Exterior calculus is a language used to express important concepts in differential geometry, which initially appear to be similar to objects from vector calculus. The exterior derivative (d) is responsible for building up many differential operators in exterior calculus, as it tells us how quickly a (k)-form changes along every step. This concept is introduced in the course notes “A Quick and Dirty Introduction to Exterior Calculus”, which aims to provide an introduction to working with real-world geometric data expressed in the language of discrete exterior calculus (DEC).
DEC is a simple, finite scheme that captures the essential behavior of the objects we are working with. It takes $p$-forms as inputs and creates $(p+1)$-forms; $d := left( frac( partial)( partial x)dx + frac( partial)( partial y)dy +.
The geometry of surfaces can be thought about in various ways, such as using Stoke’s Theorem or the Fundamental Theorem of Calculus. The geometry of a surface can be understood through various methods, such as wedge and star in Rn.
In conclusion, exterior calculus offers a useful introduction to working with real-world geometric data and differential operators. The course notes provide a concise summary of the concepts and tools used in exterior calculus, which will serve as the basis for the algorithms explored throughout the semester.
📹 The derivative isn’t what you think it is
In this video, I give a quick-and-dirty introduction to differential forms and cohomology. But as with any quick introduction, there are …
Is calculus Latin or Greek?
Calculus is a term in mathematics education that refers to elementary mathematical analysis courses focused on studying functions and limits. The term originates from the Latin word “small pebble”, meaning “stone”, which was used for counting distances, tallying votes, and abacus arithmetic. It was first used in English in 1672, before the publications of Leibniz and Newton. The term is also used to name specific methods of calculation, such as propositional calculus in logic, calculus of variations in mathematics, process calculus in computing, and felicific calculus in philosophy.
The ancient period introduced some ideas that led to integral calculus, but they were not developed in a rigorous and systematic way. The Egyptian Moscow papyrus (c. 1820 BC) contains calculations of volumes and areas, but the formulas are only given for concrete numbers and are not derived by deductive reasoning. The Babylonians may have discovered the trapezoidal rule while observing Jupiter.
What are the two types of calculus?
The field of calculus consists of two major branches: differential calculus, which deals with rates of change and curve slopes, and integral calculus, which deals with the accumulation of quantities and areas under curves. These branches are connected by the fundamental theorem of calculus. The site uses cookies, and all rights are reserved for text and data mining, AI training, and similar technologies, with Creative Commons licensing terms for open access content.
What is a wedge product in differential geometry?
The exterior product, also known as the wedge product, is a crucial operation in differential geometry that generalizes the cross product of 3-vectors. It acts on tangent vectors and is essential in differential geometry. The site uses cookies, and all rights are reserved for text and data mining, AI training, and similar technologies. Creative Commons licensing terms apply for open access content.
What is the wedge product of a form with itself?
In a vector space of dimension $n$, the wedge product of any form with itself is always zero.
What is the rule for wedge product?
The wedge product of two vectors is not a vector of the same space V, but of the exterior square Λ2V. In three dimensions, dimΛ2V=3⋅22=3. The main rules for wedge products are a∧a=0 and a∧b=−b∧a.
The wedge product is not just an array of numbers, as every finite dimensional vector space is isomorphic to the vector space of the tuples of that length. However, this is an arbitrary choice and can be represented as an array of numbers in some cases.
In the three-dimensional case, the Hodge isomorphism allows for the representation of the wedge of two vectors with a vector from the original space. However, this map requires two extra parameters, namely inner product and a choice of volume form, which introduce more structure to the space.
Is the wedge product distributive?
Geometric algebra is a branch of mathematics that emerged from the concept of vectors and the addition of vectors. It introduces the Pythagorean theorem through the multiplication of vectors, which can be used to replace synthetic proofs in Euclidean geometry. The geometric product in R 2 and R 3 is also discussed. The sum of bivectors is also a bivector, as shown in Figure 7. The concept of finding the closest point to a point and the distance from it to it is also explored using the decomposition method.
Is the wedge product commutative?
Lemma 4. 2 states that if both and are odd, then ω ∧ η = − η ∧ ω, and the wedge product is anti-commutative. The introduction of (k) -forms introduces a new operation called the “wedge product”, which is studied in this section. The wedge product of two (k) -forms in (mathbb(R)^3text(.)) can be determined, and the wedge product of two one-forms can be related to the cross-product of the associated vector fields.
Is calculus the hardest math?
Calculus is a challenging math class that goes beyond the familiar algebra and geometry concepts, requiring abstract thinking, imagination, and grappling with new vocabulary, symbols, and processes. It is often longer and more involved, taking up to a full page of written work to complete. The main reason for calculus’ difficulty is its position on an upside-down pyramid of previous math concepts, tying together everything learned in algebra, geometry, precalculus, and elementary and middle school math. This makes it a challenging subject to master, as it requires a deep understanding of the subject matter and its applications.
Why is calculus called Pebble?
The Latin term “calculus” means “pebble” and is derived from the Latin word “calx”, meaning limestone. Roman abacus counters were originally stone-made and called calculi. Calculators were known as calculones or numerarii, and teachers were known as calcules or numerarii. The Romans used “to calculate” as calculos subducere, while Late Latin “calculare” means “to calculate”, as found in the works of poet Aurelius Clemens Prudentius.
Is the wedge product antisymmetric?
In the context of differential forms algebra, the wedge symbol represents an anti-symmetric operation that is of paramount importance for defining orientation and volume.
Is calculus 2 hardest?
The difficulty level in Calc 2 depends on individual strengths and challenges. Some students find it harder due to its emphasis on integration techniques and series, while others struggle with Calc 3’s geometric and spatial reasoning. Calc 2 requires higher-level problem-solving techniques, while Calc 3 has a stronger foundation but requires more complex situations. To succeed, students should have a solid understanding of the limit concept, be comfortable working with 3D coordinates and geometric representations.
📹 Lecture 13:Exterior Calculus
CS 468: Differential Geometry for Computer Science.
Thanks for perusal! I’ve been meaning to make a article about cohomology for a while, so I’m glad I finally got around to making it. I am by no means a master on this topic, and I’m sure that you have insights into the material that I don’t – so share them in the comments below. Feel free to ask questions and recommend learning material as well. I’ve learned a ton just by hearing people’s thoughts and questions in the comments section, so let’s learn some math together! <3
This is above what I can comprehend right now, but I’m happy there’s someone making concise, beautiful articles for more advanced mathematics topics. I notice people with higher background knowledge on other websites like Numberphile commenting they want to see more equations and logic, and I think this website fills that void.
Beautiful content Aleph 0, I especially loved your explanation of de Rham cohomology. A minor nit pick, however, is that de Rham’s theorem doesn’t really state that H^k_{dR} = H_{k,sing}. This is really a consequence of the symmetry of de Rham cohomology (namely H^k_{dR} = H^{n-k}_{dR}) and Poincare duality. It’s a bit subtle so I’ll try to explain carefully. That H_{dR}^k = H_{sing},k happens to follow from the “duality” of differential forms, namely that Omega^k(X) and Omega^{n-k}(X) have the same dimension, but may not be canonically isomorphic, but if you endow your manifold a Riemannian metric, there is a canonical isomorphism called the Hodge star. de Rham’s theorem, however, is the statement that singular cohomology and de Rham cohomology yield the same answer. This is so-called a comparison theorem between cohomology theories. More precisely, de Rham’s theorem states that the integral over k-cycles (modulo k-boundaries) of closed differential k-forms (modulo exact forms whose integral always vanishes) is a perfect pairing. Symbolically, the map ∫:H^k_{dR}(X) x H_{k,sing.}(X) –> R which is a perfect pairing, hence, by linear algebra, we have de Rham cohomology = H^k_{dR}(X) ~ Hom_R(H_{k,sing}(X),R) = H^k_{sing} = singular cohomology. The fact then that H^k_{dR} = H_{k,sing.} is an “accident” that follows from the symmetry H^k_{dR} = H^{n-k}_{dR}, namely H^k_{dR} = H^{n-k}_{dR} = H^{n-k}_{sing.} = H_{k,sing.} where the last equality is Poincare duality. I’ll say the spirit of your claim is correct, namely that de Rham cohomology gives another way of measuring holes using calculus which coincides with the classical way of doing so, and your article does drive that main point home, especially for the layperson (say a math major), and for that you’ve done a superb job!
“When I first saw this, I didn’t know what to make of it. I could sense that it was profound, but it was so abstract that I couldn’t really see the point.” I saw this over a decade ago in grad school and while I could work with the symbols in a formal way successfully enough, I felt the same way. Your article cleared up a lot for me. Bringing in the curl gave me flashbacks of my vector calculus prof emphasizing the importance of simple-connectedness but I never connected the two before. Edit: alright I posted this before finishing the article. Seeing Stokes’ theorem in this new context was a freakin’ revelation. I literally choked up a bit.
This article is so satisfying and beautiful and perfect! At first I had no idea how you would fit everything into the nine minutes, but you did by presenting concepts both fast and with intuition. The punchline indeed blew my mind — I feel I need to look into cohomology more now! Best of all, the presentation did not rely on the viewer thinking at lightning speed. Somehow, even in the nine minute timeframe, you leave time to pause and ponder. And the paper and marker style felt fresh. I liked when you ripped your myths in half! Simply perfection. Inspiring. I don’t have time to enumerate the other choices you made that set the article apart, but I’m every second was planned, and it paid off. Keep up the good work!
Hi, I just really really wanted to thank you. So basically, I watched your article ‘How to learn maths on your own” and it got me really interested in abstract algebra. I got my hands on Gallian’s modern abstract algebra through moderately legal means and I started reading it about a week ago. For a bit of background, in French high-schools we have two maths classes. We have what we call “math speciality”(6 hours a week) and what we call Expert math(3 hours a week). My speciality math teacher doesn’t really like me(and he’s not particularily good at maths either) so since the beginning of the year I’ve not been as good in school maths as I was last year. My Expert maths teacher, on the other hand, is a really good teacher and seems really knowledgeable. We began learning arithmetic in Expert maths on monday and I was naturally really good at it since I got a grasp of the main concepts thanks to the book. So my teacher asked me to stay at the end of the lesson and he told me that I should consider going in a preparatory class in Paris(not uncommon for good students in France, but not what i considered either), and that he even could get me recommended. This means that I may become an advanced maths student thanks to you. I really hope that you read this comment, and I wish that you continue making your articles because advanced maths vulgarisation is too rare on the internet 🙂
Unbelievably awesome article, sir! I shared it with my colleagues in the math department before I even finished perusal it! One stupid hypertechnical point: at around the 7:55 mark, I think you mean you send the k-chain c to the unique differential k-form \\omega with \\int_c \\omega = 1, not with \\int_c \\omega = 0; in this case, we call the k-chain and the differential k-form “Poincare duals” of each other.
Hey, great to see you making articles again! Not gonna lie, but this one flew over my head just a little bit. This definitely makes me even more excited to get to the algebric topology stuff (I’m studying point-set topology at the moment), which is rather impressive considering that I was already super super excited for it. What I find interesting on this is that it shows one classic and incredibly powerful trick of a mathematician’s trick book, which is taking some key property of an object of interest (In this case the myths you talked about, and the object being the holes) and using it to define the object itself. This can be seen, at least in my experience, particularly in algebra, so I’m not surprised at all that it showed in algebraic topology. I loved too the link at the end with Stoke’s Theorem, which to be honest at this point is one of if not my favorite theorem. By the way, did you take that list of the ways of interpreting the derivative from Thurston’s letter/article (I really don’t know what to call it) “On the Proof and Progress of Mathematics”? Because I just so happen to read it a couple weeks ago, and I have a feeling that I saw it there. Now for some suggestions, considering that you just covered homology and cohomology, I think it might be appropriate to cover ther last one of the “homo” trio in topology; the homotopy. Personally I don’t really know that it’s all about, but what I’ve heard about it seems quite interesting. Another topic that might be cool to cover would be Banach and Hilbert spaces.
Dear Aleph 0, I think your article is truly beautiful! So many great ideas to visualize things, like ripping apart the implications!! Just a tiny thing. Your definition of the isomorphism between singular homology and de Rham cohomology is wrong. The integral defines a perfect pairing between the two, and hence, it induces an isomorphism only after choosing a basis on one of the spaces, by sending a basis vector to its dual basis vector. In particular, this is very different from the map you’ve written down, which, for example, would be the zero map H_0->H^0, for any connected space. This is just a very minor point, which takes nothing away from your amazing article :):)
Amazing vid! (literally gave lectures about singular homology and de Rham cohomology only last week, lol) Few comments though: – I think your part about de Rham’s theorem wasn’t entirely correct. Yes, H_k(M) and H^k(M) are isomorphic but just because they have the same dimension. As you explained correctly, the more technical statement is that integration induces a dual pairing between H_k and H^k, but this then induces an iso of H^k and the dual of H_k (which is not naturally isomorphic to H_k), so the construction of an isomorphism H^k to H_k would be to fix a basis \\omega_1, …, \\omega_n of H^k and then for each i to find a chain such that its integral of \\omega_j is 1 if i=j and 0 otherwise. – You’re using some weird cube homology… I haven’t really heard about it, does it yield the same homology groups as singular homology? Because you make it seem much more intuitive than triangulating everything 😉
Appreciation for the website: Studying physics at the moment, I was simply unaware of the joys of pure mathematics until about two years back, when I first luckily came about such abstractly intriguing stuff on social handles of passionate people in math such as yourself. By now, although I’ve taken a certain familiarity to these terms—as those used in this article—being thrown around on online math circles, I don’t understand them; for firstly, my formal courses never revealed this realm to me, nor did I manage to have enough time to study rigorously pure math from scratch myself. That said, I’ve tried endeavouring to study LinAlg and Topology from the pure mathematician’s perspectives, and from the basics I have tried developing an intuition for (and forgotten other stuff due to halted practice), these articles mean a great deal of precious explanation to me, making for a sort of wormhole opening to these advanced math concepts, that I can grasp with my (limited) lay of the land. With this small contextualisation I bestow my utmost praise upon this content! Thank you, Aleph0 ^-^
Really beautiful article! One hopefully constructive comment is that it’s all a bit too fast. I guess there’s an advantage to keeping it under 10 min but I could have used another 3 or 4, and I have some amount of familiarity with some this stuff. In any case, super impressive, thought-provoking, and aesthetic work!
8:03 Your definition of the de Rham isomorphism does not work: For any chain, the zero-form will produce a vanishing integral. If the map was well defined, it would therefore send every chain to the cohomology class of 0, i.e. be the constant map with value 0, which is certainly not an isomorphism if the space actually has any holes. I think it is easier to realize that the k-th de Rham cohomology is isomorphic to the dual vector space of H_k(X) by sending a form to the linear map that integrates the form over any given chain. To complete the argument, you need to choose an isomorphism between H_k(X) and its dual space. It’s probably most natural to assume that chains around different holes are orthogonal, and then use the corresponding inner product on H_k(X) for defining this isomorphism. Nice article though, especially from a conceptional point of view!
I shared this article with a friend (or it might’ve been the one on Stoke’s theorem) and it inspired him to start studying algebraic topology, then he told me why he likes topology, which got me interested and so I started going through a textbook on topology which I’m really enjoying. It started with your website and so I wanted to let you know it inspired us to learn more math. Keep up the great work! Do you have a patreon? If you keep making such great articles, I might join it.
So lucky this popped up on my recommended; very helpful with my topology course. Please keep making articles. Reading these comments, there’s a pretty large audience. Rightly so—there aren’t a lot of people making articles on higher level math topics. Even the ones that do seem to cater to non-mathematicians and keep things pretty cursory/basic. Which is fine, but the grad students want something too!
Amazing article, congrats! But your statement of De Rham’s isomorphism at 8min seems flawed: your omega is not uniquely defined since multiples of omega would have the same property. The actual De Rham isomorphism is into the vector space dual of chain homology: it maps a form to integration of chains over that form. This formulation would suffice for your vídeo since the dimension of a vector space and its dual coincide. -Note: The actual statement of De Rham’s isomorphism takes values in singular cohomology: cochains, which for real coefficients is isomorphic to the dual of singular homology by the Universal coefficient theorem.
This exposition was quite wonderful. I myself wrote a Twitter thread on de Rham’s theorem with the perspective of explaining how different incarnations of the equation d•d=0 ( which is one of the fundamental equations of Homological Algebra )are related. ( Here’s the link if you’re interested- twitter.com/pursuingstacks/status/1311568696758857730?s=09 ) Besides, there’s a remark I’d like add that you might already be aware of; Singular Homology detects more than just holes. It also detects ‘ twists ‘ in a space like for example in case of Mobius band or Klein bottle. On the other hand de Rham Cohomology doesn’t detect twists. So to make the Correspondence work we Tensor Singular Homology Groups with R ( field of Real Numbers ) which essentially kills the torsion part of Homology.
2:57 what does it mean for the boundary to be 0, if ∂c=0 how does that makes the boundary of a chain 0, a hole? can you maybe, provide an example, of how ∂c=0 is equivalent to a hole? In general, this is a great article, but for the non-math students examples are great for the brain, even if they’re bad for proofs.
Joan Licata recounted to me the following quote from a colleague of hers that is appropriate here — “Every mathematician is entitled to their own favourite theorem, but every mathematician’s second favourite theorem is Stokes’ theorem.” P.s. What you call de Rham’s theorem, I believe is not standard: De Rham’s theorem is the isomorphism between singular cohomology and de Rham cohomology. Your statement of de Rham appears to have some Poincaré duality and universal coefficient theorem in it. This is not worth discussing in a 10 minute article, of course. Nice work 💪
That is just the holy grail! This article should be the mandatory introduction of any course on the subject. I tried to study homology and cohomology by myself, and it was a painful experience; basically, I failed, lost in abstractions like exact sequences and functors (I kind of understood the abstract definitions, but I wondered what they were good for, why these notions were introduced in the first place), without even realizing what kind of problem these theories are supposed to solve. The big picture given in this article is just fantastic.
In which math book is the step function differentiable everywhere? The limit of f(x+h) – f(x) over h not exist at x = 0. Therefore, it is not differentiable everywhere. I also think you should explain more thoroughly the spaces you’re considering, and be more explicit in general. 1.) Who defined that spheres are hollow by default? The viewer may see the picture and think of a massive sphere. 2.) Of course, the hole in a torus is a hole if you consider the torus as a manifold, being part of our nice and cozy 3-D space. What you are assuming again is that the viewer will follow your implicit idea that the torus is a manifold all in its own. I’m not educated in topology except for some basic concepts, and found this extremely hard to follow. Then again, I might not be your target audience. 🙂 Cheers and the best for your website!
Is math created or discovered?.., is the fundamental elemental converse of instantaneous temporal superposition identification of 1-0 probability Totality, both location and trancendental non-location “Holes” of Black-body Singularity Space-time, simultaneously. (Probably why people worry about “Simulation” of holographic projection-drawing Actuality..) In the physical manifestation of sync-duration, a hole is axial-tangential orthogonality time-timing projection-drawing holographic embedding methodology, and it is methodology because of pure-math relative motion positioning in which .dt infinitesimal is closed-entangled zero-infinity sync-duration Eternity-now existence. A POV of inside-outside instantaneous trancendental i-reflection containment observation.., but this corresponds to the above discussion about Singularity positioning node-antinode “chain” links of Number Theory Sequences and pulse-evolution differentiates-> Eternity-now Quantization Interval in lines of sight meet at ONE-INFINITY. The Equivalence Principle of QM-TIME e-Pi-i sync-duration 0-1-2-3-4-etc exponentiation-ness sequences in Polar-Cartesian coordination quantization of Unity, Duality-Duration and Star-Delta Gaps in these Primes is rich territory for families of time-timing sync-duration identification conglomerations. Fun..
Hi Aleph0, around 5:20 you mention how one of the downsides of either differential forms or the exterior derivative (I wasn’t sure which one it was, but maybe you meant both) is that they have “no visual interpretation”, but I think both Prof. Keenan Crane from Carnegie Mellon (see his discrete differential geometry course) and the geometric algebra folks would vehemently disagree with that. Or even eigenchris’s content here on youtube. It’s generally true in higher dimensions, but that’s nothing to do with the exterior algebra underneath that powers these objects, and just more to do with the same reasons you can’t visualize >3D vectors in general. (really nice vid nonetheless, very happy about the algebraic parts of the content!)
Homology is dual to co-homology. Integration is dual to differentiation — the generalized Stoke’s theorem. Convergence is dual to divergence. Union is dual to intersection. The initial value theorem (IVT) is dual to the final value theorem (FVT) — optimized control theory. The dot product is dual to the cross product. Points are dual to lines — the principle of duality. “Always two there are” — Yoda.
Thank you! This was so well produced and easy to digest. Is this why is the Circle Perimeter function 2piR is a derivative of the area function PiRsquared? Is this analogous to an equivalency mapping between a discrete vs continuos function or domain? Can the equivalency presented here be related to a conformal mapping of the complex plane? Does this describe a relationship between the origin of a domain to a perimeter ring at a fixed radius from the origin. I don’t know much about topology but this article has inspired me to dig deeper.
4:04 – Your suggestion that there’s a non-constant function with identically zero derivative on R-{0} because somehow “that space has a hole at the origin” (in the sense of one hole, that you seem to identify with the missing origin) is misleading. In fact, the puncture at 0 in the R-{0} is not a 0-hole, it’s nothing at all, it’s simply not there! As an illustration: if you considered S^1 instead, and you poked one puncture somewhere, you would be left with a connected manifold (an interval), with no non-constant functions of zero derivative. It is just too degenerate to talk of “holes” in the case of 0-cohomology. You should just say what it is: connected components.
Geez… looking at this really encourages me to work hard. I’m gonna have to know all the stuff I’m studying now as something really fundamental to even come close to grasping this stuff. A lot of science/math YouTube I’ve seen is more focused to a general audience. But now that I’m actually at uni I should probably watch some higher level stuff instead. Much more motivation to me and actual relevance that way
It is very nice and informative presentation. But as far as the history of homology theory is concerned it is known the main credits for inventing it should definitely go to Euler, Betti and foremost to Henri Poincare (seminal paper in 1985 “Analysis situs”, J. Ecole polytech. 1. 1–121) instead of E. Noether as the article at 3:34 falsely claims. But – despite this error – I really appreciate the work done in this presentation which is really great!
Your examples of “myths” in differential calculus are not “myths” at all. You’ve simply redefined the derivative… You’ve replaced it with the limit of a derivative, which is by no means the same thing. In traditional calculus, a derivative cannot exist where the function itself is undefined. I would expect a little more precision from a mathematician, not a handwave.
You immediately earned a like and a subscription after about a minute and a half of this article. Such have a wonderful approach to explaining – one can immediately tell you’re really fascinated by this stuff. So am I… and very glad to have found this website to help me learn about it. After 10 years out of academia (with an MPhil in philosophy, formal logic and philosophy of science), I just began a course of studies in computational mathematics in order to get a better chance at grokking Homotopy Type Theory/Univalent Foundations – I think it might actually manage to unify the Logicist, Formalist and “Geometric/Topological” approaches from the Whitehead/Russell/Ramsey-school, the Hilbert-program and the Erlangen-program. The philosophical import is immense (especially in philosophy of science, I think) – and I feel how I imagine a young Russell or Quine must have felt – a new mathematical/logical tool in front of us, barely explored – but obviously incredibly powerful. But to understand that, you don’t just have to understand a good deal of formal logic and proof-theory, but also model-theory, category-theory, linear algebra, topology, algebraic geometry… this website seems excellent for diving in deeper! – Thank you so much 🙂
Hey, awesome article dude. Instant sub. I’m in my first masters semester and the subject of differential geometry somehow passed by me completely (specialized in PDEs). But I want to learn it. Right now I’m taking a topology course which covers homology. Would reading Lee’s introduction smooth manifolds show me the path to this awesome duality?
Looks like a great vid, thanks! But I’m stuck confused with sth. How is a k-chain is a vector space? Like, what’s the sum of two k-chains? I’d like to say it’s like adding functions, but that requires adding points in the shape, but can’t that fall outside the shape? Or is the shape a vector space itself, with addition defined so that it always falls inside it? Or is this whole question irrelevant to the rest of the article?
I really want to say thank you to this article. I took two math undergrad course: differrential geometry and smooth manifold. And both of them studies cohomology, but I had no motivations at all and I could not even have any intuitions from what I learn. This really blowed up my mind and explains most of things that I have learned so far. Thank you.
“A hole is a chain with zero boundary that’s not the boundary of anything else”. What does a “zero boundary” mean? I don’t think that’s defined here. Some quesses: 1) a boundary that consists of only single points? (A bondary consisting of only zero-dimensional objects.) Nah, that wouldn’t work with higher dimensional stuff. 2) “no boundary”, “empty set of boundary points”? I think that is what it might mean? Let’s take a filled in circle with a hole in it. Then, the a chain that corresponds to the boundary of the hole, is also the boundary of the set, i.e. the chain has no (“zero”) boundary points that aren’t shared with something else, the original set. Maybe that’s what is meant here? The original phrasing was, to be honest, a bit unclear.
Good evening! As a technical remark, De Rham’s theorem actually gives an isomorphism between De Rham comohology groups and singular cohomology groups, not homology groups. I saw an answer that you gave explaining this point, where you say they are dual to one another, but this is false in general; this is the content of the Universal Coefficient Theorem. For particularly nice spaces (for example, compact orientable surfaces) this is true, but again, this does not hold in general. In any case, very nice article 🙂
If you think it is to theoretic, nothing more far from reality. Back in Electrical Engineering School in TUM I had a professor, german engineer Peter Russer who wrote a book called: “Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering (Artech House Antennas and Propagation Library)” which used this differential exterior forms (dx ∧ dy ∧ dz ) in its entirety for explaining this very application oriented field….I am still trying to grasp how he condensed all Electromagnetics with this beautiful mathematical forms….⚡
On the card you put up at :07, the last delta in the “Logical” definition for the derivative should be an epsilon. The only reason I’m nit picking is that the definition of the limit is SO hard for many calculus students to grasp. If I had come across your definition with that error when I was learning calculus, I probably would have spent an hour trying to figure out what it means, then thrown my book across the room and switched my major to American Studies. 🤣
Great vidéo! But somehow too fast to digest (At least for me)! Here a summary that can help every person like me who needs time between concepts to successfully digest them. It is all based on what I understood from the article. Don’t hesitate to correct me if I misunderstood something (I hope I did not miss the whole thing ^^) At 1:30 Q/ What is a hole? A/ We all know from the dictionary definition that a hole is “a hollow place in a solid body or surface”. Geometrically, this means that if we consider loops (closed surfaces) on the considered space, a hole is defined by a loop that does not bound or contain anything belonging to the considered space. Q/ How are loops reprensented in Homology? A/ They are represented by mathematical objects called “Chains”. For example, a curve or line drawn on a paper is a 1-dimensional chain or 1-Chain (see the visuals at 2:03). The end-points of that curve are its boundaries. And similarly, we have higher dimension chains, with their boundaries having smaller dimension. It is important to note that fact. A 1-Chain has points (0-d) as boundaries, a 2-Chain has curves (1-d) as boundaries, a 3-Chain has 2-d surfaces as boundaries, etc. Q/ What are loops in Homology? A/ They are Chains with no boundary (because they loop onto themselves). Q/ What are holes in Homology? A/ The answer is given at 2:30 The sentence is a little bit confusing. What it means is basically the same as expressed before, i.e. that holes are loops that do not bound anything.
37. Derivative is the opposite of the boundary Holes Geom and calculus fail in weird spaces Holes in space Homology cohomology Differences about holes Test to find holes K chain map to our shape 3chain 2chain Boundary Hole 0 boundary Linear map relation between chains or vector spaces K dimensional hole 🕳️ all that fits the test Function not constant because of a hole Vector field in donut shape Unified test for dimensional holes Exterior derivative generalized Differential 3 form three chains or integrals Derivativek=k+1 Chain of zero derivative, not the derivative of anything else Rhams Theorem Isomorfic Integral Dimension cohomology grup= k dimensional holes Stokes theorem
Mate. Subbed!! I don’t even understand most of it ( I lied, I understand none of it O_O), I don’t have a base for anything you’ve explained, but I came for the curiosity and stayed for your brilliant explanation + calm voice and excitement. I want to understand!! so I can share in your excitement was wondering if you could have a series that helps build-up to help understand the complexity of these articles. Especially useful for those getting into college etc! Just food for thought. Thanks 🙂 Love the articles. You deserve way more views!
I just have a question. I tried looking up the de Rham theorem mentioned, but it didn’t appear to be as described at all. The closest thing I could find was Poincaré’s Duality, which states that k-homology groups and (n-k)-cohomology groups are isomorphic, which is different than what is stated in the article. Could anyone explain?
I’m not a mathematician and I do not understand why this should blow my mind, but I followed quite a bit more than I should have because you are very good at explaining this……………There should be free intuitive math courses on line with animations, graphs and visualizations………..sadly this may take time………but if anyone can do it……YOU can.
I am not a math expert, but there is a conflation that throws me off. At 2:45 you call k-chains as vector spaces. Indeed, mappings from (0,1)^k to R^d are vector spaces. But boundaries are just sets, I don’t see how they can be chains. I think it should be easy to prove that “there exists some (k-1)-chain whose range matches the boundary of a k-chain”. But this is most definitely not a unique chain. Multiple chains can have the same range.
a differential-form is a variation k of elements, for example the linear transformation e_i \\otimes{e_j }= 0, or for a scalar \\alpha{}>0, there is a variation -k then e_i \\oplus e_j where \\alpha It can depend on e_i but not on e_j, in which case T is a linear transformation of T-> \\alpha (e_i,e_j) and is associative as a bilinear-map
It’s not that mind blowing though. You are dualizing a complex into a co-complex. Also, why didn’t you talk about the hypothesis for these big theorems? They don’t work over any old space. You missed a big opportunity to discuss path connectedness with your everywhere vanishing non-constant derivative.
Harry Edwards wrote a book back in 1969 in which he essentially goes through all of this: “Advanced Calculus: A Differential Forms Approach”. He doesn’t get into proofs of De Rahm theory but he does devote a bit of time to cohomology. He thought he would revolutionise the teaching of calculus but pretty soon realised he was wrong!
I like the intuition you convey here, very nice. I’m sorry to say however that there is one error that I’m surprised hasn’t been pointed out: Cohomology is not isomorphic but dual to homology! They are therefore abstractly isomorphic whenever they are finite-dimensional, but not in general and certainly not canonically. In particular, the assignment you give that sends a chain to “the unique form that integrates to 0 over the chain” makes no sense–There is no unique such form! (for one the form 0 always works). The de Rham theorem properly states an isomorphism between singular cohomology and de Rham cohomology. There is one theorem that gives an isomorphism between homology and cohomology over very nice spaces (oriented closed manifolds), namely Poincare duality. This however doesn’t preserve degree: it sends k-chains to (dim M)-k forms. In even nicer situations, when you are given in addition a metric, you can extend this by what is called Hodge duality to give a degree-preserving isomorphism. This is different to what’s in the article though, the form that a chain gets mapped to will depend on the metric and defining the map uses concepts of the intersection pairing for Poincare- and orthogonality for Hodge duality.
Would it not make a bit more sense to say that the derivative IS a boundary? Suppose we integrate a 1-form along the 1D-boundary of some 2D region, i.e. a closed curve. Equivalently, we could apply the boundary operator (i.e. the exterior derivative) to the 1-form, and then integrate the resulting 2-form over the 2D region itself. So, the boundary operator can be equally applied to the region of integration or to the form you’re integrating over.
The article mentioned at the beginning is “On Proof and Progress in Mathematics” by the legendary William Thurston. The 37th definition Thurston gives is actually quite straightforward once you know all the definitions. Almost none of it relies on any theorems except perhaps that a differential 1-form of a manifold, when viewed as a section of the cotangent bundle, is closed if and only if its graph is a Lagrangian submanifold (with respect to the canonical symplectic form). So this definition does relate to forms and de Rham’s theorem. The article mentioned arxiv.org/pdf/math/9404236.pdf Also, since the tangent and cotangent bundle are canonically dual if you choose, say, a Riemannian metric, you can basically view vector fields as differential 1-forms. On Euclidean 3-space, this, along with Hodge-* is how one can interpret gradient, curl, divergence as the exterior derivative. Forms can be viewed as formal algebraic objects but one could view, for example, dx ^ dy, as giving a Lebesgue measure on the plane. Another thing to note: de Rham’s theorem demonstrates an isomorphism between de Rham cohomology and singular cohomology (not homology) with real coefficients. First, there is no such isomorphism for general coefficients, such as integer coefficients. Second, once you have the de Rham theorem, you can use the Universal Coefficients Theorem to obtain the isomorphism between singular cohomology and singular homology with real coefficients. Lastly, (singular) homology and cohomology are in some sense, dual, over the category of modules as the article states with its 38th definition.
Hi! First of all thank you for the shared knowledge, but I think there’s a mistake in the article. You said that differential forms can’t be visualized, which is completely wrong. Of course their visual intuition it’s not immediate, but there are a lot of resources on the topic who give a great visual understanding. In particular I would recommend the book “A visual introduction to differential forms and calculus on manifolds” by Fortney.
Matter is geometry, and geometry is the calculation of the forms of the space of connections of functional elements. But the connections of functional elements, the connectome, are the basis of the neurodynamic relationships of the aggregate functions of Consciousness, the system of self-organization, self-regulation and adaptation of the thermodynamic system of self-organizing criticality, the transitional form of the thermodynamic system, in the phased decomposition of connections in the evolution of the unidirectional aspiration of entropy to the maximum. This evolution is expressed in the fragmentation of forms of energy transformation, their specialization and the complication of their relationships accessible to metamorphoses of the logic of common human sense, which becomes more complicated as the array of information accumulated on physical and biological carriers of individual and collective memory of the successive generations of globalizing humanity grows (108 billion the planet of people), in their collective activity of generating the material nature of the anthropocentric universe. Let’s understand, we won’t understand, we’ll kill each other, competing in the strength of our own stupidity.
great job on this article! one note, the (non)-isomorphism you propose for De Rham’s theorem is incorrect. To see this, examine the first homology on the n-torus S1x S1 x … x S1 and choose a cycle representing a homology class, then observe that there are n-1 1-forms that will integrate to zero over a given 1-cycle. This example also gives you a hint for fixing your isomorphism.
I hope I understand this by the end of this year. All I know is basic calculus, and its application. Turns out I don’t know shit. All I understood was we can use maths to prove holes? and 2 chainz isn’t human. Not trying to be disrespectful or anything I’m just dumb even though I’ve a software engg degree
When I first learned vector calculus years ago, I absolutely hated it. Why were there three derivatives now? Why are there minus signs in the curl, what’s up with that? Why do people only ever talk about 3 dimensional vector calculus when we are simultaneously learning about n-dimensional vector spaces in the concurrent linear algebra course? And why is the curl of a gradient 0… Or is it the gradient of a curl? Divergence of a gradient…? Ahh, how was I supposed to remember all these arbitrary-looking rules!? This motivated me to study differential geometry, which unified these ideas and just cleared up so much for me. Although the minus signs in the curls were still a bit of a mystery. Mechanically it came from the determinant, but there had to be a simpler and more intuitive explanation… I’ve been self studying differential geometry for awhile now. I’m still at the intuition building stage, but I’m absolutely loving it because it’s organising all these obviously important ideas in generalised calculus that I was never able to fully digest. I’ve been aware of the generalised Stokes theorem for awhile now but I was only ever was able to understand one side of the equation (the differential form part). This article cleared so much, and has motivated understanding k-chains so much more than anything else I’ve encountered. To me, k-chains were the most frustratingly arbitrary “because it works” definition I had seen, and could find absolutely no geometric intuition (at least, ones I could understand) behind why linear combinations of them should be allowed or make sense.
I take a bit of issue with 5:53 – ‘differential forms generalize, but cannot be visualized’. Maybe in the sense of higher dimensions (> 4) as per our usual limitations with linear algebra, but I think there are visualizations in 1, 2, and 3 dimensions that capture the essential behavior in higher dimensions as well. In particular, I don’t know if you’ve taken a look at Div Grad Curl are dead from William Burke, or some of his other works, but I feel like differential forms are sort of like infinitesimal / locally varying linear forms, which can be visualized if you take the pictures here: en.wikipedia.org/wiki/Linear_form, and you cast them up in such a way that there is a linear form associated with every point of the chain, where you produce a number from the tangent to the chain and the form at that point, as visually defined in the wiki page. Do you have thoughts on this? Edit: Lovely article by the way, I deeply appreciate how thorough, clear, and distilled this is. I’ve been hoping to understand this better for ages and I think this helps a lot.
I’m just an engineer, not a mathematician, so I barely followed this. But there is one thing that really got me confused: When you said something like “it is the A that is not a B”, you write that as what looks like a division of A over B… is that so? I would have expected a subtraction instead. Should I look for a article “The division isn’t what you think it is”, or I just miss interpreted the notation somehow??
I think I can understand the end, so according to Stokes the volume can be represented as an integral over the boundary and since the integral like the inverse operation to a derivative, it’s also considered the “opposite” of the boundary? correct me if i’m wrong I just finished my vector calculus this one week ago xD
A piece of constructive criticism: The article is a good example of a problem I think runs through almost all of your articles. I feel like you don’t take enough time to make more sense of a topic. Like when you say a hole is a chain with zero boundary that’s not the boundary of anything else, it’s a fairly convoluted and opaque statement. However you didn’t take a moment to make sense of it. So unless your target audience already has some acquaintance with the subject, the article just gets more and more complicated. I understand that there’s both effort and time constraints, but this problem dampens the quality of your otherwise great articles. I’m still subbed tho!
What the actual fuck. I don’t know what i should think about this. Understanding homotopy given my background in topology was okay ish. But this is alien material to me lmao. Why is the derivative even remotely connected to the concept of holes .. and why tf does this duality with boudary means …. I’m more confused at the end of the article i think 😂
I feel like you go a little bit too fast sometimes, expecting the audience to “see” intuitions without explaining exactly what it is you’re getting at. For example, at about 1:40 when you explain how “obviously” a loop in the middle of a torus “doesn’t bound anything”, even with a degree in pure math it took me several minutes to figure out what it is you’re getting at. I think almost anyone who doesn’t have the right background is going to hear that and go “what are you talking about? sure that loop of string bounds something, it bounds the hole in the middle of the torus”. Even if you get the idea that that you mean the loop doesn’t bound *a part of the surface of the torus*, you still want to just say “okay whatever, then it bounds the *entire surface of the torus*, sort of inside-out”. Eventually I realized what you meant: when you draw a loop on a plane, there are always two maximal regions of space A and B having the following properties: A and B are both connected, and it is impossible to move from A to B without crossing the loop. For the loop of twine you stuck into that clay torus, the surface of the torus is not divided into two such regions A and B. Rather than “a loop doesn’t always bound something”, I would say “a loop on a surface doesn’t always divide the surface into two regions”. But I think if I hadn’t taken courses on topology before I would have just been very confused and given up on the whole article. This is a great article, but I suspect a lot of the people below commenting that they enjoyed it but couldn’t quite follow could be upgraded to “I enjoyed it and understood everything” if you slowed down a little to explain stuff like that.
That’s quasi a reason Markov Chain Monto Carlo is named a chain. In probabilistics all people do is exactly that: Integrating over a specific chain of a variable or parameter space. It is summing increments, modes or discrete states with certain, most probable, boudaries to accumulate an arbitary, given, distribution within all other variable or parameter spaces. Sometimes, the resulting objective function then, again, is derived w.r.t. it’s ‘free parameters’ … which then closes the whole case what to differentiate or integrate and when iot get a definitive result of the whole problem. cheers
The homology I’ve learnt uses integer coefficients.But in 2:58 Aleph 0 restricts the coefficients to be in a field. I wonder what’s the advantage of that? Is it in the real field in order to stress the connection with taking derivative? (p.s. I just started algebraic topology lessons days ago, if that reason is explained in some references, please point them out.)
A great illustration of what is the foundation of modern calculus is! A small thin is that the boundary is not the adjoint of the exterior derivative operator (the coboundary ) ?. Sincerly I’will be happy if can you describe the Hodge star operator and its role in the constitutive physical laws, and haw one can reinterpret the finite elements method using theses wonderful operators. Thank you a lot for this great work that helped me to fill some of knowledge hole.
Great article, but I have a few queries. At 4:07, you wrote that the function is differentiable everywhere. This is incorrect. Since the function f : R/{0}—–>R is not defined at the point x=0, there is a discontinuity at that point and thus the function f is not differentiable at 0. The notion that f fails to be constant at x=0 even though its derivative is zero is inaccurate because the function isn’t defined at x=0 and consequently not differentiable at 0 . Therefore, it still holds true that if the derivative of a function over an interval is zero, the function is constant over that interval.
Hey buddy… I just want to say Thanks for the work and effort you put in but the thing is it wasn’t really educational it seems like all you did was copy the definition and equations from one textbook a piece of paper onto kind of pieces of paper it doesn’t sound like you know what you were talking about it just sounds like you were repeating everything I’m a very good student and I was curious about this but admittedly I do not get in any education from perusal this. I take excitement into perusal your future projects
Saying that homology measures holes is misleading. For instance, the torus does not(!) have a hole. This is one of the most widespread misconceptions. When defining and observing properties of spaces you need to do it intrinsically in order to make it right. What you perceive as a hole is a property of the embedding of the torus (as a space) into three dimensional space. The properties of what you perceive as a hole changes with the choice of embedding (and ambient space).
This is a purely awesome article!! Being the mathematician that I am though, I think your formulation of de Rham’s theorem is wrong. At least, there is no unique omega such that the integral int_c omega is zero: the zero form will do, and if you happen to have any nonzero form that makes the integral zero, then any scalar multiple of it also works. What maybe doesn’t help is that de Rham’s theorem is talking about an isomorphism between de Rham cohomology and singular cohomology (over R), and the latter is isomorphic to singular homology (over R) by taking the dual space. Formulating it without that intermediate step maybe is what caused the confusion. But I don’t wanna end this on a sad note. Really inspiring, and very clearly laid out. You’ve earned a subscriber.
I am reminded of the annoyingly phrased and yet accurate description of how a missile guidance system works, talking about where the missile is and yet where it is not. A derivative being the opposite of a boundary rolls up in my mind as the idea that the prediction of where an object will be next is the opposite of the prediction of where the object will not be next. The prediction of what the value will be next is the opposite of what the object will not be next. It seems like this is a useful concept for computations involving precise positioning, even in contexts marred by relativity.