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
@adrianp1596
2 жыл бұрын
excellent!
@theverner
2 жыл бұрын
I am so much amazed how excited you are teaching this theorem. Wished to have teachers like you at uni too.
@buttforce4208
2 жыл бұрын
This channel has single-handedly rekindled my interest in math. Absolutely love it. Thanks so much for making these videos!
@Eigensteve
2 жыл бұрын
Awesome, so nice to hear!
@joonasmakinen4807
2 жыл бұрын
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).
@francescogiuliano9563
2 жыл бұрын
> I see a new video has been posted >> I put "like" >>> I watch the video
@AllanMontoyaVillegas
5 ай бұрын
For the Respect's sake!
@StratosFair
2 жыл бұрын
These lectures are fantastic, thank you for taking the time to produce and share them for free.
@Eigensteve
2 жыл бұрын
You are so very welcome!
@jonahansen
2 жыл бұрын
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??
@ahammedafzal7797
2 жыл бұрын
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
2 жыл бұрын
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...
@patheron7812
2 жыл бұрын
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
Жыл бұрын
have you figured out the answer to your question here? because I'm wondering the exact same thing
@wp4297
2 жыл бұрын
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
2 жыл бұрын
Thanks!!
@edmald1978
Жыл бұрын
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.
@brighttakyi9037
4 ай бұрын
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
@prodbyryshy
Жыл бұрын
its funny cus i saw this video and was like hmm this sounds interesting then i realized i already knew this from work i did like 4 years ago
@bringstarlysyierlait4164
Жыл бұрын
Sir explain why divergence of electric field line is positive,,and negative,plse
@johnalley8397
2 жыл бұрын
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
2 жыл бұрын
I'm so grateful to hear you like it!!
@c.l.6456
2 жыл бұрын
@@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.
@HarrydeBont
2 жыл бұрын
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.
@pengyuanding4228
Жыл бұрын
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
11 ай бұрын
I found this confusing as well. You can't cancel a bunch of sources.
@VladimirDjokic
Жыл бұрын
woow!cool explanation ! integral sum of dV finally make sense! Thanks!
@theawesomenerd7122
9 күн бұрын
"If it's Gauss' Divergence Theorem, it's probably the best divergence theorem."
@INFINITE_VOID_11
Жыл бұрын
Thank you sir...I needed these explanations!! Respect💯
@CBMM_
6 ай бұрын
I'm so grateful for living at this time, so I can learn this theorem in 25 minutes.
@crackyflipside
2 жыл бұрын
Fantastic lectures. Please increase microphone volume level next time.
@user-vg7zv5us5r
Жыл бұрын
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.
@johngardner4655
Ай бұрын
11:15 Why do divergences in neighboring cells cancel out? I can see why fluxes would cancel, but isn't that different from divergence? Wouldn't neighboring divergences work in the same direction?
@Martin-iw1ll
Жыл бұрын
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
@VinayakPathak-xc6kp
6 ай бұрын
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
@VinayakPathak-xc6kp
6 ай бұрын
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
@VinayakPathak-xc6kp
6 ай бұрын
Hi Steve [URGENT!] Shouldn’t F be the velocity field vector 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
@muhammadfaizanalibutt4602
Ай бұрын
In terns of computation, which integral is more effective. The surface or the volume integral?
@alexjimenez5376
8 күн бұрын
At mis 81years l'm fascinated by grate young teachers
@murillonetoo
2 жыл бұрын
Great lecture, professor! As always, very enlightening!
@Eigensteve
2 жыл бұрын
Thank you so much!
@electrolove9538
2 жыл бұрын
@@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
2 жыл бұрын
@@electrolove9538 Thank you -- that is a great question. I don't know the history of this, but I'll look into it!
@thecodegobbler2179
6 ай бұрын
@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.)
@hendriklohad631
10 ай бұрын
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.
@yashwanthcalidas6031
2 жыл бұрын
Before this I had no idea fluid mechanics can be so intuitive and interesting. Great work sir, Thank you so much for your effort.
@nicholasleclerc1583
Жыл бұрын
Wait, does…. does this guy know how to *fluently write in mirrored English ?* Or does he just write & talk to his board like if it wasn’t there & an audience was in its place, with the board being able to let invisible/special light to pass through for a camera behind it to see the guy, & his inscriptions, and then mirror it around so as to make the text visible ? Does he just add the little sounds of a marker squealing on a whiteboard just for the effect ? And he‘s just writing in the air (and is really good at it) ? Does the marker indicate its position when it crosses a certain invisible plan ? Can he see the reflexion of a light source traversing that plane to indicate he’s putting his pen in the right plane of ambient space ? Or does he just know by instinct & habit, if it’s just not that hard in the first place ? Or is it just a very clean “glassboard” ? [* Edit made 2 minutes later * :] Oh, ok, it’s both a “glass-board” (i.e. “mirroboard”) & post-production horizontal flip
@felipedepine
2 жыл бұрын
Excellent lecture, thank you for posting!
@Eigensteve
2 жыл бұрын
You are welcome!
@KalebAklilu
5 ай бұрын
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
@anonymousowl5240
4 ай бұрын
Well, I wasn't expecting to have my entire outlook on the world around me changed today but it happened.
@gerardsagliocca6292
Жыл бұрын
You are confusing me because your are constantly using Volume and Surface area interchangeably ! Your quickly drawn sketch on the board indicates to me that it is a typical Surface, typically developed in calc 3, over a domain Region in a typical x - y plane. Is this the Volume you are often referring to ?? Then you bring in the vector field , F hat,. Is this generic vector field passing over the Surface or actually through the porous surface ?? Often teachers don't state how a field passes through a Surface. If a If a vector field passes through a Surface, how is this done ? Is it radiation ? I would appreciate a response.
@user-vg7zv5us5r
Жыл бұрын
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...
@ProfFeinman
Жыл бұрын
If this were done with a physical example it would be easier to visualize. Nothing would have to be changed. The explanations are very clear but if they referred to real physical variables. That would be the dream of scientists in, say biochemistry, who love this and are watching for entertainment and need to have the physics or physical chemistry reviewed at the same time as the mathematics. Is that too much to ask? (Joke).
@dharmik2
3 ай бұрын
what id really mass energy and momentum created then how will gauss divergence theorem work?
@zack_120
Жыл бұрын
Rendering so beautiful to replace some of the MIT's teachers on its edX online teaching platform
@돌구름-t8t
Жыл бұрын
Thanks for this excellent lecture , I pray For you to be happy and long live.
@brownriceprod
2 жыл бұрын
i can't do the marker squeaking... but props for learning to write backwards
@bendavis2234
2 жыл бұрын
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.
@anasfhdelalfndi1698
8 ай бұрын
Can the Arabic language be included in the translation options please!
@fredericoamigo
2 жыл бұрын
Such a good video! Love your teaching style! Keep up the good work, I’m such a fan of it!
@calebgeballe2724
2 жыл бұрын
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
2 жыл бұрын
Love this comment!
@sanjaykrish8719
2 жыл бұрын
Very intelligent Gauss but never shared the rationale or the thought process 😒
@lunaleonem3378
9 ай бұрын
2:38 The generalized Stoke's theorem would like a talk.
@aaronlopes5256
2 жыл бұрын
Awesome explanation!🙏
@Eigensteve
2 жыл бұрын
Thank you -- glad you liked it!!
@AshishPatel-yq4xc
2 жыл бұрын
The explanation of the small volumes canceling each other out is not very clear to me- will look it up in books
@HD141937
2 жыл бұрын
Suppose that you have 2 volumes in a vector field, and the volumes share a common interface. Then the flux through this interface will have a positive contribution to the flux through the closed surface around the first volume, and a negative contribution for the second volume. Then the flux through the closed surface around the two volumes combined will be the sum of the two fluxes. The contribution of the "interface flux" to the "combined flux" is cancelled out. This idea can be extended to any number of volumes of any size and shape.
@AshishPatel-yq4xc
2 жыл бұрын
@@HD141937 this explanation makes sense. Thankyou !
@AmentasOnIce
11 ай бұрын
@@HD141937, you nailed it! I really love Prof. Bruton's lectures on this channel, but the picture he drew is not a good representation of this elegant idea. Instead of having two arrows hit each box wall, it might be better to imagine a single arrow going through the wall.
@seekingtruth9304
2 жыл бұрын
What a great lecture!! I am truly looking forward to more videos from Professor Brunton.
@bumeegabentharavithana2572
4 ай бұрын
wonderful explanation thank you.
@lhliu5264
2 жыл бұрын
Very excited to watch every update on this series!
@Eigensteve
2 жыл бұрын
Awesome, I'm excited too!
@ChristAliveForevermore
2 жыл бұрын
Great explanation! Gauss truly was a super genius for figuring this out.
@_1708atem
26 күн бұрын
How is he writing backwards ❔
@MauriC-i7i
10 ай бұрын
amaaaaazing
@lioneloddo
2 жыл бұрын
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
2 жыл бұрын
“physicians”? “physicists” sound more likely!
@lioneloddo
2 жыл бұрын
@@sib5th Sorry, I'm french ... ;)
@himanshuraj1482
2 жыл бұрын
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.
@AudioScript152
Жыл бұрын
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.
@Shrira123
Жыл бұрын
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.
@shahabtariq2479
Жыл бұрын
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 ❤
@d7ffab979
2 жыл бұрын
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.
@mariuspopescu7543
4 ай бұрын
the name of the theorem is Gaus-Green Theorem
@antesikiric3782
2 жыл бұрын
Brilliant explanation , thank you
@gabrielbelmont8691
2 жыл бұрын
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).
@PTGaonkar
2 жыл бұрын
ನಿಮಗೆ ಅನಂತ ಧನ್ಯವಾದಗಳು... ಗುರುಗಳೇ ಇದೊಂದು ಅದ್ಭುತ ಪ್ರದರ್ಶನ
@sekus
2 жыл бұрын
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
@AceBlockey
7 ай бұрын
Such great explanations and a highly quality channel. Great for building a strong intuition of concepts rarely explained in a straightforward manner.
@curtpiazza1688
6 ай бұрын
Your lectures are so inspiring! 😊
@pierrot-baptistelemee-joli820
2 жыл бұрын
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 :)
@claytonestey767
2 жыл бұрын
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?
@marcelb6214
Жыл бұрын
Thank you so much! I don't even know what to say. You did an amazing job explaining this!
@theonlinezone6904
9 ай бұрын
this video is helping me a lot, thanks
@samirelzein1978
2 жыл бұрын
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
2 жыл бұрын
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.
@chilivaryvishal6037
Жыл бұрын
Great visualized explanation of Gauss's Divergence theorem
@iniyanmdr5504
Жыл бұрын
This is a treasure worth 1M views. I learnt this in my college days. Understood 15 years later.
@fernandojimenezmotte2024
2 жыл бұрын
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.
@immortaljanus
2 жыл бұрын
Sure, you're smart. But are you smart enough to casually write complex math in mirror image?
@vahiddanesh4661
8 ай бұрын
I was wondering how good can be someone in explaining complex subjects in an easy way.
@JoaoLima-pq1hm
8 ай бұрын
Beautiful content, professor. Brilliant channel, thank you.
@anshik567
2 жыл бұрын
Sir you put d/dt inside integral but if volume is changing with time then also can we put d/dt inside integral
@sanjaykrish8719
2 жыл бұрын
I could feel the flux emanating from this person.. Beautiful
@lavieestlenfer
2 жыл бұрын
This is how AMATH 501 should have been taught.
@teatea5528
2 жыл бұрын
He has 4k quality lectures, my eyes are so comfortable watching it.
@paul_gradenwitz
2 жыл бұрын
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!
@felemezhasturk559
2 жыл бұрын
A good refresher. Could you also show contuinty eqn for deformable control volume (i.e. V=V(t))?
@VictorJunyiWang
2 жыл бұрын
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
@mariovrpereira
2 жыл бұрын
Remembering in such a good way...thank you so much
@pierreafoutou7368
2 жыл бұрын
Watching this while waiting for my next flight. Thanks
@c.l.6456
2 жыл бұрын
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).
@PTGaonkar
2 жыл бұрын
You have no idea how much gratitude i have towards you... Thank you soo much for uploading this...
@nointerference11
8 ай бұрын
I like how you write in mirror texts along with teaching.
@silverbullet007
Жыл бұрын
This is the best explanation of the Gauss's Divergence theorem I have heard till now. ☺ Thanks, Steve.
@katej392
7 ай бұрын
I just realized he is writing mirrored. That's insane!
@majorfallacy5926
2 жыл бұрын
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
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