is denoted by ##\partial V##. At any point of an infinitesimal surface area at this boundary surface you can define the surface-normal vector ##\mathrm{d}^2 \vec{f}##, which has the direction perpendicular to the surface pointing by definition out of the volume ##V##. It's magnitude is the area of the surface element.

If the boundary surface is parametrized by two parameters ##u## and ##v## via the function ##\vec{r}(u,v)## then
$$\mathrm{d}^2 \vec{f} =\mathrm{d} u \, \mathrm{d} v \partial_{u} \vec{r} \times \partial_{v} \vec{r},$$
provided the order of ##u## and ##v## is chosen such as to make this surface-normal vectors point out of the volume (if not, simply interchange ##u## with ##v##).

Now we like to know, how much fluid flows into or out of the volume ##V##. Let's define ##\rho(t,\vec{x})## as the mass density of the fluid and ##\vec{v}(t,\vec{x})## the fluid-velocity field, which tells you how the fluid element at position ##\vec{x}## momentarily moves at time ##t##. Then, in an infinitesimal time ##\mathrm{d} t## there'll flow a mass
$$\mathrm{d} m = \mathrm{d}^2 \vec{f} \cdot rho(t,\vec{x}) \vec{v}(t,\vec{v}) \mathrm{d} t$$
out of the volume through the surface element since the volume of this outgoing fluid is given by ##\mathrm{d} V=\mathrm{d}^2 \vec{f} \cdot \vec{v} \mathrm{d} t##, and the corresponding mass is ##\mathrm{d} m = \rho \mathrm{d} V##. Note that ##\mathrm{d} m>0## if the angle between ##\vec{v}## and ##\mathrm{d}^2 \vec{f}## is in ##[0,\pi/2]##, meaning that then the flow through the surface element is indeed outgoing; otherwise the fluid element flows into the volume and is thus counted negative.

Now the total mass per unit time is
$$\dot{m}=\int_{\partial V} \mathrm{d}^2 \vec{f} \cdot \vec{j}.$$
Here I've interduced the current density ##\vec{j}= \rho \vec{v}##. Which is defined such that for any surface element ##\mathrm{d}^2 \vec{f}## the amount of mass flowing through this surface element per unit time is ##\mathrm{d}^2 \vec{f} \cdot \vec{j}##.

If you want to know how much mass per unit time flows through the surface of an infinitesimal volume at a given point you must just shrink the volume ##V## to this point. Of course this amount will go to 0. So a better measure is to ask for how much mass will run out of the smaller and smaller volume at this point per unit of the volume. This leads to the definition of the divergence of a vector field.
$$\text{div}\, \vec{j}(t,\vec{x}) = \lim_{V \rightarrow \{\vec{x} \}} \int_{\partial V} \mathrm{d}^2 \vec{f} \cdot \vec{j}.$$
In this definition you can choose a little cube with its edges parallel to the coordinates axes of a cartesian coordinate system. Then analysing the surface integral and taking the limit leads to
$$\text{div} \, \vec{j}=\vec{\nabla} \cdot \vec{j}=\partial_1 j_1 + \partial_2 j_2 + \partial_3 j_3,$$
where
$$\partial_j=\frac{\partial}{\partial x_j}$$
denotes the partial derivative wrt. the cartesian coordinate ##x_j##.

From the definition it's also clear that Gauß's integral theorem holds:
$$\int_{\partial V} \mathrm{d}^3 r \mathrm{div} \, \vec{j} = \int_{\partial V} \mathrm{d}^2 \vec{f} \cdot \vec{j}.$$
This is easily seen using the definition of the operator ##\mathrm{div}## and then evaluating the volume integral by putting a finer and finer spatial grid to evaluate the volume integral. In using the definition on each tiny volume you'll see that the corresponding surface integrals over the inner surfaces of the grids cancel pairwise from the contributions of two adjecent grid cells, and only the surface integral over the boundary surface of the volume ##V##, i.e., ##\partial V## remains.

• Hawkingo, Delta2 and Dale
Staff Emeritus
If you're looking for physical meaning, it always reminds me of this picture. #### Attachments

• TachyonLord and vanhees71
zul8tr
Great example of a Morning Glory spillway, designed several. Very interesting hydraulics not so much at the circular weir but once you get down into the receiving pipe and the down stream conditions.

Also called a Glory Hole and Bell Mouth spillway

Wouldn't want to get stuck in one, not likely to survive

https://www.bing.com/videos/search?...D1595D409C8E9C3DC4A1D1595D409C8E&&FORM=VRDGAR

Certainly is a interesting flow system also could be an analogy to a black hole with the weir lip as the event horizon and the region of no return unless radially outward local velocity greater than the critical velocity for the local water depth.

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alan123hk

visualizing the Divergence and Curl...

I am confused with all these definition.Can someone give me a proper visualizable definition which also satisfies in some way all the above definition and descriptions? Thanks for your response.

Starting at 3:27:

alan123hk
I really can't understand what the third statement "The divergence of a vector field is the rate at which"density"exists in a given region of space" is saying, so I search it on the internet and find the information as follows: -

http://www.chabotcollege.edu/faculty/shildreth/physics/curl.htm

"The physical significance of the divergence of a vector field is the rate at which "density" exits a given region of space.
The definition of the divergence therefore follows naturally by noting that, in the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into or out of the region.
By measuring the net flux of content passing through a surface surrounding the region of space, it is therefore immediately possible to say how the density of the interior has changed.
This property is fundamental in physics, where it goes by the name "principle of continuity." When stated as a formal theorem, it is called the divergence theorem, also known as Gauss's theorem"

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Homework Helper
Gold Member
The way I see it (it might be oversimplification but I believe it holds an important part of truth) is that if the divergence of a vector field is not zero at some point, then this means that there is a source ( or a sink) of the vector field at that point. This is verified by the first Maxwell equation where we have $$\nabla \cdot \vec{E}=\frac{\rho}{\epsilon_0}$$ so this means that if the divergence of electric field is not zero at some point, then there is charge density (which essentially means electric charge which is the source of the electric field) at that point. It is also verified by the continuity equation $$\nabla \cdot \vec{J}=-\frac{\partial \rho}{\partial t}$$.

So this is all about "coupling" the notion of divergence with the notion of the source of the vector field.

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• Hawkingo
Homework Helper
2022 Award
I really can't understand what the third statement "The divergence of a vector field is the rate at which"density"exists in a given region of space" is saying
If you replace "density" with "source density" does it make it clear? For gravity it would be mass density; for electric field charge density; for flow it would be influx density etc etc. Not complicated.

• Hawkingo
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