Glad it helped!
“…the amount of flux entering or leaving a point.” How big is the point?
You wrote “Divergence = Flux / Volume” then gave a similar equation with a limit as volume approaches 0. So what’s the limit for? Thanx
Hi Simon, great question. Basically, limits are used to find a reasonable estimate for an “impossible” situation, such as the amount of flux at a point. (Technically, points don’t have any size, right?)
This is similar to finding the density at an x,y,z coordinate in space. We figure out how much mass is in a volume surrounding that spot (a little cube?), and let it shrink. Hopefully, the limit as we shrink the volume converges into a meaningful result. There’s some more on limits here: http://betterexplained.com/articles/an-intuitive-introduction-to-limits/
I think in the general math definition of a point a point is 0D. I think you should make sure readers know what you mean by a point. By a point I think you mean an infinitely small volume. If a volume is not infinitely small couldn’t you find the divergence of the volume? If so the limit isn’t really needed. I like the site a lot by the way, simplicity is great, thank you for replying too. @ made easy: you wrote: “How flux is more when the volume is less?” I don’t see where it says that. I think it’s kind of like finding density, Say if you found the density of a volume of water, if I found the density of a smaller volume of water it would be the same as the bigger one. Is that correct anyone?
This article is a great service to humanity. Very rare piece.
Why can’t the teachers explain stuff like this. The gradient and divergence could never be so clear! Thank you. In college it looks like rocket science. xD So simple it is! Thank You. =)
Great stuff, big fan of the site, although I only recently discovered it!
I was wondering whether you plan to cover some intuition regarding the Laplacian operator [tex] \Delta [/tex] and its connection to divergence/gradient (as it is, of course, the divergence of the gradient). I’ve worked through some derivations that show that it can be thought of as a local averaging operator in the sense that the Laplacian of a function [tex] f [/tex] at a point [tex] p [/tex] is proportional to the average value of [tex] f(x) - f§ [/tex] on the surface of spheres of radius [tex] r \rightarrow 0^{+} [/tex] centered at [tex] p [/tex], but it isn’t totally clear what the connection is between this construct and divergence/gradient. It is, however, a really useful way of looking at the heat equation: [tex] \frac{\partial u}{\partial t} = \Delta u [/tex] simply means that the rate of change in time of heat is governed by its average rate of change in space. That is, there is more heat exchange in regions of highly variable temperature than regions with smoothly varying temperature. The ‘averaging’ interpretation also works nicely for understanding harmonic functions (those that have zero Laplacian).
Great work mannn… Thank you a lott>>
Keep it up… greattt…
yoyo
consider f(x)=xi+yj,when u take divergence of this function the value is 2
now a positive value shows the flux is moving away.i would like to now how do we graphically account for this value of 2 when we draw a plot of the function.
assalamu alaikum brother
great job indeed.the point source is similar to a tiny cube it hits the point.
hope some more article especially on vector calculus concepts like line/volume/surface integral
cross /dot product
vector field
""another question burns my neurons for several years since I first met vector addition.
the concept /intuition of triangle law.
it contradics geometry though direction matters.but how direction parameter change that sum of two sides of traingle greater than the rest one.
Hey, just want to say that these descriptions are great! They really helped me get an intuitive feel for some very common math equations. 
Hi,
Could you please consider covering the laplacian operator ? It is connected to this concept and would help a great deal !
_thank u …u just shortened the working hours of millions of students
Hi!
Your articles are well written, clear, and comical. This piece on divergence made it easier for me to understand my class notes on E&M - and somewhat increased my confidence in the area of differential calculus. It’s no-longer a matter of students not understanding their coursework…speaking on my own part, reading your intuitive treatments is plain old fun!
Great job, and I’m looking out for a discussion on the Laplacian.
Thanks, and have a great day!
- Oswald.
Thanks Oswlad, glad it helped! Added the Laplacian to my topics list :).
@Kazz: Thanks so much! It’s awesome to hear when the site is helpful. I’d like to do more on physics down the line :).
I wish if i had words to express my gratitude towards u sir. I recently took admission in engineering college and our teacher taught us about divergence, gradient and Curl. I didn’t even understood a bit from her lecture, i thought that may be the fault is in me. Than i randomly came across ur article and now i feel confident on my hold of concepts. Plzzz keep posting engineering physics concepts posts, that would be very helpful for guys like me who find teachers boring and bookish stuff useless. Once again, thanks a lot.
I’m taking a Automation and Control technology course and this stuff might be a little advanced for me but it has helped me understand magnetism better. Do you have any helpful links on A C circuits or Solid State (electronic principles). Any and all help would be appreciated. Thanks.
Unfortunately I don’t have anything on electronics. But, I’ve found the hydraulic analogy helpful. The original names for things like voltage were “electric pressure”. Sometimes the historic understanding helps figure out the key intuition.
Hi Kalid
Again, I have to thank you. What a brilliantly clear way you have explained this! I am very happy - because I can get formulae, etc. from my textbook and I do believe that what you’ve said is also explained in there but not in a really memorable way. It always helps to see something explained differently. And in your case, so clearly! Anne, UK