Gauss's Divergence Theorem

Gauss's Divergence theorem is one of the most powerful tools in all of mathematical physics. It is the primary building block of how we derive conservation laws from physics and translate them into partial differential equations.
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This video was produced at the University of Washington
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0:00 Introduction & Overview
8:55 Why Gauss's Theorem is True
14:38 Gauss's Theorem for PDEs: Mass Conservation
24:11 Recap

[@buttforce4208]

This channel has single-handedly rekindled my interest in math. Absolutely love it. Thanks so much for making these videos!

[@Eigensteve]

Awesome, so nice to hear!

[@theverner]

I am so much amazed how excited you are teaching this theorem. Wished to have teachers like you at uni too.

[@francescogiuliano9563]

> I see a new video has been posted >> I put "like" >>> I watch the video

[@AllanMontoyaVillegas]

For the Respect's sake!

[@johnalley8397]

Outstanding lecture, professor. Defining first in words, providing an intuition and then releasing the math! Shock and Awe. Anyone can deliver the symbols. Gifted educators deliver intuition and genuine understanding.

[@Eigensteve]

I'm so grateful to hear you like it!!

[@c.l.6456]

@@Eigensteve such a blessing to have your intuitive explanations. Even with the lecture notes from Oxford Uni, the systematic proofs and equations were insufficient for a student to fully appreciate the utility of the material. Every university (even the top ones) shall learn from your pedagogy sir.

[@joonasmakinen4807]

At 20:25 Steve oversimplified by moving the time derivative inside the volume integral just like that. It can only be done if the volume being integrated wont change over time, invalid assumption in fluid dynamics. Taking this into account leads us to another beautiful theorem called Reynolds Transport Theorem (RTT), which interestingly naturally leads to the right-hand-side on Steve’s board (if F is a velocity field).

[@CBMM_]

I'm so grateful for living at this time, so I can learn this theorem in 25 minutes.

[@StratosFair]

These lectures are fantastic, thank you for taking the time to produce and share them for free.

[@Eigensteve]

You are so very welcome!

[@biogrisha4433]

Every math teacher feels it is his duty to say that he is not a fancy artist or so when he draws some kind of diagram

[@wp4297]

Great video. Excellent for showing the intuition of the volume built as a union of smaller volumes, for divergence theorem. Just few comments: - Divergence Thm has some assumptions. Broadly speaking, everything inside of the statement of the theorem must be meaningful, as an example if you write a divergence of F, that function F must be regular enough (differentiable) for the divergence to exist; REMEMBER that PDEs of Physics translate into "regular enough local regions" all the general Principles of Physics holding for all the physical systems, differentiable or not; - Mass equation example: > when you put time derivative inside the volume integral, you're doing right only if that volume does not change in time (i.e. you're implicitly considering a fixed control volume, otherwise that manipulation is WRONG). Anyway the conclusion you reach is right, but your derivation only holds for a steady volume for integration. Integral laws and then differential laws can be easily translated from the very statement of the Physical Principles if you firs consider Lagrangian volumes (i.e. those volumes moving with the continuum), and then transformed to fixed control volumes (some math required) > maybe I missed that, but the physical meaning of F is not explicit here. In order to have the right physical dimension, it must have the dimension of a velocity. Indeed, it is the velocity field of the continuum under investigation in most of cases (few times it's a bit more tricky, i.e. in diffusion problems that vector field contains both a local average - averaged on the species velocity, maybe - velocity contribution and a drift velocity, likely due to gradient in the specie concentration, see Fick's law for diffusion). Keep going. I'm very curious how this series evolves.

[@Eigensteve]

Thanks!!

[@hendriklohad631]

It is thanksgiving eve and I am learning some quality vector calc from these lectures. They are so greatly made!! Every detail is explained and is wrapped so elegantly together. A joy to watch.

[@HarrydeBont]

Watching this video, I remembered being totally fascinated (for the first time in my life) by theoretical electrical theory. Thanks for the passion you bring presenting this math.

[@yashwanthcalidas6031]

Before this I had no idea fluid mechanics can be so intuitive and interesting. Great work sir, Thank you so much for your effort.

[@user-hw7iy7ls6y]

Such great explanations and a highly quality channel. Great for building a strong intuition of concepts rarely explained in a straightforward manner.

[@longyang8964]

The lecture is very well organized and superbly delivered!

[@fredericoamigo]

Such a good video! Love your teaching style! Keep up the good work, I’m such a fan of it!

[@bouipozz]

Thank you so much Dr Brunton for making such high quality content freely available. I have recently left work to return to uni and without this channel I would be seriously underprepared. Your ability to take a difficult subject and make it seem almost like common sense in incredible. I hope that my lecturers will have half of your passion and skill.

[@pavangaonkardonigadde]

You have no idea how much gratitude i have towards you... Thank you soo much for uploading this...

[@silverbullet007]

This is the best explanation of the Gauss's Divergence theorem I have heard till now. ☺ Thanks, Steve.

[@seekingtruth9304]

What a great lecture!! I am truly looking forward to more videos from Professor Brunton.

[@majorfallacy5926]

just came here to say that while i'm currently not watching most of your videos as you upload them, i'm still very thankful because i'm 99% certain that i'll need them again at some point in the future

[@c.l.6456]

I'm so lucky to have discoverd your channel while self-learning multi-variable calc! Abosolutely recommend to anyone (even non-math majors who hasn't touched calculus in 4 years).

[@lhliu5264]

Very excited to watch every update on this series！

[@Eigensteve]

Awesome, I'm excited too!

[@bendavis2234]

What an outstanding explanation! I'm so surprised that my Calc textbook left out the Mass Continuity Equation when going over the Divergence Theorem. It's really motivating to hear how powerful this equation is in applied math and physics. I love hearing the real-world applications.

[@curtpiazza1688]

Your lectures are so inspiring! 😊

[@ChristAliveForevermore]

Great explanation! Gauss truly was a super genius for figuring this out.

[@JoaoLima-pq1hm]

Beautiful content, professor. Brilliant channel, thank you.

[@chilivaryvishal6037]

Great visualized explanation of Gauss's Divergence theorem

[@iniyanmdr5504]

This is a treasure worth 1M views. I learnt this in my college days. Understood 15 years later.

[@VladimirDjokic]

woow!cool explanation ! integral sum of dV finally make sense! Thanks!

[@briandwi2504]

That was really interesting. Thanks for such a fascinating lecture.

[@Martin-iw1ll]

Great lecture again, you are treasure for mankind! I find the most interesting with the mass continuity equation is the physical interpretation that we can derive for div (F) by rewriting the continuity equation in terms of material derivatives

[@INFINITE_VOID_11]

Thank you sir...I needed these explanations!! Respect💯

[@pavangaonkardonigadde]

ನಿಮಗೆ ಅನಂತ ಧನ್ಯವಾದಗಳು... ಗುರುಗಳೇ ಇದೊಂದು ಅದ್ಭುತ ಪ್ರದರ್ಶನ

[@VictorJunyiWang]

非常感谢！这个讲解让人印象深刻，过目难忘！这是我见过的向量微积分原理最好的讲解，再次感谢

[@antesikiric3782]

Brilliant explanation , thank you

[@fernandojimenezmotte2024]

Great , very neat, clear and didactical explanation Professor Steve of Gauss´s Divergence Theorem. I really enjoy it !! I am following You on the networks and also in the University of Washigton UW Internet Sites. I am thinking about going back to Graduate School [second round from 58 to 100 !!] and besides the quality of the university I believe the Advisor is crucial. Not only that He has abroad knowledge and background on the subject matters but also his ability to motivate. Your lessons are highly motivational.

[@sansh72]

really broadened my mind . thanks !

[@bumeegabentharavithana2572]

wonderful explanation thank you.

[@mariovrpereira]

Remembering in such a good way...thank you so much

[@felipedepine]

Excellent lecture, thank you for posting!

[@Eigensteve]

You are welcome!

[@DJ-yj1vg]

Thats a brilliant intuitive explanation

[@user-vg7zv5us5r]

23:57 dro/dt - Density can't become smaller or be infinitesimally small - it's a constant property of a mater even idealized for the sake to keep the conversation theoretical and abstract.

[@marcelb6214]

Thank you so much! I don't even know what to say. You did an amazing job explaining this!

[@vahiddanesh4661]

I was wondering how good can be someone in explaining complex subjects in an easy way.

[@nointerference11]

I like how you write in mirror texts along with teaching.

[@murillonetoo]

Great lecture, professor! As always, very enlightening!

[@Eigensteve]

Thank you so much!

[@electrolove9538]

@@Eigensteve I loved 💖💖💖the example of all volume integrals cancelling except the outer skin. Wish I had this visualization in class. I always used Gauss's thm as simply a mathematical tool. 1. What I am wondering is did they call it "divergence" before Gauss's thm? Or when Gauss proved it did they coin the term "divergence". 2. Gauss doesn't get as much recognition in statistics even though it's called a Gaussian distribution. For example, if I Google 'who is the father of statistics' it says Fisher, not Gauss. Why is this? Thank you Steve!

[@Eigensteve]

@@electrolove9538 Thank you -- that is a great question. I don't know the history of this, but I'll look into it!

[@theonlinezone6904]

this video is helping me a lot, thanks

[@user-lq7ks3ei8j]

Thanks for this excellent lecture , I pray For you to be happy and long live.

[@bradfoster5643]

I am sooooooo grateful for this video!!!

[@varunahlawat169]

I can't thank you enough, this playlist alone made taking the first step to start serious mathematics and physics for me so damn easy!!

[@Eigensteve]

I'm glad to hear it! Thanks for watching :)

[@aaronlopes5256]

Awesome explanation!🙏

[@Eigensteve]

Thank you -- glad you liked it!!

[@willyh.r.1216]

MVP....Most Valuable Professor of the year.

[@AudioScript152]

Dr. Brunton, you used "Gauss's Divergence Theorem"(GDT) to derive the conservation of mass(CVM). Could you show how GDT relates to the "Reynolds Transport Theorem"(RTT) & also derive the CVM using RTT? Thank you! Dr. Brunton, for taking the time to teach all of us.

[@JP-wl2nz]

Awesome lecture sir...🙏

[@vladmartian]

Can't wait to watch the next one.

[@user-zv7dl6bp7v]

amaaaaazing

[@himanshuraj1482]

I am a fluid dynamics researcher at IIT Bombay. I want to do my Ph.D. at WashU. Now I am modeling fluid vortex around a Mobius Theorem.

[@JohnSmith-qp4bt]

Makes a complicated subject clear and attractive.

[@jonahansen]

There is something I seriously don't see starting at 10:00 to about 13:30 when talking about the little boxes filling the volume. The claim is that with an interior filled with boxes with positive divergences, only the boxes on the surface contribute to the surface integral, with the internal ones cancelling along their apposing surfaces with other adjacent boxes. This can't be true; here are two lines of reasoning which, for the sake of the first argument assume a constant, positive, divergence throughout the volume. One: Compare two cases that have the same surface, but different volumes. The integral of the divergence over the volume is proportional to the volume, but the surface integral of the flux would not increase to match if there was a cancellation of the field F at the apposing adjacent surfaces of the interior boxes. Two: Since the integrals are linear, the surface integral flux for a volume with a single box with positive divergence must be half of that with two interior boxes with the same positive divergence. But if they are adjacent, and something cancelled, it wouldn't be. I'm thinking of this a la Gauss's law for the electrical field at the surface as proportional to the enclosed charge. Am I missing something??

[@felemezhasturk559]

A good refresher. Could you also show contuinty eqn for deformable control volume (i.e. V=V(t))?

[@teatea5528]

He has 4k quality lectures, my eyes are so comfortable watching it.

[@thecodegobbler2179]

@stevebrunton Is the gauss’s divergence theorem also relevant to transitions of chemical states? (Water turning to ice, dry ice to co2… etc? As it does flux mass to and from the volume through the surface area.)

[@sekus]

I'm not disrespecting my professor, but I wish I had you teaching vector calculus concepts to me. I enjoyed your machine learning series. I'm looking forward to your next videos

[@ANTGPRO]

Perfect, please more calculus lectures.

[@wp4297]

What are you really interested in? Calculus is quite wide topics. Are you interested in some application/example with differential operators and evaluation of integrals? Three-dimensional space is enough for you? Are you interested in time derivative of integrals over time-dependent domains?...

[@paul_gradenwitz]

One issue with this theorem is that we have to define how much flows through the surface at the same time. That mans that we integrate over the surface for one moment in time. After completion we can see how that result evolves over time, but we can't use the left part of the surface values of one time moment and add that to the right part of the surface for a later time moment. But this means that we have to know what simultaneity means in that case. Thanks!

[@claytonestey767]

Hi Dr. Brunton. As obscure as this seems is it scientifically useful to somehow perturb Guass's Divergence Theorem with an arbitrary differentiable function to see what would happen if non-conservation were to ever take place, and the consequence on the derived PDE?

[@pengyuanding4228]

Hi Steve, I have to say the tiny boxes analogy is a bit confusing. Because when you integrate over the volumn, you are integrate the divergence, so at each point the integrand is positive since each point is a source, which does not reflect any 'cancellation'. (If it does, then at the points in central region the integrand should become 0 since they are 'cancelled'.) Whilist the 'cancellation' happens between the vector field F itself. So it might not be the right intuition for the theorem.

[@AmentasOnIce]

I found this confusing as well. You can't cancel a bunch of sources.

[@crackyflipside]

Fantastic lectures. Please increase microphone volume level next time.

[@patheron7812]

I'm confused regarding a basic idea. If there are no sources in the volume, I don't understand why the flux as defined isn't always equal to zero. Where the field enters the volume the flux contribution F dot n would be negative and where the field exits the volume F dot n would be positive. So, the net flux would be zero.

[@English1108]

have you figured out the answer to your question here? because I'm wondering the exact same thing

[@shahabtariq2479]

Sir love you I am immensely thankful to you you are great My teacher didn't give me any concept any amount of concept of that topic❤ Love from Pakistan 🇵🇰🇵🇰 may Allah bless you ❤

[@pat2715]

gotta love this man

[@KalebAklilu]

Some thing I didn't understand is why we are assuming that each tiny boxes would work identically either as a source or sink. Isn't the divergence induced by the vector filed on the surface going to change from section to section

[@Divergent_Integral]

Thank you for posting this awesome lecture! I was wondering, though, what hardware and/or software did you use to animate your handwriting?

[@RuthBingham]

I think he is standing behind a pane of glass and literally writing backwards.

[@francescoghizzo]

@@RuthBingham I think that it's actually simpler than this. He's writing normally and then the video is inverted

[@anonymousowl5240]

Well, I wasn't expecting to have my entire outlook on the world around me changed today but it happened.

[@ahammedafzal7797]

Hi, I think there is some problem with the "explanation' of canceling off between fluxes in the video. The cancelling off takes place because the flux through an interfacial control surface will have different signs when taken in two adjacent control volumes.Anyone thought the same?

[@jonahansen]

Ahammed - I also have a problem with the cancellation, and just posted a comment about it today (Sep 24, 2022), and after doing it I thought I should see if anyone else saw this too. I'm not sure if we have the same issue, but it sounds similar...

[@manfredbogner9799]

very good

[@pierrot-baptistelemee-joli820]

At 21:35 or so, does anyone know how we formally justify changing the total derivative in respect to time with a partial derivative with respect to time when we move the derivative operator inside the triple integral? Also, in this particular exemple, I get the feeling that this could only be true if the volume is not a quantity that depends on time, but I know that conservation on mass is always true and does not rely on such assumptions... How can I convince myself that this is true no matter what happens to the volume? And this is closely related to my last question : what does happen if we consider that the volume does depend on time? Can we still switch the integral with the total derivative? Thank you Steve Brunton for these videos! It's been a long time since I saw these topics (if ever for some of them!) and I really appreciate your enthusiasm and the quality of your work :)

[@user-vg7zv5us5r]

26:39 Rho can't be continuously varying even if that term adheres to us looking how mass enters end exits particular volume. Maybe I collide density with hardness together, yet still...

[@Shrira123]

Amazing video, sir. Any1..Correct me if I'm wrong : He was able to interchange the order of derivative and integral because of the following>>> derivative of an integral= integral of a derivative. Just thought it might help some1 like me cause i was wondering how twas possible for a couple of minutes.

[@Bruh-vp6qf]

Thank you

[@brighttakyi9037]

Nice video but I do not understand the concept of the divergence in the volume, how they cancel out and how there was a surface without a divergence

[@gabrielbelmont8691]

Dear Sir, I cannot understand the part at 12:35 sec. If there is a perimeter which has continuous outward emerging arrows, then what about the arrows that are in +z direction (emerging in 3D), as flux F is coming/flowing out (not expanding).

[@bringstarlysyierlait4164]

Sir explain why divergence of electric field line is positive,,and negative,plse

[@adrianp1596]

excellent!

[@d7ffab979]

I love your content. Your followers are brainy people. They love ur style. They are bored by Netflix, autodidacts. I love your lectures about Compressed Sensing.

[@VinayakPathak-xc6kp]

Hi Steve [URGENT!] Shouldn’t F be the velocity field V in this case, I think that is intuitive from dimensional matching and have also seen it written in the standard text book instead of F. Please correct me if I am wrong. Otherwise awesome lecture. Thanks and Regards Vinayak

[@JHS-gu4lw]

This is so amazing!! 🥹👍

[@pierreafoutou7368]

Watching this while waiting for my next flight. Thanks

[@giro808]

thank you

[@samirelzein1978]

Being slow to get it Will watch again A 3D simulation would be perfect for full visibility Cant thank you enough for the giant efforts

[@vesselofmercy6988]

Is this guy writing backwards, or is there some kind of postprocessing effect that makes it have the correct orientation to the viewer? Great video btw, takes something complicated and makes it pretty intuitive.

[@anshik567]

Sir you put d/dt inside integral but if volume is changing with time then also can we put d/dt inside integral

[@lunaleonem3378]

2:38 The generalized Stoke's theorem would like a talk.

[@edmald1978]

In this equation I see that units do not match: ∫∫_S ρF • n dS = ∫∫∫_V div(ρF ) dv on the LHS: [kg/m^3][m/s][m^2] = [kg/s] on the RHS: [kg/m^3][m/s][m^3] = [kg m / s] Please let me know what I am missing.

[@mariuspopescu7543]

the name of the theorem is Gaus-Green Theorem

[@calebgeballe2724]

Some facts about Gauss: Gauss could divide by 0 Gauss squared the circle He knew the last digit of pi He could construct lines with a compass and circles with a straight edge

[@Eigensteve]

Love this comment!

[@sanjaykrish8719]

Very intelligent Gauss but never shared the rationale or the thought process 😒

[@anteperica3662]

Could you please explain how total derivative d rho/dt suddenly becomes partial derivative? Thanks.

[@Shrira123]

Because it might be a function of space too, along with just time as assumed previously, i believe

[@brownriceprod]

i can't do the marker squeaking... but props for learning to write backwards

[@someonestolemyname]

There are other divergence theorems?

[@lioneloddo]

So, it means that, thanks to this equivalence between what happend in a volume and its surface, we can intuitively feel what is the conept of continuity. It's not the coninuity of the mathematcians, but rather of the physicians. What is the continuity ? To check at every scale and at every shape, this equivalence. If this rule is true then it means that the medium is continous. It's very surprising that for knowing something locally, we need to look at globally.

[@sib5th]

“physicians”? “physicists” sound more likely!

[@lioneloddo]

@@sib5th Sorry, I'm french ... ;)