@@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.
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!
@Ali Van thanks so much for your reply. I found the site through google and Im in the hacking process now. Takes a while so I will get back to you later with my results.
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.
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.
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?
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.
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 RU-vid. I'm glad someone did.
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.
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?
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.
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'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
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.
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
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 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
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!
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!
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
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 :)
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.
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.
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.
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.
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
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
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
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.
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.
Great content! I wish I saw this right before taking differential geometry. Maybe it could have made me appreciate the subject, or at least not despise it.
You should make a video about de Rham's theorem. That makes the _duality_ of boundaries and derivatives a concrete theorem, by showing that de Rham cohomology is the dual of singular homology. Maybe take a couple of videos to set of up singular homology and de Rham cohomology though! Incidentally, this duality between singular and de Rham cohomology has rough analogies in p-adic cohomology. The statement that algebraic de Rham cohomology, etale cohomology, and crystalline cohomology coincide is analogous to de Rham's theorem, part of Fontaine's program of p-adic Hodge theory, an essential ingredient in the proofs of Falting's theorem and Fermat's last theorem.
That is such a coincidence -- I am literally working on a video about De Rham Cohomology and De Rham's Theorem right now! Should be out in a week or so. That last bit you mentioned is super cool, I've always been curious about how cohomology shows up in number theory but honestly, I have barely any background in algebraic number theory ... so that'll have to wait I guess. Thanks for the comment!
@@Aleph0 There are some excellent resources to learn algebraic number theory. To get the "feel" for it, make sure you're up to speed with algebra, particularly rings and fields, some basic Galois theory. Marcus' book is a fantastic introduction but has nothing p-adic. Milne's notes might fill this gap. For more advanced material, Neukirch is simply beautiful but very challenging, it may be profitable to have some more commutative algebra. Search around though there are hundreds of books and lecture notes so I'm sure you'll find something that fits your style and level. Cohomology enters number theory in many circumstances, one of the first historical examples being that Gauss sums are secretely cocycles (see Ireland and Rosen Chapter 14 Exercise 9 on pg 226). Generalizing this observation leads to class field theory; the Artin reciprocity law generalizes all the abelian reciprocity laws and it's an isomorphism of Tate cohomology groups! The type of cohomology that appears in p-adic Hodge theory however is much more general and subtle (see e.g. etale cohomology). The role of cohomology in this setting is as a tool to compute obstructions to lifting (more accurately _deforming_) Galois representations. It takes quite the background in scheme theory though before venturing there. Best of luck with your studies!
@@theflaggeddragon9472 This is so exciting! I just started looking at Neukrich's book and it seems at the right level (at least so far ... I'm on chapter 1 :P). Thanks for the recs.
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.
I assume these are related to the video: The derivative of the volume of a sphere is its surface area. The derivative of the surface area of a sphere is the circumference of a circle. The derivative of the circumference of a circle is constant length.
The part that tripped me up that I want to clarify is that when you are integrating over the boundary in 1d space, like the fundamental theorem of calculus, you are only considering 2 points even though I commonly though of integration as applying only to infinitely many points, you can integrate over just 2 points
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.
The total change on the outside is *not* the sum of all the little changes on the inside. From 2 to 4 is 2, but from 2 to 2.5 is .5, from 2 to 3 is 1, from 2 to 3.5 is 1.5. and we've got more than 2 already without even adding in the change from 2.793124 to 3.91874521 (1.1256212099999998 it sez here, but I suspect machine representation error at work here...), which would make it even greater.
