The Divergence Theorem, a visual explanation

This video talks about the divergence theorem, one of the fundamental theorems of multivariable calculus. The divergence theorem relates a flux integral to a triple integral.
Green's Theorem: • Green's Theorem, expla...
Line Integrals: • The Line Integral, A V...
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This video was animated using manim: github.com/3b1b/manim
Source code for the animations: github.com/vivek3141/videos

[@drrajendrap2202]

At 12:14, The Gauss divergence statement should be ∇ · F instead of ∇ X F.

[@bhgomes]

exactly! easy to see it, because left side results in a scalar and the rigth one in a vector.

[@cwaddle]

Also, the surface integral should be over F.n ds rather than F.ds

[@isxp]

Thank you! I kept rewinding it to see what happened. Should have checked comments first.

[@Dhanush-zj7mf]

@@cwaddleit is not simply " ds " it is " ds-bar " which is a vector representing " (n-cap).ds " and it means n-cap scaled by ds which is also right to use......

[@JohnVKaravitis]

You are absolutely correct.

[@TheTKPizza]

I do have to say, the quality of your videos is quite amazing, close to 3blue1brown. Keep on delivering quality like that and this channel is going to grow fast (as you can already see with a few of your vids). The visualization of Green's and Divergence Theorems really helped me a lot with actually understanding them (and not only being able to calculate on their basis). Thanks a lot, dude!

[@stevekeller7696]

I really enjoyed learning this some decades ago. Here it's better explained, and without the exams.

[@adityagiri2084]

Great job buddy , u really explain it in depth

[@ElZenom]

I'm surprised that I only stumbled upon this channel when I misclicked, thinking that it's 3blue1brown's video. One of the best mistake of my life. One suggestion I have is for you to slow down in some transitional parts. For example, when you're calculating 2D Flux integral for F=[xy+x, x+y], you can show the third step in which you input in the {xy+x} portions before inputting the boundary values (cos, sin). Those small inputting steps might mean little once you know it but will help more in visualization if you show it.

[@vcubingx]

Good point, thanks for the input 😊

[@binishjoshi1126]

@@vcubingx Yes, I recommend that too, the videos need more visualization. Anyways, I enjoyed the content!

[@NovaWarrior77]

Awesome work sir!

[@jwfundsd]

Excellent!!! Congratulations!!!

[@vcubingx]

Thank you!

[@mathOgenius]

Hi , Can you please tell me which software are you using to make these awesome videos , Please !?

[@josht7238]

great explanation thanks so much!

[@darinarieko]

This came up on my recommended page 3 years after graduating college. I am not complaining.

[@MrThemastermind88]

At 5:20 there's a mistake in the formula: in the left hand side you have the line integral of a vector field F over a curve C with a parametrization r, that is the "work" integral. On the right hand side you have the expansion of the line integral over a curve C of a SCALAR field f, in which you multiply f evaluated at r(t) by the magnitud of the derivative of r(t). The reason why you need the formula of the right hand side (the expansion of the line integral over C of a scalar field) is because the dot product of the vector field F times the n hat vector is in itself an scalar field. Sorry for any spelling mistakes, and great videos man. Keep it up, will subscribe

[@saurabhsingh-ow7ue]

thank you sir.....

[@zahraakhalife9150]

Thank you !!!!!!

[@girishgarg2816]

Damn! You are just 16!!!!

[@hrkalita159]

Means??

[@girishgarg2816]

@@hrkalita159 he was only 16 years old when he made this vdo

[@alannolan3514]

Thank you.

[@ProfeJulianMacias]

Excellent Problem

[@aadhuu]

Just discovered your channel today! Absolutely amazing! How did you learn all the partial differentiation, divergence and stuff?

[@NovaWarrior77]

Check out Khan academy for calc 3 taught with this specific animation style, or my HIGHLY RECOMMENDED professor Leonard on KZbin, both offer a full calculus 3 course. Super excellent instructor! Also this channel has a couple of videos on assorted topics from calculus 3.

[@gianlucacastro5281]

@@NovaWarrior77 I second the recommendations!! Today I'll do my last test on what is covered by both these playlists and they helped me A LOT with multivariable calculus. Absolutely wonderful content from them.

[@NovaWarrior77]

@@gianlucacastro5281 Right!

[@khan.shadab]

Watch Calculus series by 3Blue1Brown

[@joefuentes2977]

Internet

[@alanioth5388]

At 5:11 you say that you're rotating the tangential vector by 90deg. Then you show an expression in radians that includes 2*pi. How does this represent rotation by pi/2?

[@stampai2305]

Carry on

[@Jirayu.Kaewprateep]

In case of Electric flux, that is not only the electric field BUT the random high potential electric discharge ( Vander Graff ). What if it is magnetic field, in imbalance shape magnetic force is stronger near by the magnet or at the pointing area? ( Spherical shape, average force is reasonable )

[@bmet001]

At 6:37, surely in a linear flow field the divergence is zero? Advection into and out of the region F are identical, no? Would love to know why it is grad.F > 0

[@vendettawasd4516]

Do you have an answer now?

[@OlliFritz]

This is funny, I subscribed when you were making cubing content, and now there's advanced math videos that are relevant in my University courses. What do you study?

[@vcubingx]

Haha, I'm still 16 and in high school.

[@vcubingx]

That's funny, I see you on cf a lot also and I remember subscribing to you a couple of years ago as well.

[@kanewilliams1653]

@@vcubingx Jeez, doing much better than me, a humble viewer, keep it up!

[@kanewilliams1653]

If you don't mind, I'm making educational videos myself on another channel, what software do you use?

[@ster2600]

@@vcubingx haha do you study this at school? Why don't you do some Olympiad stuff? You could probably get into the IMO

[@pbj4184]

In a line, the amount of "fluid" flowing out of a volume is equal to the fluid flowing out of its surface if it has a closed surface. Edit: As Nikita Kipriyanov has pointed out below, the amount of imaginary fluid flowing out of the volume is equal to the amount entering it PLUS what is created/sucked inside/into it

[@nikitakipriyanov7260]

Well, not quite. You forgot to add at the end: "plus a fluid that is created inside that volume". This "creation out of nowthere" is essentially what divergence is. You sum all creation... wait, that's a triple integral by the enclosed volume, the right part of the formula.

[@pbj4184]

@@nikitakipriyanov7260 Oh yes, I had imagined a light bulb emitting a "light fluid" but I forgot to add that detail in my comment. Thanks for mentioning that 👍👍

[@pbj4184]

@@nikitakipriyanov7260 It does make sense that way, the amount of "fluid" emerging from a volume must be coming out of its surface and if it has a closed surface, then they must be equal because then the fluid coming out must be coming out of the surface of the volume

[@pbj4184]

@@nikitakipriyanov7260 Would you agree with this intuition? I'm not entirely sure about it since I haven't learned much about multivariate calculus. When I first learned this equation, this is what I could Intuit for myself

[@nikitakipriyanov7260]

@@pbj4184 Again, the amount of the fluid leaving the volume equals the amount of the fluid entering the volume PLUS the amount in the fluid that is created in the volume. How the fluid could ever be created? I always understood that through electrostatics. Let's suppose there is electric charge in some volume V. It has some spatial density, which is often specified as ρ(x,y,z). The charge generates electric field. Then, the total flow of electric field vector Ē ("the fluid" is electric field here) through some area S enclosing that volume equals equals the charge in the volume. So, some of our electric field "fluid" might enter the volume through our chosen surface, some might leave leave, but the amount of total Ē leaving the volume is the amount of that entering plus the amount of the charge inside (because the charge "creates" our "fluid"). Triple integral of the ρ(x,y,z) around all the volume (the total charge) equals the (double surface) intergral of the flow of the vector Ē, which is (Ē dS), around all surface (the total flow). The elementary flow here is dot product of electric field and a elementary surface element, which is the vector pointing outside of the volume, perpendicular to the surface in that point. This was the statement of Coloumb law in the integral form. There is also a differential form of the same law, which is: div Ē = 4π ρ. (4π here stands for a unit sphere surface area). In words: the divergence of the vector Ē in the some point is the amount of the charge in that point. To move from one form to another you use, surprise, the theorem from the video. This is, by the way, one of the equations of the Maxwell's system, the basis of the classical electrodynamics. UPD: what you wrote is analogous to the magnetic field. There are no magnetic charges (monopoles), so the amount of flow of magnetic field entering some area equals the amount that is leaving. In total, the flow of magnetic field around the complete surface is zero. This was your formulation, the integral form; in the differential form this is simple div B = 0 (the density of the magnetic charge is zero, there are no charges). The (double surface) integral of the magnetic field flow (B dS) around complete surface is zero. Again, vector flow is dot product of (ā S), where S is a surface element, as a vector perpendicular to the surface, pointing outside of the volume. The elementary flow in the point is (ā dS), you sum that around all the surface. And this is another equation of Maxwell's system :)

[@nikitakipriyanov7260]

In Russia we know this theorem as Ostrogradsky-Gauss theorem. For me, this better serves to explain what divergence is, rather than to "explain a theorem".

[@RealLifeKyurem]

At 6:34, the divergence is 0, since the flux going in the circle/surface is equal to the flux going out the circle/surface. So ∇ · F = 0, not ∇ · F > 0.

[@TheViolaBuddy]

That confused me for a moment, too, but the difference here is that the magnitude (color) of the vector field changes as you go across horizontally, which makes the divergence nonzero.

[@vcubingx]

^

[@RealLifeKyurem]

@@vcubingx Whoops, my bad. I wasn't paying much attention :p

[@ThomasHPuzia]

@12:16 it should read nabla dot F on the right

[@CR-by4ky]

I need to subscribe

[@douglasstrother6584]

Check out "Vector Calculus" ~ Marsden & Tromba www.macmillanlearning.com/college/us/product/Vector-Calculus/p/1429215089 We used the Second Edition when I took this course from Tony Tromba at UC Santa Cruz in the early 80s; Chapter 7 "Vector Analysis" has a section on "Applications to Physics and Differential Equations" which gives a detailed presentation on constructing Green Functions as solutions to boundary-value problems. The current 6th Edition has a different layout.

[@MrJdcirbo]

vcubingx: to get normal vector you take the tangential vector and rotate it by 90°. Cross product: Am I a joke to you?

[@dlmacbr]

it should be the div F (diverence of vector field F) in the triple (volume) integration instead of the rot F (rotational of vector field F), Thus, divergence theorem. Otherwise, great video.

[@NyaloinhomAcholMorwel-oh2vr]

Given that

[@gaaraofddarkness]

10:35 12:22 in one its divergence, another its curls?

[@anilsharma-ev2my]

Any app over it ?

[@danny.math-tutor]

מעניין

[@ppugalia9000]

Mistake at 12:20 Showing curl instead of divergence on right side

[@derfelix54]

On 5:48 "2D Divergence Theorem" shouldn't it be cross Product and not dot product?

[@vcubingx]

Nah it should be dot product

[@hybmnzz2658]

You may be thinking of Green's theorem. In that case the integral over a closed curve is indeed the curl (cross product) over the area. The difference is subtle: In the cross product case you are integrating a vector field over differential area vectors. In the case in this video you are integrating scalar values over differential patches of area. Also in this video you are not exactly integrating over the path C but integrating it like flux; you are integrating the normal vectors to the curve C.

[@InfinityLRDL]

Do you go to VCU?

[@monishreddy1797]

I'm confused at one point..My lecturer told the flux formula as integral of {F.dS} over the surface. Now after watching this video I interpreted it should be {F.n dS} (!?) . So, are n.dS and dS vector the same? No right, I am pretty sure the n.dS represents normal vector and dS vector is more likely to be a positional vector/tangential vector!.. Which one should I consider in the Divergence formula..n.dS or dS?

[@vcubingx]

They're the same! tutorial.math.lamar.edu/classes/calciii/surfintvectorfield.aspx here's an article to help you out

[@matthewjames7513]

at 12:31 you write p/epsilon_0 and then q/epsilon_0 on the next line. I'm guessing that's a typo?? :O

[@trihasta4229]

Advance Calculus Murray R Spiegel

[@prathameshsirmalla8324]

Pretty cool video ! Needs a lil bit of work on the explanation, but otherwise its great.

[@douglasstrother6584]

"Divergence and curl: The language of Maxwell's equations, fluid flow, and more" ~ 3Blue1Brown kzbin.info/www/bejne/qHObZHemd6-Eqac Grant Sanderson does some nice graphics as well.

[@ripsad1847]

Why am I even going to the lectures, if I can just learn it visually from home?

[@douglasstrother6584]

Videos like this remind me to visualize like Michael Faraday and crunch analysis like James Maxwell.

[@agrajyadav2951]

hey isnt the example flux wrong?

[@robmarks6800]

At 8:43 what does |r| represent?

[@vcubingx]

A tiny piece of area (one of those red squares)

[@bon12121]

GAMMA FUNCTION VIDEO PLEASE

[@agrajyadav2951]

Hey man! The video was great, and the animations were awesome! But, you didnt elaborate too much, and sort of over-referenced greens theorem video...

[@govamurali2309]

Please do z transform

[@gaaraofddarkness]

5:30 i got answer as pi, not 2pi

[@dprx1066]

Are you taking MVC right now, and this is how you study?

[@vcubingx]

Not really, this isn't how I study. I make the videos because I enjoy making them. Although, yes I do take multivariable calculus rn. Our course is still doing double integrals rn.

[@AbhishekKumar-jg7gq]

I think he has stutter in his accent it becomes difficult for me to understand but overall he is doing great 👍

[@agrajyadav2951]

12:15 bruh

[@agustinsaenzanile1900]

You say "The divergence is a better aproximattion of the flux integral of the curve as the curve gets smaller and smaller" Why? How would you proof this amazing fact?

[@daddy7973]

I know even more complicated way of explaining that.

[@federicopagano6590]

5:30 the first formula you wrote it can be solved by the green theorem and the answer is pi. This value is a circulation not a flux The formula below (flux 2D)the n vector is perpendicular to the curve and its the radii itself as shown ok the answer is 2pi(notice you didnt take the derivative coz there are 2 different formulas not equal! First one is a circulation and second one its a flux 8:14 that equation is wrong the flux integral(2D) is approximatly the divergence at the poit times the Area arround the point !! As this Area goes to zero 12:18 that equation is wrong my God that should be a diverence not a rotational x!!

[@ericsu4667]

The divergence theorem requires a differentiable vector field but electric field from Coulomb's law diverges at the origin. Consequently, Gauss's flux theorem is not applicable to the divergence of the electric field. sites.google.com/view/physics-news/home/updates

[@douglasstrother6584]

The divergence of the electric field is proportional to the charge density at that point. Coulomb's Law applies for the special case of point charge distributions represented by the Dirac Delta Function mathworld.wolfram.com/DeltaFunction.html which should be thought of as a limit of spikey functions. In general, a charge distribution can be decomposed into a set of multipoles: monopole, dipole, quadrapole, etc. en.wikibooks.org/wiki/Mathematical_Methods_of_Physics/The_multipole_expansion There are comparable generalizations for current distributions and magnetic fields. Check out "Classical Electrodynamics" by J.D. Jackson for a ton of applied mathematics in the context of Electromagnetism. Get a used 2nd Edition.

[@ericsu4667]

The divergence of any function following inverse square law is equal to zero. This is a mathematical identity. You should be able to verify it your self in any coordinate. It has nothing to do with mass, charge, or any physical quantity. It is pure mathematics.

[@MR-kk5bf]

You ain't a good teacher and I found lots of flaws on explanation

[@GeorgePiskopanis]

Dude, if you can't talk, use a digital narrator.