Vector Calculus: Understanding Divergence

@mihu: Awesome comment, thanks for the pointer. I had to take a look at those links, I think the intuition still holds but I should phrase it differently.

A “point” in this case is really a tiny cube, and flux is coming in or leaving in the x, y and z directions. It’s possible that two sides are positive and the third is negative so the results cancel (or one side overpowers the other).

That’s essentially what’s happening with the sphere in the tricky case… there is more flux (by surface area) leaving, but the strength of the field is less. This perfectly balances the incoming (fewer but stronger) flux lines.

So, a better phrasing may be “The point looks in all directions: x, y, and z. Is more flux entering or leaving overall?”.

This is really great. I’ve been going through a lot of books on divergence and curl and this is by far the clearest explanation.

@aha: Glad it helped! It took me a while to find an analogy that worked for me.

Liked the physical intuition too. I guess math prof are very much in love with abstraction. Perhaps for pure maths majors, it is ok. Cos it gets much worse, sometimes there maybe no easy physical analogies, just a set of laws or axioms and rules to follow and then investigation. But for most of us in engineering, we would love to know why? Why is it relevant? Analogies are important.

I really liked d concept of physical intuition of mathematical concepts. Gud job.

hey can you tell me where divergence of vectore field is bieng used please. reply asap. email:qasim_mnawaz@yahoo.com

@Drew: Thanks for the note - gradients, flux, and divergence really bothered me in school also. I have a general purpose book but would like to make one on vector calculus eventually – really appreciate the encouragement!

Excellent Resource. I am an engineering physics major who would certainly recommend this site to any of my colleagues. I Read the articles on gradient vector, flux and divergence and consequently gained a much more thorough understanding. The authors should write a textbook if they haven’t already. Plain language explanations with practical examples are crucial to the understanding of this material. Write a textbook and market it to the professors. It would sure beat out all of the piece of shit books that I’ve been forced to use thus far. Thank you.

@Green: Thanks so much, really glad it’s helping! Thanks for spreading the word too :).

Hi Kalid, it’s a great fortune for me to come across your website :slight_smile:
I feel so lucky that i click it by accident! :slight_smile: It’s an awesome place where a lot of my math doubts are cleared! THANKS A LOT. Keep it up with your good work and we will all gain benefits from it! I’m sharing it with my friends who are struggling with college math too!

A chinese word for you, JIA YOU! :slight_smile:

Hi Kalid… Honestly, I was very much impressed by your explanation. My question is , Does gradient always acts on Scalar quantities and Divergence on Vector quantities. It would be grateful if you can explain the divergence of a gradient. I know we get a Laplace Operator , but can i have a physical explanation? ( If Possible)

“i Like it.
But could not understand the perfect defination of the divergence which is
” Divergence is a vector operator which measures the magnituade of a vector field’s source or sink, at a given point, in terms of a signedl scalar."

we need to see the larger number of examples and derivetion concerning vector calculus

Hey, what about vector integration and green’s theorem?

Good work…! i have few doubts…

  • why the divergence is explained with respect to unit volume basis… and even particularly why as volume decreases to zero?

  • How flux is more when the volume is less?

Thnak u in advance.

Thanks for your fantastic articles on flux and curl and now divergence. I have two calculus textbooks neither of which was particularly clear on the intuition of these topics. At least not to someone who hasn’t done a certain amount of practice on the breadth of the material (I’m reading the textbook so I can understand a few formulas I’m working with). Your descriptions were very clear and helpful!

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?