Gauss Divergence Theorem. Get the DEEPEST Intuition.

[@fufaev-alexander]

More: en.fufaev.org/advanced-maths Books by Alexander Fufaev: 1) Equations of Physics: Solve EVERY Physics Problem en.fufaev.org/physics-equations-book 2) Alexander Fufaev and His Never Ending Story: en.fufaev.org/lifestory

[@tyrian007]

Can you do this for Navier Stokes and RANS turbulence modeling?

[@hossamhafez9826]

​@@tyrian007may it be possible as President of Arab Republic of Egypt has said

[@jobebrian]

The very essence of clarity. As a city bus driver, I must say this is easily the most straightforward explanation of GDT I have ever heard. Many thanks!

[@compegord07]

As I read this, all of the connotations of these words arranged this way is almost absurd in what it says: It starts with the hyperbolic “very essence of clarity.” What do you or anyone else generally assume about someone who is a bus driver? If it isn’t that why is it relevant enough to mention? In addition to above, would we assume that any random or even average bus driver would have heard of, knows or thought about GDT?

[@jobebrian]

@@compegord07 I was just having some fun with some of the other commenters here. Several have said things like, “I’m an Electrical Engineer and this is best explanation …” and “I am a PhD student in Mathematics, and this clearest video …” I really am a bus driver, and although I do enjoy watching videos about math, this was just above my level. I certainly don’t pretend to speak for all bus drivers.

[@pandakso3365]

As a city bus driver you really have time to think

[@jobebrian]

@@pandakso3365 Yep.

[@malefetsanekoalane4549]

You set out to give "the deepest understanding" You achieved this magnicently. Thanks a million.

[@KsmithBlackstar]

You actually explain things. Most of my professors speak as if I should already know the topic I am learning. You are a savior.

[@alphalunamare]

Have Faith in yourself! If you do not understand then ASK! Any Prof who looks down at your question is a failed Prof. We all have to learn, it's just that some prof's get so comfortable that they forget their own troublesome journey to tenure. I came from the wrong side of the tracks and did mathematics and upset a few with my up front, no respect, questioning. Luckily there were enough Profs around to shield me from expulsion for being rude in response to a ridiculous answer. What I am saying is that You have the ability to achieve, don't let that be curtailed by a numpty with a superiority complex.

[@piotr1175]

That is probably the clearest explanation I've ever had in my life. This is truly brilliant 👏

[@tennisfreak312]

In CFD, Gauss divergence theorem is very important for FVM. Thank you so much for this clear and intuitive explanation.

[@petercheng7890]

It is so amazing!!!!!!!!!!!. Unlike the book just teaching us how to put the number into it, it explains what exactly we are doing so we can be more easy to understand and accept the concept. Thank you so much.

[@marcelb6214]

I am just amazed. After 6 years of studying this was the best video I have ever seen.

[@fufaev-alexander]

thank you!!

[@renesperb]

This is really how one should explain the divergence theorem . It plays a central role in partial differential equations and has many applications. Well done !

[@shobanaa6936]

Sir, I am a physics student, please teach the green theorem and stoke theorem ,I am always support for u,thank u for your teaching,thanks a lot 😊😊

[@umbraemilitos]

They are all versions of the same thing: "The Generalized Stoke's Theorem." If you learn that, you can use those and many other examples.

[@savejobar]

Pls tell me what course you are in and what university you study at. In Russia, these theorems are studied in the third semester of education

[@Haveuseenmyjetpack]

Please explain gauge theory.

[@joefuentes2977]

​@@savejobarIn US it's generally called Calculus 3 or Multivariable calculus or vector calculus. The generalized Stokes' Theorem is usually covered in a course called Differential Geometry

[@savejobar]

​@@deusexmachina3091 I learned it on my third semester of undergraduate in the direction of plasmic synthetic

[@WenGao-ys9rw]

Thank you for your explanation! It's the clearest explanation of the divergence theorem I've ever seen. When reading the calculus textbook, it just tell me the theorem and a mathematical proof, but the connection between the divergence theorem and the green theorem is still ambiguous to me. Your animation helps me understand the connection between them! Thank you very much!

[@newsgo1876]

Mathematics is a concise description of what we take for granted.

[@mayonakao2488]

Wish I could like this video twice. I know nothing of the intuition behind Electrodynamics, and yet this made it max’s equations seem as clear as day. Great work!

[@fufaev-alexander]

Glad you liked it!

[@haroonrasheed3844]

The best explanation I have seen yet. Thank you bro 🙏

[@vadiquemyself]

the idea is that it sums up to only what’s on the boundary surface of a volume, because all internal small volume parts dv mutually compensate (as to 0) each other on adjacent neighboring surfaces

[@JakubS]

I was at a taster lecture where a Cambridge professor tried to explain this formula, and I was entirely dumbfounded. But now that I've watched your video, it all makes sense!

[@fufaev-alexander]

Awesome!! :)

[@saeedelghaziri2385]

Amazing explanation, I understood it from the first 30 seconds of you speaking. Quality explanation !

[@joeponsford3503]

absolutely amazing explanation to do in 9 minutes is brilliant

[@manarlab84]

This is a gem - I came across it as a suggestion from KZbin while viewing tutorials about Transformers. The equation in this video gave me a deeper understading of the use of scalar and dot product in understanding the similarties between features and the relationships between features. Thanks a lot!

[@Epoch11]

It's nice when you have someone who can explain things in a way that are intuitive. Please keep up the good work.

[@fufaev-alexander]

I will! Thank you very much for your positive feedback!

[@chuckforke9785]

This is an awesome video. Great intuitive explanation. This helped me understand Divergence theorem for my calc class.

[@ShizhouLuo]

A very explicit and intuitive explanation. Thanks.

[@djredrover]

I am an EE engineer and I must say, this is THE BEST illustration of GDT I have seen, and I have seen MANY! Well done. (Same with your Maxwell Equation video)

[@fredthechamp3475]

You are an Electrical Engineer engineer?

[@djredrover]

@@fredthechamp3475 ahaha I guess I am an EEE! LOL

[@PatrickDonaldson]

Thank you for posting this, it was a great presentation and helped me review this material to help my nephew who is struggling with it just like I did 30 yrs ago. I now understand this (the math) better than ever! The physical concept has been clear long before tackling the formal math, but it's nice to be able to articulate the concept using the language of math.

[@sjonnithors1986]

Oh man! I wish I had this video back in 2012 during my first year in Engineering.

[@jeffreyhowarth7850]

This helps me understand more clearly some of Maxwell's equations. I'm also interested in closed forms. Thank you

[@Integral_of_x_i_987]

So clear and concise.😍😍😍😍

[@alexlo7708]

If this was to be taught in my undergrad electro magnetic field subject , it'd make us much more understanding than we were then.

[@rx5514]

It would be better if you could explain more about divergence, but overall this is a highly undervalued channel.

[@vanshmishra7119]

Makes sense now! Thanks for the visual intuition.

[@Tamir_Karniely]

Amazing video! So much explained and illustrated beautifully and detailedlly. A real deep explain for each component in the writing convention. Keep on the exellent work!

[@manpreetsingh-yh5iu]

Thanks @Universaldenker for this perhaps most intuitive and meticulous explanation of the divergence theorem.

[@trippstreehouse]

Great explanation.

[@1bite-]

the thumbnail does a really good job of relating the divergence integral with the closed loop one i imagine the cubes within the larger cube to represent the infinitesimal points at which you test divergence (it is based off of a derivative after all), and if you just project the surface faces of the infinitesimals to the larger box you get the flux thru the entire surface

[@ThatBigGuyAl]

Hi, just to let you know, before the 2:00 mark, you have da_y listed twice in the vectors (you forgot the z component). Great video though!

[@Francinun]

So much clearer after this, THANKS

[@Zamperla]

This is such a good and visually pleasant explanation! thanks a lot!

[@fufaev-alexander]

Thank your for your positive feedback! :)

[@Krishnanjan_Sil]

More on Mathematical Physics please

[@theQuantumPenguin]

Error in the graphic of the da vector @1:10 - 1:25 and @1:36 - 1:54, does not matter too much to the presentation which is very well done.

[@1008OH]

Yes I love this intuition as shown on the thumbnail, all of the inner surfaces cancel out and it's just the outermost ones left

[@gastonsolaril.237]

This is much easily understood while studying fluid dynamics... "The amount of water that flows through a closed surface, is equal to the change of volume inside the enclosed body"

[@Britishbaptist]

You have a talent for clarity in your explanation . thank you for this.

[@maxqi3729]

Pretty epic dude, very underrated

[@SanzAlfredo]

Sorry... I will use my language to express my opinion about the video.... GRACIAS!!!!! Esto me llevó mucho esfuerzo de entender en mi época de estudiante de ingeniería. La claridad del video es perfecta. Muchas Gracias. Saludos desde La Ciudad del Humo.

[@manoprithviraj8668]

Very well illustrated and explained :)

[@meccamiles7816]

Really great video. Well done on the visuals.

[@dolbodolb6173]

Excellent. What prevents the textbook authors to explain things this way with lots of insightful illustrations instead of a wall of formulas!

[@robertdavie1221]

Very good explanation.

[@Soundsaboutright42]

Dude I wish you were my math teacher growing up 😂

[@sashamarie3712]

Exactly the explanation I was looking for, thank you!

[@fufaev-alexander]

Hey Sasha, thank you very much for the comment!

[@Amansingh-nl4ev]

Killed it. Now i wonder how flux is a scalar.

[@JobsTech-y6n]

Man!! your contribution will be remembered.

[@fufaev-alexander]

oh thank you!

[@MusicEngineeer]

very well done! thanks you! ...i have the strong hunch that stokes' theorem will be coming soon

[@sidpandey2762]

Amazing, Sir. Thank you. 🙏😊

[@marshmelows]

Very well explained, good speaking rhythm, the visuals were great too, very easy to understand! Loved it I must confess I am revising this information years later I studied this in my college years, so it's easier to understand now. Good luck to my thermodynamics bros and braettes!

[@lukmanhdrjaya]

Best explanation i've seen so far. Many thanks for the video dude!

[@fufaev-alexander]

no problem !

[@blackhole1222]

you earned a new subscriber!

[@Cardaverr]

Very clearly explained, thanks.

[@Native_love]

Beautifully explained! Subbed to your awesomeness!

[@isarojdahal]

OMG. THIs is so crystal clear explanation. Thankyou so much .

[@fufaev-alexander]

Glad it was helpful!

[@newsgo1876]

Surface to the volume, is just like what the two end points are to a segment of line.

[@ismagine]

Great video! Btw, there’s a typo on the vector da at 1:41 there are two dy’s missing a dz term

[@gauravchandra1585]

Lucky are those who have great teachers or mentors

[@AdelElsayed-qr9dg]

Excellent explanation❤

[@ww4011]

Thank you, I just had my exam a week ago, I hope I can know this video before the final ...

[@nako7569]

basically, what goes out, or comes in, must pass through the surface, am I right?

[@fufaev-alexander]

yes

[@Olaf_Schwandt]

brilliant explanation

[@cakefactoryy]

AMAZING video!!!👍

[@moticohen6785]

Excellent!

[@muhittinselcukgoksu1327]

thank you so much.

[@eterty8335]

so my guess is that from the perspective of the volume integral, you're splitting the enclosed volume into like infitesimally small cubes, and when you're adding them up, every two adjacent cubes share one face, and both bring it up into the computations but with opposite effects (facing opposite directions), so the inner stuff all gets cancelled out and you end up only with the outer faces, which is just the surface?

[@greatgrumble]

You don't even need to got that far. The volume integral sums up the sources and sinks that "create" positive and negative flux. Inside that volume. The remaining flux has to flow out of (or into respectively) the volume. Picture each source as a tap and each sink as a drain. If all taps combined carry more water than the capacity of all drains combined, the amount of water increases (And of course the mathematical drain acts as a negative tap so this example also works for a decrease).

[@samisiddiqi5411]

Awesome channel. Thanks so much.

[@fufaev-alexander]

Thank you, Sami! #### Want more videos? #### As a channel member you have many cool benefits: ✅ Unlock ALL Physics Videos ✅ Channel Badges For Your Nickname ✅ Unique Channel Emojis ✅ Your Vote Counts 10x in Polls ✅ Immortalization in the Hall of Fame ✅ And that's not all! CLICK -> kzbin.infojoin

[@ERFARINGyoutube]

Perfect explanation for revision

[@frun]

Reminiscent of the holographic principle and ADS/CFT

[@a.s.l711]

If only my prof teaches like this

[@fufaev-alexander]

oh, thank you mate!

[@y.8901]

Thank you for your explanation ! Could we say that Green's theorem is exactly analogous to this theorem but with curl instead of divergence ?

[@bishalsarkar73]

Very well explained ❤

[@atakan716]

I wish professors knew how to explain concepts better like this😅 how could one not get this theorem after watching this!🎉

[@lufax]

Great step by step explanation

[@tanvirmahtab7666]

God bless you man

[@danielarisa1610]

Great job man!

[@rounakrizvi2915]

Thank you Sen sei

[@chongwang7768]

So it's basically like saying, when you add up everything produced in a country and subtract everything consumed, the result is the exports of the country.

[@bisrattegegn1686]

subscribed by the 4th minute, just mind blown

[@fufaev-alexander]

[@RalKrish]

Please provide an intuition for cross products and determinant

[@fufaev-alexander]

I'll do it!

[@surendrakverma555]

Very good lecture Sir. Thanks 👍

[@fufaev-alexander]

Thanks for liking

[@ernj85]

Physicists: “The sum of the forces entering or leaving an imaginary spacial boundary is equivalent in magnitude and direction to the sum of the forces at the boundary itself” College students: “This is getting complicated.”

[@alphalunamare]

I am not surprised because that is totally incorrect, whom ever said that should be sacked.

[@thomaspham2752]

great graphical explanations !

[@jeetubais5507]

Lovely wow 😍😍😍😍😍superb that's excellence work wow 🤩🤩🤩🤩

[@Channel-zb1fi]

Why is a single integral being used for the closed surface integral? Should it not be the closed double integral over the surface A of the dot product between F and the normal vector function to A times a surface element given by the length of the cross product times an area element.

[@quantum_mechanics253]

It's not really a single integral, some people only write one integral, but it could be a double integral or a triple integral. What matters is over what surface you integrate. If you integrate over V, you know that it's gonna be a triple integral.

[@joe_mama92]

That's exactly my question too 😢

[@mathsmaths8210]

Thanks, beautifully explained.

[@nicolasfernandes9987]

Great vídeo, congradulations 👏

[@fufaev-alexander]

thanks!

[@imadebouftini316]

Important clarification !! this transformation originally belongs to Green and Ostrogradski but not to Gauss

[@afsarshaikh9654]

That was amazing!!!!❤

[@fufaev-alexander]

Thank you very much!

[@Ben786]

Very well explained, thanks!

[@fufaev-alexander]

thank you!

[@learning_with_irving4266]

Best explanation of calculus and physics

[@fufaev-alexander]

Thank you very much!

[@theblitz1687]

I have a question. Didnt the tangential F components also came from the cube? But we omitted them through dot product. Why? Since F is a vector field, can it be perpendicular to cube at one side but be diagonal to another side? Your video made the consept a bit clearer so thank you.

[@mktsp2]

I enjoyed the explanation. How did you make the graphs?

[@fufaev-alexander]

Thank you!!! Adobe Illustrator + Premiere :)

[@denniswandera8582]

Easy and simple explanation

[@savagehalt1555]

Brilliant