Vector Calculus: Understanding Divergence
Physical Intuition
Divergence (div) is “flux density”—the amount of flux entering or leaving a point. Think of it as the rate of flux expansion (positive divergence) or flux contraction (negative divergence). If you measure flux in bananas (and c’mon, who doesn’t?), a positive divergence means your location is a source of bananas. You’ve hit the Donkey Kong jackpot.
Remember that by convention, flux is positive when it leaves a closed surface. Imagine you were your normal self, and could talk to points inside a vector field, asking what they saw:
- If the point saw flux entering, he’d scream that everything was closing in on him. This is a negative divergence, and the point is capturing flux, like water going down a sink.
- If the point saw flux leaving, he’d sniff his armpits and say all flux was existing. This is a positive divergence, and the point is a source of flux, like a hose.
So, divergence is just the net flux per unit volume, or “flux density”, just like regular density is mass per unit volume (of course, we don’t know about “negative” density).
The bigger the flux density (positive or negative), the stronger the flux source or sink. A div of zero means there’s no net flux change in side the region. In plain english:
Divergence = Flux / Volume
Math Intuition
Now that we have an intuitive explanation, how do we turn that sucker into an equation? The usual calculus way: take a tiny unit of volume and measure the flux going through it. We need to add up the total flux passing through the x, y and z dimensions.
Imagine a cube at the point we want to measure, with sides of length dx, dy and dz. To get the net flux, we see how much the X component of flux changes in the X direction, add that to the Y component’s change in the Y direction, and the Z component’s change in the Z direction. If there are no changes, then we’ll get 0 + 0 + 0, which means no net flux.
If there is some change in the field, we get something like 1 -2 +5 (flux increases in X and Z direction, decreases in Y) which gives us the divergence at that point.
In pseudo-math:
Total flux change = (field change in X direction) + (field change in Y direction) + (field change in Z direction)
Or in more formal math:
Assuming F1 is the field in the X direction, F2 in the Y and F3 in the Z.
A few remarks:
- The symbol for divergence is called “del” and is an upside down triangle.
- Divergence is a single number, like density.
- Divergence and flux are closely related – if a volume encloses a positive divergence (a source of flux), it will have positive flux.
- “Diverge” means to move away from, which may help you remember that divergence is the rate of flux expansion (positive div) or contraction (negative div).
Divergence isn’t too bad once you get an intuitive understanding of flux. It’s really useful in understanding in theorems like Gauss’ Law.
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Have a question? Know an explanation that caused your own a-ha moment? Write about it here.
i came across a piece of info saying that the equation of conitinuity comes from divergence theorem(i will give the statement here:”for any arbitrary region of volume v coverd by surface area s,the flux of the current density over the surface s is equal to the rate at which mass / charge leaves the volume v”)
is there a derivation for this…? ..or could anyone tell me an intuitve approach..where i can atleast visualise what is happening?
kirtika — September 26, 2007 @ 1:33 am
Hi kirtika, I’d have to see some more on this, but I think in this context “flux of current density” means “change in current density”.
This statement may be saying the amount of current you see passing through a surface depends on the amount of charge leaving the region (a moving charge can induce a magnetic current for example). This is a bit tough to visualize, but as the charge moves the flux through the surface will change — imagine a firehose (constantly spitting out water) moving through an invisible sphere. As the hose goes along, the amount of water passing through will change.
Kalid — October 16, 2007 @ 10:52 pm
would be nice if you could tell us something about Gauss & Stokes Thm! nice stuff btw!
kalimentes — November 29, 2007 @ 3:20 pm
Thanks — I actually have some stuff on Gauss and Stokes at my old site here:
http://www.cs.princeton.edu/~kazad/resources.htm
Which I need to revise and update. But it’s on the list
Kalid — November 29, 2007 @ 4:44 pm
Hiy’all!
I wish to tell you a story.
As a young kid and in the high school I had few to no problems with math; it was easy to understand and the examples in the maths books were quite intuitive and visual.
What a shock it was to attend the first math courses in the university. I don’t know about the ones for math majors, but at least the math course materials for us engineering students were conjured from some fiery bowels of hell.
With no clarifying pictures and even less explanations (”divergence measures the change of vector function in its direction and the spreading of the direction. Now do the exam.” is pretty much all we get), the typical brute-force technique among tech students here is to memorize the ten or so calculations that the exam questions are picked from each year and vomit them on the test paper. Formula after formula. Equation, equation, equation.
Somebody just forgot to tell what the hell these formulas do and what they are used for anyway.
All this bitter rambling is here for a reason… I really wish to thank you for your explanations! This is the first time that instead of memorizing some stupid upside down triangles and strange-looking d’s with no comprehension of them whatsoever I really do understand, what the hell a “curl” actually is.
I am going to recommend your site to every tech and math student I know. Something like this is really missing from the teaching of mathematics and I don’t know whether the professors are too jaded, indifferent or too alienized from the real world to notice this.
In case you haven’t noticed, there are some neat animations of divergence and curl in
www.math.umn.edu/~nykamp/m2374/readings/divcurl
also.
Once again… A big great thanks to you, Kalid! Keep up the good work!
Jaakko Seppälä — June 2, 2008 @ 11:36 pm
Hi Jaakko, thanks for the comment — it’s something I would have written a few years ago! I had the same exact problem with regurgitating formulas in engineering classes, which motivated me to create this site. It really bothers me to know how without knowing “why”
.
Thanks for that link, the animations look really cool! Visualizing these concepts makes them so much clearer.
Again, appreciate the comment!
Kalid — June 5, 2008 @ 11:11 am
hi nice post indeed. it clarified a lot of stuff regarding divergence. but tell me one small thing.
Is it possible when i consider an infinitesimal small volume from the actual volume, and the divergence might be flowing out in the infinitesimal volume but in the overall volume the divergence might be flowing in towards the volume?
janakiraman — October 8, 2008 @ 10:27 pm
Hi and also it would be extremely good if you can also tell something similar about the Tensors as well.
janakiraman — October 8, 2008 @ 10:37 pm
i whould like someone to explane me curl what is mean in simple or what is mean in fluid flow
simple example with a culculation number,iam intersted in but i need help ,regardes basheer
basheer — June 15, 2009 @ 4:04 am