Stokes' Theorem on Manifolds

Stokes' Theorem is the crown jewel of differential geometry. It extends the fundamental theorem of Calculus to manifolds in n-dimensional space.
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This video aims to give an intuitive discussion of Stokes' Theorem, without the complicated equations and formalism. For those interested in the details, here's a thorough treatment of the topic:
arxiv.org/abs/1604.07862
(Be warned, this rabbit hole goes very deep!)
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Pictures:
Boundary and Green's Theorem - "Calculus: Early Transcendentals - 8th Edition" by James Stewart
Exterior Derivative - "A Geometric Approach to Differential Forms" by David Bachman
Music:
Relaxing Guitar Music - Acoustic - Calming Music for Stress Relief, Studying

[@dylanparker130]

this was wonderful! I sort of had a sense that Green's Theorem was a 2D version of Stokes' Theorem in 3D, but I didn't appreciate most of the connections you highlighted here - thank you!

[@Aleph0]

Thanks so much! I, too, was mindblown when I first saw the connection - it's surprising that we don't learn about it in schools! I'm glad you enjoyed it :)

[@AyoTayParks]

@@Aleph0 any chance you could make a video on the process of how you go about making both the visual representation of the ideas as well as your process of crafting a well organized and concise summary/explanation of a particular concept, in this case, stokes’ theorem. What I’m really trying ask is if you are able to attempt to depict your way or manner of thinking as you move through your process of creating your content. Thank you!!

[@abhisarma7249]

Not quite true. Green’s theorem and the (non generalized) Stokes theorem (the one with curl) is not a generalization to 3 dimensions, it’s more like a generalization to different embeddings of 2d manifolds (aka surfaces) than the simplest embedding (embeddings in the 2d Euclidean plane)

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[@bassdasdwadsasdasw]

This is a really awesome video, but I couldn't help but notice "pissmaking" at 6:11 lol

[@monojitchatterjee3185]

Same🤣🤣

[@vtrandal]

I wasnt bothered by that at all. I dont get it. I hope i never get it.

[@hadr0nbeam]

The best video about mathematics I have ever seen. You flipped my understanding of Calculus on its head. Bravo Aleph! Keep up the great work!

[@Aleph0]

@TurboGTR thank you! glad you found it helpful :)

[@lordfnord5768]

@@Aleph0 He didn't say he found it helpful. He said you flipped his understanding on its head. Without knowing, or even wanting to know, what his understanding was before, we don't know whether this was a good or a bad thing. Your graphics are superb, so if you didn't spend so much effort trying to be cute you could probably do some good work in math education.

[@LittleWhole]

@@lordfnord5768 This is the most non-sequiturish response to the above interaction you could have possibly made

[@lordfnord5768]

@@LittleWhole Sorry, I've always thought that a contradiction direct was pretty sequitish.

[@cboniefbr]

Just discovered your channel, and I have to say I'm really glad I did so.

[@Aleph0]

Thank you!! Glad to have you join us! (And super sorry for the late reply :P)

[@meahoola]

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@@Aleph0 ∂∂=0 dd=0

[@kalebgirmay1724]

Dude... Ur the reason I applied to Applied Math school

[@EderTrujillo]

This is great! But it is misleading to say that this is the "truth about calculus". This is one of many generalizations to calculus. One can study the derivative and integral operators in linear algebra context. Other possible generalization comes as complex analysis. Maybe it is useful for storytelling purposes (I think calculus on manifolds is a deep and really beautiful topic), but referring to the stokes theorem as 'the generalization' instead of 'one of many' may be a bit too much.

[@jasonparker6138]

I thought complex analysis was pretty much an application of Stokes' theorem in 2-D.

[@andreantoine8005]

@eder can you recommend some good vids about calculus generalized by linear algebra?

[@mastershooter64]

@@andreantoine8005 "Functional analysis" by Peter Lax is a good video, well it's a book, but if you flip it through really fast it'll be a video ;)

[@monny1815]

Complex analysis is still pretty much the same language as calculus on manifolds and most of it can be translated in terms of it. But otherwise, I agree, this is definitely not the only generalization worth knowing/exploring

[@jowillll]

I just started my first year as an undergraduate, but if there are various generalizations of calculus on different fields, could all of those fields in mathematics be related to one another then? Something like the modular form bridge, but with calculus?

[@gabrielandrade8022]

6 minutes. 6 minutes to explain and make me understand what hours of studying and other videos couldn't. thank you. you are a saint and a scholar.

[@AerocityMusic]

This is incredible. I’m about to start Calc 3 and a lot of the ideas I’ve seen on the horizon have felt scary, but this just makes me excited for what’s to come.

[@some1rational]

this is honestly the best video I've encountered that provides the intuitive understanding of the exterior derivative of differential geometry, I honestly don't know if & how it can be explained any clearer at least within the scope of our current framework(s) - well done, I wish this material was available during my undergraduate studies

[@AmoghA]

Having taken a course in calculus where I studied Green's and Stokes' theorem, you explained what my professor took a semester to explain in a very clear manner. Good job.

[@anthonyexplains]

This really made me think about derivatives, integrals, and stokes theorem from a new perspective; awesome content and keep it up man!

[@TheEpicGod111]

This is beautiful. Period. Sometimes in class we get so deep into the formulas that we miss the good stuff. Thanks man

[@Jesin00]

I've been excitedly explaining this to every student I tutor in multivariable calculus for years, but I never had the confidence to put it on KZbin. I'm glad someone did.

[@alang.sanchez671]

Such an amazing video, Aleph-sensei!

[@HazemA1]

This channel is amazing. I really hope you keep making more videos at a faster pace!

[@admiralhyperspace0015]

Dude, I am here before this channel blows up. Insane quality. I deal with these in physics and have never found such an explanation, especially of green's theorem.

[@jeffersonhuynh941]

Straight to the point and explanations and the visualizations are very intuitive! It took me quite a while to build some understanding with Green's Theorem, but I feel like I have a better grasp of the concept after watching this video. Thank you so much.

[@hewwo3743]

I sure am lucky to be studying this when you’re posting these videos. Feeling a passion for math again that died a long time ago

[@olivermorrison7127]

You've given a neat summary to the most mind blowing thing that my maths degree taught me :) I remember the lecture mentioned this as some trivial formula before moving on to other things while I was there completely blown away

[@LouMazero]

This is one of my favorite videos of all time.. thank you

[@pranaviyer8178]

Holy hell this channel is a goldmine.... Though i sometimes find it hard to follow you as im still learning pure math, could you please make a series on tensors and differential Geometry?

[@carloselfrancos7205]

Incredible video. You’re really pulling off one of the best math educational channels right here. Thanks a lot.

[@PaulMurrayCanberra]

One of the things that makes this difficult and misleading is that we typically draw one-dimensional (scalar?) fields on one-dimensional manifolds as 2-dimensional graphs. It might help if before moving to the 2-d case, the one dimensional case was shown as being arrows drawn along the line: positive becomes left-to right arrows, negative becomes right to left arrows. I still don't grok "the derivitave is the opposite of a boundary", though. Need to view this again.

[@tupoiu]

I think saying the derivative is the opposite of the boundary is a bit misleading - it's only the case within integrals. If you know the derivative of a function everywhere in a region, you can find the integral over a function everywhere (which effectively gives the function, up to some constant) but if you know the function everywhere in a boundary that doesn't give you the derivative everywhere. I guess they are parallel in that knowing the derivative of a function everywhere in a region is the same as knowing the value of a function over ANY boundary.

[@generalezaknenou]

this really is outstanding, I'm uppset cuz we study math with francophone wich gives me some difficulties understanding this content but most of it is too straight to human mind to be missed . Trully thank you and I hope you dig more on the coming videos and give more time for small details

[@jaeimp]

Absolutely outstanding! I am on my feet clapping, and cheering! The depth of your presentation is only matched by your tactful decision to try to transcend the usual "and you can't understand it because there is a thing called tensors, and another thing called forms, and well, you are just too young for it" underlying condescension in the vast majority of presentations of Stokes' theorem - which by the way, it is complicated even in remembering where to place the apostrophe!

[@Aleph0]

Thank you! Your comment really made my day :) Can't help but agree with you about the condescension in presenting Stokes' Theorem -- when I learned it in class, we had to wade through thirty pages of definitions about tensor products, forms, differentials, chains ... and when we finally arrived, I couldn't help but think: "Really? All those definitions were just drama! This is so much simpler then it was made out to be."

[@hyperduality2838]

Transcendental logic is dual to transcendental aesthetic (sensory) -- Immanuel Kant. Concepts are dual to percepts -- the mind duality of Immanuel Kant. Generalization (boundary) is dual to localization (derivative). Convergence is dual to divergence Integration is dual to differentiation -- Generalized Stoke's theorem. Vectors are dual to co-vectors (forms). The dot product is dual to the cross product. "Perpendicularity in hyperbolic geometry is measured in terms of duality" -- Professor Norman Wildberger. "Reflections preserve perpendicularity (duality) in hyperbolic geometry" -- Professor Norman Wildberger. Homology is dual to co-homology. The initial value theorem (IVT) is dual to the final value theorem (FVT) -- optimized control theory. The time domain is dual to the frequency domain -- Fourier analysis. Positive curvature is dual to negative curvature -- Gauss, Riemann geometry. Curvature or gravitation is therefore dual. Apples fall to the ground because they are conserving duality. Potential energy is dual to kinetic energy. There appears to be a pattern here? "Always two there are" -- Yoda. The big bang is a Janus point/hole (two faces) = duality! Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics.

[@paulo123brasil]

That explanation is amazing. I graduated in electrical engineering in 2017. At college I knew "how to", but I never understood the real meaning of this. Congratulations for this great explanation.

[@elcotera8042]

Just finished a course on vector calculus this semester but never got introduced to all the theorems like this, this is amazing, my mind is still spinning.

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You deserve at least as many subscribers as 3blue1brown. This channel is a gem.

[@luis5d6b]

Absolutely amazing videos you have, keep it up, thanks a lot for the insight.

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Literally one of the best videos I've ever seen!!

[@vidishapandey91]

So comprehensive, thank you! Looking forward to more of this stuff!

[@gabitheancient7664]

oh boy, another amazing channel to watch all videos, this is beautiful

[@rajibsarmah6744]

Please make video on Differential Geometry

[@Aleph0]

Will do! Thanks for the suggestion.

[@sheungmingchoi6804]

What a legendary intuitive insight of this major theorem, salute!!!

[@billob1305]

that is so satisfying to see the fundamentals come so vividly

[@fabiocarletti6084]

I will be forever grateful for this video. Keep it up man!

[@cacimbinha5131]

This is exactly what I was looking for, thank you so much

[@rishiraj8738]

Your explanation attracts me to pursue maths.🤓 Though there is few months left in that decision.👍

[@Aleph0]

I'm honored! Hope you choose maths - definitely a wise choice :)

[@frun]

Think twice before you do this

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@@frun we shouldn't,

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[@rhke6789]

I watched this and also the one quintic impossible (so far). What insight you impart...now I really understand both. I hope you keep doing more math topics... congrats on your insight and on your ability to teach that insight.

[@DeanCalhoun]

this exact thing jumped out at me when I learned stokes theorem/divergence theorem /greens theorem in calc 3. it’s all the same thing: an integral over a boundary is the same as the integral of the whole if we take the derivative. math is so beautiful

[@jimmorgan6213]

stunningly, incredibly good. I took an entire course on Stoke's Theorem - and got an 'A' - without ever grasping this.

[@ruggerobenetti6213]

Thanks for sharing such amazing content! You are a big source of inspiration, keep it up!

[@mohamedlotfi982]

This is just wonderful!

[@ayushkumarjais2483]

Beautifully explained

[@amaarquadri]

Great video. I had always heard of Stokes theorem in my university calculus classes, but I never really understood how it was a generalization until this video!

[@Aleph0]

Thanks! Gotta agree with you there -- we're not taught how all the theorems of calculus are just special cases of one big theorem!!

[@hyperduality2838]

Generalization (boundary) is dual to localization (derivative). Convergence is dual to divergence Integration is dual to differentiation -- Generalized Stoke's theorem. Vectors are dual to co-vectors (forms). The dot product is dual to the cross product. "Perpendicularity in hyperbolic geometry is measured in terms of duality" -- Professor Norman Wildberger. "Reflections preserve perpendicularity (duality) in hyperbolic geometry" -- Professor Norman Wildberger. Homology is dual to co-homology. The initial value theorem (IVT) is dual to the final value theorem (FVT) -- optimized control theory. The time domain is dual to the frequency domain -- Fourier analysis. Positive curvature is dual to negative curvature -- Gauss, Riemann geometry. Curvature or gravitation is therefore dual. Apples fall to the ground because they are conserving duality. Potential energy is dual to kinetic energy. There appears to be a pattern here? "Always two there are" -- Yoda. The big bang is a Janus point/hole (two faces) = duality!

[@TheStringKing7]

Amazing insight. Exactly what I was looking for. Thank you :)

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This guy is GOOD! I'm impressed with your explanation.

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Thank you very much for making such a nice video.

[@ogginger]

Wow. That was brilliantly presented. Thank you.

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This is so beautiful, I could cry ❤️

[@curtisp471]

This video is so good. Many thanks. I will be sharing.

[@gauravmaindola4571]

This video is pure GOLD.

[@salsasafi]

breath taking video... now I am super convinced that I want to take calculus on Manifolds

[@TheMauror22]

Damn I just discovered this channel and I'm loving it. You have great content! Keep it up!

[@nitishupadhyay3972]

Now, I'm gonna share this happiness with my whole class.

[@locksh]

I swear this for some reason made me emotional. Great content

[@none6986]

Thank you for immersing such a thoughtful insight into me

[@IproCoGo]

That was really helpful! Good work.

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Man your commentary is amazing ! keep posting more videos !

[@vangelisrocks]

amazing video, thank you very much

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Glad I found this channel!

[@adarshkishore6666]

Absolutely beautiful. This is what I love about mathematics, it is the art of generalizations

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One of the best things 2020 brought This channel

[@josechang7450]

Keep up the great content!

[@arthsojitra53]

Beautifull Explanation !!! Really enjoyed it!

[@Aleph0]

thank you Arth!!

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I hecking LOVE your channel

[@mikewasinger9029]

You have a gift. Thank you for sharing it!

[@rajabhattacherjee5141]

Superb....learning new....please continue...I think never even FEYNMAN thought about it..

[@MikhailBarabanovA]

I would lie if I tell that understood it all. But videos like that are giving directions on where too look. And that sometimes is Very helpful.

[@eduardoandrescontrerasrome6703]

As an electrical engineering student currently learning Vector Calculus in my Physics 3 course while suffering (AND loving as well) with all of these Stoke Theorem and Divergence Theorem problems, this was BEAUTIFUL.

[@pimcoenders-with-a-c1725]

This is absolutely beatiful! We worked with this in my tensor calc & general relativity classes, but I didn't understand the profoundness of it back then; You made the exterior derivative and stokes' theorem more intuitive than the entire tensor calc course could! I'll be doing topology, manifolds and differential geometry in the coming year, and I'm looking forward to it even more now

[@MessedUpSystem]

Rewatching this for the 3rd time, it's just so elegant how this one theorem brings all of calculus together in the language of differential geometry

[@meccamiles7816]

This is a damn good video. Very, very well done.

[@12jgy]

Lovely stuff! I specially liked this one, I don't know what it's about it, but I find it quite beautiful. I would like to see you cover some abstract algebra(group and ring theory), topology, or some parcial diferencial equations(maybe some specific ones like for example Navier-Stokes?), as I think it would be very interesting. Anyway, good job again, and I'm looking forward to seeing whatever you decide to post in the future :)

[@Aleph0]

Thank you for the comment and the suggestion! I'm currently working on a series of videos on topology, so stay tuned :)

[@toadmove3774]

BEAUTIFUL VIDEO!!!

[@beimein3244]

awesome way of describing stokes theorem right off the bat. a connection of how it jumped to spin then flow would've been nice

[@BlackbodyEconomics]

That was just fascifuckinawesomnating! I've never thought of the integral and the derivative as opposites - even so, you still blew my mind :)

[@peterd5843]

You are very good at making educational videos. Please make more

[@depressedguy9467]

When i was in high school i had that doubt of integral and derivative , i had a feel about them both are not exactly same today it is cleared to me that they are not same at all

[@kruksog]

It's just "Stokes" not "Stokeses." I know I sound pedantic, but I can see how educated you are and I don't want anyone to dismiss you for your pronunciation. Amazing video.

[@darksecret6050]

"So it's all Stoke's Therorm" "Always has been" *STOKESCEPTION*

[@thomaswatts6517]

I love your voice :) it makes me feel calm

[@add-mt5xc]

I really love this! Thank you

[@marcosguillermoisunzaalvar3868]

This is incredible !! Thank you !

[@nolifeonearth9046]

i think i finally can understand it. Thank you!

[@jaikumar848]

By your logic ..you are genius . You explained complex thing in simple way

[@davialefe7646]

This is one of the most brilliant explanations in math I ever watched. Congratulations 👏

[@ohanneskamerkoseyan3157]

Three separate flashes of light inside my head in one video. I definitely will go deeper into this. Thank you!

[@rajibsarmah6744]

Please made a video on differential forms

[@NoNTr1v1aL]

Amazing video!

[@sergiolucas38]

Truly great video :)

[@82Muntasir]

Amazing, you r genius... the way how it should be told

[@azathoth00]

this is mind blowing

[@user-th3tg4fb5g]

super underrated channel

[@TyronTention]

Differential Geometry is so far my favorite field of study within math. You are definitely helping to prove that for me.

[@hyperduality2838]

Generalization (boundary) is dual to localization (derivative). Convergence is dual to divergence Integration is dual to differentiation -- Generalized Stoke's theorem. Vectors are dual to co-vectors (forms). The dot product is dual to the cross product. "Perpendicularity in hyperbolic geometry is measured in terms of duality" -- Professor Norman Wildberger. "Reflections preserve perpendicularity (duality) in hyperbolic geometry" -- Professor Norman Wildberger. Homology is dual to co-homology. The initial value theorem (IVT) is dual to the final value theorem (FVT) -- optimized control theory. The time domain is dual to the frequency domain -- Fourier analysis. Positive curvature is dual to negative curvature -- Gauss, Riemann geometry. Curvature or gravitation is therefore dual. Apples fall to the ground because they are conserving duality. Potential energy is dual to kinetic energy. There appears to be a pattern here? "Always two there are" -- Yoda. The big bang is a Janus point/hole (two faces) = duality! Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics.

[@tomasmanriquezvalenzuela5909]

Your channel is amazing

[@rajibsarmah6744]

Most informative

[@technowey]

After watching this video, I am stoked about calculus. :)

[@nihanth9145]

i Watched this video 3 months ago, didnt understand rigorously , now i am back after spending time learning actual topology and differential geometry, it feels good but still more to learn

[@stefanotorelli3688]

That is fascinating! full stop.