Stokes' Theorem on Manifolds
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- Опубліковано 30 тра 2024
- 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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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 - Наука та технологія
This is a really awesome video, but I couldn't help but notice "pissmaking" at 6:11 lol
Same🤣🤣
I wasnt bothered by that at all. I dont get it. I hope i never get it.
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!
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 :)
@@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!!
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)
I dont mean to be offtopic but does anyone know a method to log back into an instagram account??
I somehow forgot my login password. I appreciate any help you can give me!
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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!
@TurboGTR thank you! glad you found it helpful :)
@@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.
@@lordfnord5768 This is the most non-sequiturish response to the above interaction you could have possibly made
@@LittleWhole
Sorry, I've always thought that a contradiction direct was pretty sequitish.
Just discovered your channel, and I have to say I'm really glad I did so.
Thank you!! Glad to have you join us! (And super sorry for the late reply :P)
@@Aleph0 I come three months behind Carlos, but I'm on the same track.
@@Aleph0 👍
@@Aleph0 ∂∂=0 dd=0
Dude... Ur the reason I applied to Applied Math school
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.
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.
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.
I thought complex analysis was pretty much an application of Stokes' theorem in 2-D.
@eder can you recommend some good vids about calculus generalized by linear algebra?
@@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 ;)
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
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?
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.
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!
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."
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.
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.
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 UA-cam. I'm glad someone did.
Please make video on Differential Geometry
Will do! Thanks for the suggestion.
This is beautiful. Period. Sometimes in class we get so deep into the formulas that we miss the good stuff. Thanks man
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
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.
Your explanation attracts me to pursue maths.🤓 Though there is few months left in that decision.👍
I'm honored! Hope you choose maths - definitely a wise choice :)
Think twice before you do this
@@frun we shouldn't,
S....t...a..y!!!
@@daphenomenalz4100 Think Twice ;) ua-cam.com/channels/9yt3wz-6j19RwD5m5f6HSg.html
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
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
This really made me think about derivatives, integrals, and stokes theorem from a new perspective; awesome content and keep it up man!
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
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.
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
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!
Thanks! Gotta agree with you there -- we're not taught how all the theorems of calculus are just special cases of one big theorem!!
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!
Incredible video. You’re really pulling off one of the best math educational channels right here. Thanks a lot.
Such an amazing video, Aleph-sensei!
This channel is amazing. I really hope you keep making more videos at a faster pace!
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?
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.
You deserve at least as many subscribers as 3blue1brown. This channel is a gem.
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.
Beautifully explained
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.
This is one of my favorite videos of all time.. thank you
that is so satisfying to see the fundamentals come so vividly
So comprehensive, thank you! Looking forward to more of this stuff!
Now, I'm gonna share this happiness with my whole class.
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.
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
stunningly, incredibly good. I took an entire course on Stoke's Theorem - and got an 'A' - without ever grasping this.
oh boy, another amazing channel to watch all videos, this is beautiful
What a legendary intuitive insight of this major theorem, salute!!!
Literally one of the best videos I've ever seen!!
This is exactly what I was looking for, thank you so much
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.
Wow. That was brilliantly presented. Thank you.
Absolutely amazing videos you have, keep it up, thanks a lot for the insight.
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.
I will be forever grateful for this video. Keep it up man!
Damn I just discovered this channel and I'm loving it. You have great content! Keep it up!
Please made a video on differential forms
This guy is GOOD! I'm impressed with your explanation.
Amazing insight. Exactly what I was looking for. Thank you :)
This is so beautiful, I could cry ❤️
By your logic ..you are genius . You explained complex thing in simple way
Beautifull Explanation !!! Really enjoyed it!
thank you Arth!!
This was really good. I have to say though, I don't think it's wrong to say derivatives and integrals are inverses -- it's true in single variable calculus and sometimes true in vector calculus (gradient theorem). When we generalize the idea of integration beyond that, it doesn't make as much sense to say it's inverse of differentiation anymore, but that doesn't invalidate its truth for single variable calc. It's like how it's perfectly fine to say that multiplication is just repeated addition when we're only dealing with the integers, even though that doesn't really work once we get to the rationals and reals.
Furthermore, while the way you wrote the FTC is arguably more fundamental, presenting it to calc 1 students like that would be a recipe for confusion. It is something that could be presented towards to the end of calc 3 though.
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.
Thanks for sharing such amazing content! You are a big source of inspiration, keep it up!
Superb....learning new....please continue...I think never even FEYNMAN thought about it..
breath taking video... now I am super convinced that I want to take calculus on Manifolds
One of the best things 2020 brought
This channel
Absolutely beautiful. This is what I love about mathematics, it is the art of generalizations
This is just wonderful!
This video is pure GOLD.
This video is so good. Many thanks. I will be sharing.
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
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.
Man your commentary is amazing ! keep posting more videos !
I hecking LOVE your channel
That was really helpful! Good work.
Keep up the great content!
You have a gift. Thank you for sharing it!
Glad I found this channel!
Thank you very much for making such a nice video.
I swear this for some reason made me emotional. Great content
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
The total change at the boundary equals the sum of the changes on the inside.
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!
@@hyperduality2838 wtfff is all of this why is duality just like everywhere ?? I studied real analysis until Riemann integration and vector calculus and Fourier series ... How can I come to understand what duality is !!?
@@generalezaknenou It is physics.
Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).
Energy is dual to mass -- Einstein.
Dark energy is dual to dark matter.
Space is dual to time -- Einstein.
Certainty is dual to uncertainty -- the Heisenberg certainty/uncertainty principle.
Waves are dual to particles -- quantum duality.
There is a load more, but there is a pattern here!
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics.
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
Teleology is not encouraged in physics so there is a reluctance to accept duality and hence the concept of a 4th law of thermodynamics.
Thesis is dual to anti-thesis creates converging thesis or synthesis -- the time independent Hegelian dialectic.
Being is dual to non-being creates becoming -- Plato.
Mind is dual to matter -- Descartes.
Absolute truth is dual to relative truth -- Hume's fork.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
"Philosophy is dead" -- Stephen Hawking.
Physics has a problem with philosophy if you believe Mr Hawking, which means teleological thinking is not allowed, but teleology is required if you want to understand duality.
Target tracking = teleology = syntropy.
@@hyperduality2838 a bunch of mysticisms that have nothing to do with reality. You're using the term "dual" in a bunch of different meanings that make the entire thing unclear.
I hate it when people ascribe mystical significance to a concept this way.
I'm going to tell you that positive integers are dual to negative integers.
Now I'll tell you that positive integers are dual to fractions with unitary numerator.
Nice, now we have two dualities, rendering the entire concept useless. Duality is a very specific thing that is comprehensively studied in category theory and none of what you've written has anything to do with that.
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
awesome way of describing stokes theorem right off the bat. a connection of how it jumped to spin then flow would've been nice
You keep reminding me why math i love math so much. Ima get that doctorate. One day.
All the best!
That was just fascifuckinawesomnating!
I've never thought of the integral and the derivative as opposites - even so, you still blew my mind :)
amazing video, thank you very much
BEAUTIFUL VIDEO!!!
Amazing, you r genius... the way how it should be told
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 :)
Thank you for the comment and the suggestion! I'm currently working on a series of videos on topology, so stay tuned :)
1:15 I think that this is work rather than force, since this is a flux integral. Great video
This is a damn good video. Very, very well done.
Most informative
Three separate flashes of light inside my head in one video. I definitely will go deeper into this. Thank you!
this was one of my favorite topics in university, the other was Lagrangians/ Hamiltonians
super underrated channel
This is one of the most brilliant explanations in math I ever watched. Congratulations 👏
Amazing video!
Holyshit first time I saw this I didn't appreciate it for what it was. This is beautiful, you put it so elegantly.
This is incredible !! Thank you !
If there were a Fields Medal for Math UA-cam Educational videos, this would be my winner. Thanks for your work as a whole.
Truly great video :)
You are very good at making educational videos. Please make more
Absolutely loved it !!! I wish you could have tutorials on basic calculus too . What kind of digital device and software are you using ?
I really love this! Thank you
I love your voice :) it makes me feel calm