I just realized a stupid mistake I made: at 8:20, this should actually be that the inner product is this length times ||u||. I don't know how I didn't notice this in all of my times watching this video while making it. EDIT: There is now an addendum to this video here: ruclips.net/video/0bOiy0HVMqA/видео.html. It clears up a few misleading things in this video and then focuses on actually understanding the geometric product. EDIT 2: Another correction: at 29:00, the equation should be i a · (b × c) = a ∧ b ∧ c, not a · (b × c) = i a ∧ b ∧ c.
No worries You got your point through...that is more than enough, Thanks a lot for such an explanation. I want to apply geometric algebra to viscoelastic equations. Whenever viscosity is introduced in any media, it is associated with a complex number and this results in the rotation of the velocity vector. Please let me know if you are interested in Collaborating.
I wish I could upvote thousand of times. You are a hero thank you for the great work. I have tried to understand this for like 10 years and I always thought 3d rotation is a magic box just copy&paste and it works, however, it works don't even try to think about it but after seeing rotors as bivectors I can understand what all these were about. Thank you a lot keep it up this is the only channel I have clicked the bell icon,
I made exactly that same mistake once while I was showing it to someone. I think it is because so often you are used to working with an orthonormal basis that you go into auto pilot.
Makes one wonder if 19th century mathematicians or physicists were too spooked by Clifford algebras that it took another century for someone to rediscover its utility.
As soon as you brought up "i", I thought "he hasn't..." but as the vectors were rotated through 90° I was like "he is!!!" then you dropped the big reveal and my face was all Ö That was stunning and brave! Bravo!
geometric algebra feels like something that was missing from mathematics, it kinda explain things that were weird but proven, now everything makes sense
From as far as I got... It does have some interesting things to it. It does make for a more compact approach too. The only thing I'm worried about is dimensionality of the entities being geometrically produced(?). There are quite a few things in physics that have weird behavior because the way we handle them is mathematically strange, but you get used to it. I knew I could use the external product instead of the cross product and get much the same things out, but the geometric product solves some notation problems we noticed but had to fix afterwards.
I work in applied electromagnetism and never learned so much in 45 min. You've made sense out of concepts that were always just out of reach. You've changed my life
There’s a lot of great literature about this field and it’s applications. Check out Geometric Algebra for Electrical Engineers or Geometric Algebra for Physicists.
@@IanBLacy Applied Electromagnetism is probably in reference to transmission line theory and similar radio applications. You don't need it to understand Maxwell, in fact it's in reverse. You need Maxwell to understand transmission line theory and radio frequency circuits. However, you might not see it depending on your undergraduate curriculum.
I don’t normally comment on RUclips, but as someone trying to understand higher level math concepts, this is one of the best videos for education I’ve ever seen, at least for the way my brain works; thank you so much!
@@olegt962 E=MC2 IS F=ma, as time is NECESSARILY possible/potential AND actual IN BALANCE; AS ELECTROMAGNETISM/energy is gravity. This explains the term c4 from Einstein's field equations. Time dilation ultimately proves ON BALANCE that E=MC2 IS F=ma, AS ELECTROMAGNETISM/ENERGY IS GRAVITY. Gravity IS ELECTROMAGNETISM/energy. Balance and completeness go hand in hand. It all CLEARLY makes perfect sense. By Frank DiMeglio
If you want to understand a higher level of mathematics you need to study the book of “advanced engineering mathematics” That is about vectors in 2d, 3d; double and triple integrals, surface area, Curve, Green’s theorem, Stokes theorem, surface integrals,, Gradient, differentiate equation in different orders....that would be higher levels. Good luck with your study 📚
I always had this problem with torque and angular momentum. I remember how after a Classical Mechanics class, I started searching for a more profound meaning to those concepts, since I was already in third year of Physics and still got no feeling of understanding the actual concept completely. Now I watched this video, and all of a sudden my soul can finally rest. I mean, even Feynman, when trying to explain the conservation of angular momentum, talked about how the area was preserved. It makes total sense.
Torque, angular momentum, area, volume... all bi and tri-vectors. For regular vector algebra, it felt like, directionality is property that appears and disappears at will. Why the hell an area is a vector, volume is not? This makes perfect sense.
After learning GA, I've started using the term "Kepler's Law of Quantum Mechanics" in a few places, because bivectors as areas (while not actually the most natural interpretation of them from what I've seen) trivially equates Kepler's Law to the Law of Conservation of Angular Momentum. Treating bivectors as areas also gives another interpretation behind the angle doubling: exp(aB) for a unit bivector B gives a rotor that covers a sector of the unit circle with an area of a. This way of viewing the bivector exponential extends to hyperbolic rotations and translations as well.
@@Utesfan100 actually.... at 3D, it's not really quaternions. Quaternions are missing the 3D 1D vectors and pseudo-scalar. Octonions are a different direction entirely; a wrong-turn IMHO. As you go into higher dimensions with GA, octonions are not a thing you run into. Octonions are not associative; etc. I think the O(2^n) growth of the calculations as dimensions go up is why we did not go this direction when it was discovered. It's almost too tedious without computer assistance.
@@robfielding8566 I was merely attempting to point out that if you think geometric algebra has a cult following, you should look to the composition algebras :) As a devotee, I am compelled to address your comments more fully. I concede that the ability of vector notation to scale indefinitely lead to its wide adoption across all of mathematics. Which is a more fitting property for a geometric algebra, associativity or compatibility of the product with length? This latter property is what defines a composition algebra. Merab Gogberashvili has done some interesting things using the split-octonions in place of Cl(0,3), analogous to paravectors. This algebra is the unique real composition algebra that contains Minkowski 3+1 space. (There is even a Cauchy integral formula for this algebra.) The relationship between the octonions and the exceptional Lie groups is deep. If one encounters the latter, they would be well served to understand the former. Speaking of cult followings, E_8 comes to mind :) While not a Clifford algebra, the composition property gives the octonions a very geometric flavor. Indeed, Advances in Applied Clifford Algebras frequently includes articles dedicated to the octonions, suggesting they are at least in the same family of algebras. Even if just as the crazy, eccentric uncle.
I'm completely amazed. I found a Wikipedia article on Geometric Algebra a while ago, seemed interesting, but I didn't quite get it. The past few days I've been working with quaternions and now your video comes along. I almost cried when the "i" entered the stage, it's all so incredibly natural. Thanks for blowing my mind.
I actually paused the video to laugh for a good minute when 'i' appeared in the 2D case, then again when the comparison to quaternions appeared, then several times again for Maxwell's equation(s). For some reason this is funnier than comedy.
Sietse Have got to the point of expressing the rate of change of unitary quaternions to the angular velocity pseudo vector of the rotating body? Essential for attitude determination.
@@animowany111 For me, comedy is when you really should've seen something coming, but didn't. Seeing familiar elements pop up unexpectedly when doing something completely different and not noticing at first definitely counts as comedy in my eyes.
@Paul Wolf Except with Imaginary numbers, linear algebra, calculus. Also why you even need some explanation, is math you just define something and works with it, unless it is useless or contradictory, who cares about explanations?
I have noticed many people has had the same actual reaction that I had. I really cried, and I didn't understand why at first. In my case I believe it is because I experienced something so potent that it almost dwarves everything I know in terms of physics. I can't help but feel for something so beautiful, so deep that my soul finally rests on knowing that those doubts I had and my inability to sometimes grasp concepts is perhaps justified. I am almost finishing mechanical engineering and I really felt there was still too much to many knowledge I lacked, not because of not knowing the existence of such topics but feeling I wasn't on top of many subjects to the degree my curiosity demands. This was only mentioned briefly but the connection between tensors and complex numbers was something not even in my wildest dreams I could have come up and here it is JUST SO NATURAL. I always felt there was something missing from my education and I have looked desperately everywhere. I can't even begin to explain how relieved I am to finally arrive at this, not because it has solved all of my doubts, but because it gives me hope that I can perhaps eventually solve them. It's a light on a path I thought was shoruded in darkness. I just want to scream, we should all have been introduced to this as kids. And I believe all of the physics education programme should be built around it. It's something you can immediately and intuitively relate to.
Well said. But the reason that the subjects are not taught to everyone isn't that they are too hard. They are kept difficult precisely because the powers that be are not interested in a large population of mathematically capable individuals who are not under surveillance by institutions of government, education, or corporation. National security is why we cannot have nice things.
I don't usually leave a positive comment on a video, but I'm making an exception. This is a masterpiece, the knowledge to produce this must be so utter. I haven't seen a book or video about the topic more enlightening. My most sincere congratulations, you really have a gift for this, consider making a career out of it.
I love that Maxwell's equation is basically the same as the tensor notion, which also condenses the 4 equations down to 1. It would be cool to see how geometric algebra relates to tensors
Michael Penn has a video about Clifford Algebras and there he describes them as subalgebras of Tensor Algebras, and I don't know what Tensors are, they look too terrifying to me. One my friend studied tensors in latest years of normal high school. Now, after 4 years of university he still doesn't know what Tensors are. He is smart. He understands really complicated math. He studies cybernetics and applies for the job in the best it firm of our country. But he didn't get the Tensors. This terrified me.
@@РайанКупер-э4о no need to be scared, just like how the video described Geometric Algebra as a ‘special rule’ for multiplying (multi)vectors, tensors are a ‘special rule’ for transforming (nested)matrices. It’s convenient result that these transformation rules can also be written as the nested matrices that they transform
Best set of videos on the algebras I have thus far viewed. congrats. Very easy to understand and presented clearly. Viewer like me feel rewarded after viewing and very grateful
this feels like a video that would be unimaginably useful in the future, but now I just lost it around the 20 minute mark. It feels as if I will return to this in the future, and find it really helpful. Edit: I remembered this video again now 5 months later, and I feel like I understand most of it! I've massively improved in maths in the past few months, and I'm even starting a bachelor's degree in just a few weeks.
I.... My mind is effin' blown. I'm trying to finish my PhD in mechanical engineering this semester, at MIT. This is my 26th semester at MIT, counting previous degrees. And somehow, I had never even heard of geometric algebra until this week. Suddenly, the math I was stuck on for the last two weeks seems trivial. THANK YOU
I've been looking at the Spacetime Algebra formulation for electromagnetics and it gets even crazier. When you add a separate basis vector for time, the charge-current is no longer a multivector with a vector part and scalar part. The charge-current in Spacetime Algebra is a pure vector, with charge being nothing more than the timelike component. Similarly, the electric field gains a timelike factor, turning the electromagnetic field into a single bivector with 6 components in 4D spacetime. Even with these modifications to the formulation, Maxwell's Equation still remains ∇F = J. Because WTF is up with this wizardry‽
To quote William Kingdon Clifford from his founding article on geometric algebra: „ I may, perhaps, therefore be permitted to express my profound admiration of that extraordinary work, and my conviction that its principles will exercise a vast influence upon the future of mathematical science“ Clifford was refering to Graßmann, I would like to extend the compliment to your great video!!!
This video genuinely feels life changing to me. This is like you have finally given access to all the colors as an artist. This is absolutely gorgeous... I'm stunned
Thanks for the assist... I've been working on this for awhile, because I knew it had to exist! Incredible, we can ALL breath now and refocus, the chains are off now friends...
I so wish I'd had explanations like this when studying physics 40 years ago! The intuitions are rendered apparent and the "just because" statements eliminated.
Your teaching style carried me a novice through your presentation, didn't understand hardly anything but somehow had enough grasp to eagerly finish without boredom enjoying every word. Such a powerful tool. Thank you for clarity of concepts, 79 yrs senior, non math major.
Easily the most helpful algebra (mathematics) explanation video I have ever seen on RUclips! I will definitely watch this channel whenever I have a math problem to solve.
Nice presentation. Some remarks as support: 1. Note that multiplication rules for orthonormal vectors (at 14:40) are the same as the rules for the Pauli matrices, which gives a direct connection to quantum mechanics. The Pauli matrices are just a representation of Cl3 (geometric algebra of 3D Euclidean vector space). 2. We can express classical mechanics in Cl3, calculations are much faster (up to 10 times), the Kepler problem can be reduced to a harmonic oscillator problem, we do not need tensors (like inertia tensor), etc. Then, we can use the same mathematics (Cl3) for EM theory (as in this video), the special theory of relativity (we do not need the Minkowski 4D spaces), quantum mechanics (without the imaginary unit), Dirac theory, even the general theory of relativity. Briefly, all main theories of physics with the same language, without imaginary numbers, matrices, and tensors. 3. As for the possibility of generalization and unification, just two examples. a) Once you write down a rotor formula in 3D, you can apply it immediately in, say, 121D. b) In geometric algebra, all integral theorems of the vector calculus (including those from complex analysis) are united in just one theorem (the fundamental theorem of calculus). 4. Geometric algebra likes programming; we almost do not need "if-then-else". 5. Without matrices, but with oriented numbers (k-vectors), geometric algebra gives us a geometric clarity at each step of calculations. This is very important, because it employs the tremendous visualization power that human beings have. 6. Geometric algebra sheds new light on the concept of numbers (oriented numbers). Note that Cl3 contains real numbers, complex numbers (but without the imaginary unit), hyper-complex numbers (say, vectors), dual numbers, quaternions (the even part of Cl3), spinors (they are just quaternions here), etc. See also www.springer.com/gp/book/9783030017552 (a book recommended by Hestenes), www.researchgate.net/publication/343382199_Dirac_theory_in_Euclidean_3D_Geometric_algebra_Cl3
Hello, thank you for the two references. Do you have any reference for in depth explanation of point 2: expressing Classical Mechanics in _Cl_ 3? Thank you in advance, or thanks to anyone contributing to the answer.
I love the fact that you used Tau instead of Pi... these are part of the essence of a revolution in math, where we need to make it shine in its intuitiveness.
This seems like the future of understanding physics. I'm hoping you make more videos about geometric algebra 🤞. Your ability to explain the subject matter has been the best I've found so far. And your use of manim is top notch.
Thank you for the corrections in the description. As someone who's into abstract algebra, some things in the video struck me as strange but intuitive, so I let them slide so that I could get an intuitive sense of what you were discussing. But after the intuitive comes the specifics - and pointing out where intuition goes wrong. And this is exactly what you did in the description! Thank you for the very well-taught broad and intuitive overview of geometric algebra, it seems very fascinating. And for the specificity and footnotes in the description!
This video caused me to go on an unhinged rant to my non-math friends, which is always a fun time. This genuinely blew my mind over and over, and was fascinating and informative. Thank you so much for making this, and I pray to God you make more videos like it
It should be noted that Pascal’s Triangle seems to tell how many of each component there are; in 3D it has 1-3-3-1 scalar-vector-bivector-trivector terms. I find this interesting
I am a nerd,I love science and math, I love learning about things even when I don't need to. I've seen a lot of math videos because of this. I have never felt so invested into a price of media in my entire life. This video has single handed changed my view of geometry. This is a masterpiece
Middle-school teachers aren't the only ones who have criticized the "addition" of scalars and vectors -- it's also been criticized by physicists. David Hestenes has given good explanations (summarized below) of what's involved, but he's also given flippant responses that cause trouble for us GA proponents. I wish the uploader hadn't indicated that we can "just add" scalars and bivectors. Hestenes' "good" explanation is that Grassman and Clifford (the 19th-Century originators of GA) identified and defined a certain operation by means of which the information expressed individually by scalars and bivectors can be combined in a very useful symbolic form. Grassman and Clifford chose to call that operation "addition" (what else?), and to represent it via the symbol "+". As Hestenes explains in _New Foundations for Classical Mechanics_ (2nd edition, pp. 28-29), Grassman himself had balked at the idea of "adding" scalars and bivectors, but finally recognized (too late in life) that ... _"The absurdity [of "adding" scalars and bivectors] disappears when it is realized that [that operation] can be justified in the abstract "Grassmanian" fashion which [originated with Grassman, and] has become standard mathematical procedure today. All that mathematics really requires is that the indicated relations and operations be well defined and consistently employed."_ Similar comments apply to the "addition" of bivectors. To answer the criticisms directed at such ideas, I made the video *Answering Two Common Objections to Geometric Algebra* , and wrote a document entitled *Making Sense of Bivector Addition* (available at vixra). The video is part of my playlist "Geometric Algebra of Clifford, Grassman, and Hestenes". Such topics are also under discussion currently in the LinkedIn group "Pre-University Geometric Algebra".
Mathematicians regularly add things that one might feel should not be added. One can form what is called "formal linear combinations". My whole comment here is to justify this and give some examples. This happens in the constructions of objects similar to that of geometric algebra: the exterior algebra of a vector space gotten by formal linear combinations of exterior products of vectors: en.wikipedia.org/wiki/Exterior_algebra Note that k-form is by definition an alternating multilinear map that takes k vectors and returns a scalar. This means that adding a k-form and a j-form does not really make sense, but it does algebraically (that is, abstractly) if you form direct sums of the vector spaces of all k-forms and all j-forms. As the comment that Jim mentions, this abstract construction has nice properties (it is not done just for abstraction sake) because the exterior algebra formed by all such linear combinations of k-forms for different values of k is interesting. You can also do something similar (adding things that do not seem should be added) when forming the tensor space: en.wikipedia.org/wiki/Tensor_algebra The tensor product of k vectors in V is technically a vector in the k-fold tensor product of V. However, since there is a natural way to multiply tensor products of different sizes, it is natural to put them all together into a large space. This is what is done above in the exterior algebra. This construction with tensor products is actually useful in physics (if the Grassmannian theory is complicated and seemingly unmotivated). If you have a Hilbert spaces H (think roughly: C^n an n dimensional complex vector space, where the unit vectors give you a state of a particle in quantum mechanics). Then you can form the k-fold tensor product of H with itself k times to get the space characterizing k independent particles. Now, if you want to consider the space of all states where you might not know the number of particles, if the particles are allowed to change, or something than you can consider the common construction in mathematical physics: Fock space which is the tensor algebra for H. Here is a video about it: ruclips.net/video/jAw9WMkcCj0/видео.html Formal Linear combinations / Sums: math.stackexchange.com/questions/1029851/understanding-the-meaning-of-formal-linear-combination-and-tensor-product math.stackexchange.com/questions/996556/why-formal-linear-combination Often the term "formal" is used in math. This means that usually the math written has some meaning, but you are to ignore the technicalities and focus on some other aspect of it. A famous example of this is the dirac delta "function", which is an example of physicists using a formal expression for an integral. ruclips.net/video/SQwyLjVQwF8/видео.html ruclips.net/video/SxNVcCVj-3c/видео.html The mathematical theory of Distributions was developed to address "functions" like this. Physicists are currently doing something similar with what is called the Feynman path integral, but we are not currently aware if there is a logical justification of it using mathematics... interesting unsolved question. In knot theory, one sometimes does add things that one "probably should not" add. This is a way of turning combinatorial objects into algebraic objects where invariants are sometimes more easily introduced. For instance, knot theorists form linear combinations c_1v_1 + c_2v_2 where c_1, c_2 belong to Z/2Z. That is, c_1 and c_2 are either 0 or 1 and arithmetic is done modulo 2, i.e. bit arithmetic like in a computer. This is done in such a way that you can form linear combinations of literally anything: 1 square + 1 pentagon. If you don't want to worry about a fraction number of cubes (but have multiplicity), then you form linear combinations with scalars in Z (integers). This is done in de Rham cohomology (the d is the exterior derivative of differential forms, related to the exterior product, and the differential \partial gives you the boundary of the region you are integrating over with some signs attached based on orientation) and the extension of Stokes' Theorem for higher dimensions, as generalizations of The Fundamental Theorem of Calculus in1 dimension, Divergence Theorem in 3 dimensions , Green's Theorem in 2 dimensions, Stokes' Theorem in 3 dimensions. Introduction to Knot Theory: ruclips.net/video/zNffZ3UcARs/видео.html Another example of Z-linear combinations is at 57:23 of ruclips.net/video/aY18jGeim38/видео.html If you don't want to worry about negative numbers, you can form linear combinations over Z/2Z as I introduced it above. You can see ruclips.net/video/RArAHA3Oe7M/видео.html If you start at 42:20 and go to the end of the presentation and keep an eye out for sums of things that do not have a natural sum, then you can see that it fundamental to the construction. He starts with making some paths into an algebra: so you can multiple by adjoining trees and can add formally. Then he does a lot of algebra involving these trees. He later takes an infinite sum of these graphs wen defining "the differential" At 45:23 the lecturer says "My algebra is all over Z/2Z out of some horrible laziness" referring to the idea that algebra involving scalars in Z/2Z is much, much simpler than over Z since arithmetic only involves 2 numbers, as discussed above.
You can't add scalars and vectors. In fact, you can't even add vectors and vectors! Because original definition of addition - is "scalar+scalar", and that's it. Every other "addition" - is in reality a different function. This whole deal always revolved around how we write down stuff.
@@blinded6502 You can't add two vectors, addition is an operation between scalars and scalars, what you should do with vectors is addition. On that hand you can add vectors with vectors, but you can't add two scalars. Clearing that misunderstanding up :)
Master's in electrical engineering here. Absolutely floored. This is a great distillation of a topic that I have been always wary about touching. I am eager to step through this reformulation of electromagnetics to see what I might gain in the new viewpoint. Excellent!
38:00 a current that is moving through time instead if space Yes! This intuition is really missing from so many textbooks and it really makes relativity so much more obvious. Density is a flow in time!
Man, for some reason I shed a tear at the end. What a great video, thanks a LOT for sharing. As an undergrad in physics, this just adds even more motivation to pursue more knowledge. Such a beautiful topic, I will definitely read more on it.
This is by far the most impactful math video I've seen all year. I bought a book for game engine development and it touched on Clifford Algebra, but this video brings everything together. Thank you so much!
Am I right in suggesting that this also perfectly explains why the cross product with Nabla/Del gives us a measure of Curl? It should actually be an outer product which would naturally results in bivectors, that are rotation objects! If I’ve got it right it’s absolutely gorgeous!!
I am totally amazed by this. It was during quantum mechanics when I was thinking to myself that physics just gets too complicated and abstract for my brain to keep up, but in 45 minutes you've shown that with the right language, all natural phenomenon can be expressed in a shockingly simple way. When you pulled out the pauli spin matrices, I was caught totally off guard.
This might be the best math video I've seen on RUclips. Suddenly everything about vectors and complex numbers just clicks and makes perfect sense. I always had a hard time conceptually understanding complex numbers, cross products etc., I was always taught about them in a disjointed way. Now I see it's all a beautiful interconnected web.
I did a postdoc in geometric algebra applications to 5 axis machine milling. We used a version of geometric algebra which no one other than two people in the world use.
I am finishing a dual math/physics masters and I can't believe I've never seen this. My head exploded when the bivector/imaginary number connection was made. I love physics, but math never ceases to amaze me.
You're an incredible teacher. The presentation of this is brilliant and so well laid out! It somehow managed to keep my attention for the entire 45 minutes. I'm a physicist and never had to learn this formalism, but now that I know it it is so useful. Thank you!
You are a GOD amongst men !!! i appreciate your time and your mathematical vizualisation capabilities and your explanation of pure maths !! god bless you my brother !!!!
I have read up on these topics for years and never really understood them fully. This was the most informative thing I've ever encountered on any of these topics! WELL DONE!
As a Physics student my mind was blown again and again during this video... Knowing a mathematical object that connects so many physical concepts (pseudo scalar, cross product, Pauli matrices, spinor...) that seem to be unrelated is just amazing
i major physics, and recently been wanting to study some math. My math nerd friend suggested me to start from geomtric algebra, and i came across this video. Almost had heart attack when the scalar adding vector shows up, but after that, every his "wait a minute" blows my mind. This is such a great video! and I could never view Maxwell equation the same way again.
This video was incredible; truly a service to humanity. I hope someday we start teaching these concepts to much younger students so they can absorb the unity of geometry, algebra, and physics while their minds are still plastic. It really makes all obscure concepts that students struggle with so much more intuitive! You've changed the way I think with this video, I can't thank you enough. I wish you nothing but good things :)
Looks like a proper formalism to describe both, field theories and quantum mechanics. And it's so easy and illustrative that it even could be taught at high school.
This is the best thing ever happened. I literally watched entire video with goose bumps. I desperately needed some strong unification of electrodynamics under vector calculus and this has just made my life probably. Thank you so much 💓
All my physics classes and struggles flashed before my eyes lamenting how simple could it be if a knew this elegant and beautiful representation of geometric algebra.
Some videos manage to blow your mind. A small subset even does so twice. But this video is unique: it blows your mind at LEAST five times. Thank you sir for teaching a mildly aging engineer/associate professor an INCREDIBLE lesson ❤
Thanks for this. I have a bachelors in electrical engineering so I was familiar with some of this, but this video brings it all together beautifully in a way that makes me wonder why it isn't taught in college this way.
this video has inspired me so much, i think the point math existing is that it can simplify the real world problems into numbers, looking at the complex calculations of vectors that the school taught, and plenty of functions, this is a much elegant way of expressing
Absolutely mind blowing, this has fundamentally altered my perspective of physics math and how they relate. Please keep doing what your doing, maybe even delve into some of these topics like the Pauli matrices ect.
Hands down, this has been perhaps the best youtube video I have seen in the past 20 years. So many Ahhha moments! Beautifully explaining the original of many relationships that I took for granted or didn't understand where they come from.
My god. I remember watching this video 2 years ago and thinking "hey thats pretty cool". Now I'm doing a masters degree in mathematics, trying to wrap my head around differential geometry, and seeing the way you've explained the wedge product has blown my mind, and helped me through this absolute slog of a degree 😂. Many thanks for this amazing video
Thank you SOOOOO much for this video! I'm currently taking a differential geometry course and this video gave me insanely valuable intuitions for working with differential forms!
This is so amazing!!! I have a more or less basic knowledge on vectors, matrices, complex numbers and quaternions and never heard about geometric algebra, this is beautiful!
I watched this video for over 1 hour, because my mind get getting blown and I needed a few minutes to appreciate the simplicity in this mathematical approach.
This is a damn good video! I had to pause, go back 20 seconds, and rewatch several times, and each time I would end up understanding what was going on. I'll be revisiting this video for sure; you did an amazing job.
This is way too much information all at once and I love it. Geometric Algebra seems to provide highly useful results for facilitating conceptual understanding in physics and it seems like one of the most profound connection networks for multiple crucial fields of mathematics. Amazing video, I'll be looking forward to diving head first into the intricacies of this field in the future!
You've sent me on an absolutely wild goose chase. I've found lectures by David Hestenes and worked through his books (or at least tried). Thank you for introducing this!
for so many videos in RUclips about Clifford algebra I watched so far, this one is the most comprehensive one without starting the algebra space based on coordinate and basis algorithm, This approach shows the nutshell clearly of the Clifford algebra and it’s powerful influence in both maths and physics
I like to think of blades geometrically by extending the geometric description of vectors. We identify a vector *v* by its magnitude and direction. That is, the geometric properties | *v* |, Span{ *v* }, and a choice of orientation (+/-) determine *v* . In this view, a vector is an oriented quantity associated with a particular linear subspace. This is readily extended to all k-blades B = *v1* ^ ... ^ *vk* as an oriented quantity ±|B| associated with a linear subspace Span{B} = Span{ *v1* , ..., *vk* }. I feel like this perspective is more conceptually flexible, as blades are used to represent a variety of geometric and physical relationships, such as areas/volumes/etc., rotations, torques, and even linear subspaces themselves. Thinking of blades generally as quantities associated with a subspace, rather than something more specific like an oriented area/volume/parallelepiped/etc., helps unify these seemingly disparate ideas.
@@hyperduality2838 stop spamming this duality junk everywhere, it isn’t actually insightful. There are things about dualities that can be said and which are insightful, but none of the things I’ve seen you say about (alleged) dualities are at all insightful, and the ones that aren’t trivial are generally wrong. If you want to say some somewhat more interesting things about dualities, learn some category theory at least to the point where you understand the precise sense in which products and coproducts are dual concepts. There are valuable things to say about dualities, but the stuff you have been saying just sounds like someone who maybe did some recreational drugs and mistakenly concluded that they gained some great insight.
I finally "Grokked" Geometric Algebra thanks to your visual presentation. I don't even think 3Blue1Brown could enlighten Geometric Algebra this well. Many thanks!
@@depression_isnt_real Don't get me wrong: 3b1b videos are amazing, but what's even better is that he made the tool he created for creating those videos open source. I can't wait to see how many great math videos in the 3b1b style like this one will be out there in a few years created by people like you, inspired by 3b1b. I believe that this is 3b1b's biggest gift, even bigger than his videos.
i literally cried After watching this video and i NEVER cry, this was a masterpiece and geometric algebra Is One of the most beautiful things i've ever seen, you did a fantastic job in explaining It and you probably changed my Life, I'm about to start a physics university and this Will surely help me SO MUCH, thank you!!!
I am a physics major and I sincerely thank you for your great work on this video. You solve my confusion lasting for many years when I encounter with multi-dimensional calculus. Although sometimes we just get used to the "rules" we maneuver the terms, I somehow feel uncomfortable to the interpretation of the equations. The concept of geometric algebra you introduced to me just take off the frame which trapping my mind for so long. I wish I could see this video earlier.
I'd like very much to see treating the Navier-Stokes full equation set (mass, momentum and energy) by this tool. The tensor treatment of those equations is too tedious.
It is just the missing lesson in the university, which I wanted to know for years, connecting linear algebra to higher level math concepts and dozens of physics theories.
I just realized a stupid mistake I made: at 8:20, this should actually be that the inner product is this length times ||u||. I don't know how I didn't notice this in all of my times watching this video while making it.
EDIT: There is now an addendum to this video here: ruclips.net/video/0bOiy0HVMqA/видео.html. It clears up a few misleading things in this video and then focuses on actually understanding the geometric product.
EDIT 2: Another correction: at 29:00, the equation should be i a · (b × c) = a ∧ b ∧ c, not a · (b × c) = i a ∧ b ∧ c.
No worries You got your point through...that is more than enough, Thanks a lot for such an explanation. I want to apply geometric algebra to viscoelastic equations. Whenever viscosity is introduced in any media, it is associated with a complex number and this results in the rotation of the velocity vector. Please let me know if you are interested in Collaborating.
I wish I could upvote thousand of times. You are a hero thank you for the great work. I have tried to understand this for like 10 years and I always thought 3d rotation is a magic box just copy&paste and it works, however, it works don't even try to think about it but after seeing rotors as bivectors I can understand what all these were about. Thank you a lot keep it up this is the only channel I have clicked the bell icon,
@@daemond8093 I would like any public updates from your collaboration with sudgylacmoe....if that’s possible 👍
I made exactly that same mistake once while I was showing it to someone. I think it is because so often you are used to working with an orthonormal basis that you go into auto pilot.
can make a video about "pseudovector and pseudoscalar"?
Every "wait a minute" I have to pause to wipe tears from my eyes so I can see the screen
Does doing this happen to take about a minute?
"wait a minute"
>takes a shot
All I think of is Kazoo Kid
This video is like deeply unsettling
SAME
There’s something really beautiful about the fact that simply choosing to ignore “you can’t multiply vectors” leads to ALL OF PHYSICS
Makes one wonder if 19th century mathematicians or physicists were too spooked by Clifford algebras that it took another century for someone to rediscover its utility.
@@zemoxian Clifford died when he was 35 years old, so that might have been one of the reasons.
As soon as you brought up "i", I thought "he hasn't..." but as the vectors were rotated through 90° I was like "he is!!!" then you dropped the big reveal and my face was all Ö
That was stunning and brave! Bravo!
geometric algebra feels like something that was missing from mathematics, it kinda explain things that were weird but proven, now everything makes sense
From as far as I got... It does have some interesting things to it. It does make for a more compact approach too. The only thing I'm worried about is dimensionality of the entities being geometrically produced(?). There are quite a few things in physics that have weird behavior because the way we handle them is mathematically strange, but you get used to it. I knew I could use the external product instead of the cross product and get much the same things out, but the geometric product solves some notation problems we noticed but had to fix afterwards.
I work in applied electromagnetism and never learned so much in 45 min. You've made sense out of concepts that were always just out of reach. You've changed my life
Share it with others at work
Right?!
There’s a lot of great literature about this field and it’s applications. Check out Geometric Algebra for Electrical Engineers or Geometric Algebra for Physicists.
WHAT THE FUCK IS APPLIED ELECTROMAGNETISM
IM STUDYING ELECTRICAL ENGINEERING AND ID BETTER NOT HAVE TO LEARN THIS TO UNDERSTAND MAXWELL
@@IanBLacy Applied Electromagnetism is probably in reference to transmission line theory and similar radio applications. You don't need it to understand Maxwell, in fact it's in reverse. You need Maxwell to understand transmission line theory and radio frequency circuits. However, you might not see it depending on your undergraduate curriculum.
I don’t normally comment on RUclips, but as someone trying to understand higher level math concepts, this is one of the best videos for education I’ve ever seen, at least for the way my brain works; thank you so much!
@@hyperduality2838 who are u
@@olegt962 E=MC2 IS F=ma, as time is NECESSARILY possible/potential AND actual IN BALANCE; AS ELECTROMAGNETISM/energy is gravity. This explains the term c4 from Einstein's field equations. Time dilation ultimately proves ON BALANCE that E=MC2 IS F=ma, AS ELECTROMAGNETISM/ENERGY IS GRAVITY. Gravity IS ELECTROMAGNETISM/energy. Balance and completeness go hand in hand. It all CLEARLY makes perfect sense.
By Frank DiMeglio
If you want to understand a higher level of mathematics you need to study the book of “advanced engineering mathematics”
That is about vectors in 2d, 3d; double and triple integrals, surface area, Curve, Green’s theorem, Stokes theorem, surface integrals,, Gradient, differentiate equation in different orders....that would be higher levels. Good luck with your study 📚
@@cht5086 I think he meant higher level as more abstract
I can recommend you 3blue1brown. His vids are even better.
Imaginary Numbers being Pseudo-Scalars is better plot twist then some movies 😂
The fact that geometric algebra and algebraic geometry are completely different things just proves that mathematical language is non-Abelian.
Very😂
I think it's just not commutative like those multivector in geometric algebra
@@appybane8481 that’s what non-abelian means
The relevant xkcd here is 2028 (title text).
I always had this problem with torque and angular momentum. I remember how after a Classical Mechanics class, I started searching for a more profound meaning to those concepts, since I was already in third year of Physics and still got no feeling of understanding the actual concept completely.
Now I watched this video, and all of a sudden my soul can finally rest. I mean, even Feynman, when trying to explain the conservation of angular momentum, talked about how the area was preserved. It makes total sense.
Torque, angular momentum, area, volume... all bi and tri-vectors. For regular vector algebra, it felt like, directionality is property that appears and disappears at will. Why the hell an area is a vector, volume is not? This makes perfect sense.
@@daniyelplainview Yeah, volume isn't because trivectors in 3D are pseudoscalars.
Can anyone link where Feynman talks about that (or tell me what to search bc links on youtube are usuallly a no-no)?
@AKAMI
Probably “Feynman’s lost lecture” on Kepler’s laws .
After learning GA, I've started using the term "Kepler's Law of Quantum Mechanics" in a few places, because bivectors as areas (while not actually the most natural interpretation of them from what I've seen) trivially equates Kepler's Law to the Law of Conservation of Angular Momentum. Treating bivectors as areas also gives another interpretation behind the angle doubling: exp(aB) for a unit bivector B gives a rotor that covers a sector of the unit circle with an area of a. This way of viewing the bivector exponential extends to hyperbolic rotations and translations as well.
Just realised that the number of components of a K-vector is described by the Kth row of Pascal’s triangle. Beautiful
wowww whattt
I swear I haven't had more surprises in 44 minutes in my entire life! This is one of the most underrated video on all of RUclips.
This sounds like initiation in a cult, and i'm all for it.
To become truly initiated you must learn the octonions.
Many mathematicians see geometric algebra as a cult, so you are right :D
@@Utesfan100 actually.... at 3D, it's not really quaternions. Quaternions are missing the 3D 1D vectors and pseudo-scalar. Octonions are a different direction entirely; a wrong-turn IMHO. As you go into higher dimensions with GA, octonions are not a thing you run into. Octonions are not associative; etc. I think the O(2^n) growth of the calculations as dimensions go up is why we did not go this direction when it was discovered. It's almost too tedious without computer assistance.
Lol - good initiation. I am all for it as well.
@@robfielding8566 I was merely attempting to point out that if you think geometric algebra has a cult following, you should look to the composition algebras :)
As a devotee, I am compelled to address your comments more fully.
I concede that the ability of vector notation to scale indefinitely lead to its wide adoption across all of mathematics.
Which is a more fitting property for a geometric algebra, associativity or compatibility of the product with length? This latter property is what defines a composition algebra.
Merab Gogberashvili has done some interesting things using the split-octonions in place of Cl(0,3), analogous to paravectors. This algebra is the unique real composition algebra that contains Minkowski 3+1 space. (There is even a Cauchy integral formula for this algebra.)
The relationship between the octonions and the exceptional Lie groups is deep. If one encounters the latter, they would be well served to understand the former. Speaking of cult followings, E_8 comes to mind :)
While not a Clifford algebra, the composition property gives the octonions a very geometric flavor. Indeed, Advances in Applied Clifford Algebras frequently includes articles dedicated to the octonions, suggesting they are at least in the same family of algebras.
Even if just as the crazy, eccentric uncle.
I'm completely amazed. I found a Wikipedia article on Geometric Algebra a while ago, seemed interesting, but I didn't quite get it. The past few days I've been working with quaternions and now your video comes along. I almost cried when the "i" entered the stage, it's all so incredibly natural. Thanks for blowing my mind.
Same!
I actually paused the video to laugh for a good minute when 'i' appeared in the 2D case, then again when the comparison to quaternions appeared, then several times again for Maxwell's equation(s). For some reason this is funnier than comedy.
Sietse
Have got to the point of expressing the rate of change of unitary quaternions to the angular velocity pseudo vector of the rotating body? Essential for attitude determination.
@@animowany111 For me, comedy is when you really should've seen something coming, but didn't. Seeing familiar elements pop up unexpectedly when doing something completely different and not noticing at first definitely counts as comedy in my eyes.
@Paul Wolf Except with Imaginary numbers, linear algebra, calculus. Also why you even need some explanation, is math you just define something and works with it, unless it is useless or contradictory, who cares about explanations?
I literally shouted when spinors came up out of no where. This is an absolutely amazing algebra and I am excited to see more of it.
Wait a minute ,that double turn reminds me off something ...?!daaaamn...quantum mechanics...
If you like that, try Geometric Calculus.
Lmao me too
@@ixion2001kx76 Oh my gosh are we about to learn in a natural way why derivatives, integrals, and infinitesimals work
Sudgy: *casually starts using tau
Me: wait a minute
RUclips wouldn’t shut up with recommending me this video. And now I know why. This is a freaking gold mine. Nice work 👍
I have noticed many people has had the same actual reaction that I had. I really cried, and I didn't understand why at first.
In my case I believe it is because I experienced something so potent that it almost dwarves everything I know in terms of physics. I can't help but feel for something so beautiful, so deep that my soul finally rests on knowing that those doubts I had and my inability to sometimes grasp concepts is perhaps justified. I am almost finishing mechanical engineering and I really felt there was still too much to many knowledge I lacked, not because of not knowing the existence of such topics but feeling I wasn't on top of many subjects to the degree my curiosity demands.
This was only mentioned briefly but the connection between tensors and complex numbers was something not even in my wildest dreams I could have come up and here it is JUST SO NATURAL. I always felt there was something missing from my education and I have looked desperately everywhere. I can't even begin to explain how relieved I am to finally arrive at this, not because it has solved all of my doubts, but because it gives me hope that I can perhaps eventually solve them.
It's a light on a path I thought was shoruded in darkness. I just want to scream, we should all have been introduced to this as kids. And I believe all of the physics education programme should be built around it. It's something you can immediately and intuitively relate to.
♡♡♡
Well said. But the reason that the subjects are not taught to everyone isn't that they are too hard. They are kept difficult precisely because the powers that be are not interested in a large population of mathematically capable individuals who are not under surveillance by institutions of government, education, or corporation. National security is why we cannot have nice things.
This video is not a part of your "Favorites" playlist.
Why is this?
I don't usually leave a positive comment on a video, but I'm making an exception. This is a masterpiece, the knowledge to produce this must be so utter. I haven't seen a book or video about the topic more enlightening. My most sincere congratulations, you really have a gift for this, consider making a career out of it.
I love that Maxwell's equation is basically the same as the tensor notion, which also condenses the 4 equations down to 1. It would be cool to see how geometric algebra relates to tensors
bivectors are also antisymmetric tensors
Multivectors are antisymmetric tensors.
Michael Penn has a video about Clifford Algebras and there he describes them as subalgebras of Tensor Algebras, and I don't know what Tensors are, they look too terrifying to me. One my friend studied tensors in latest years of normal high school. Now, after 4 years of university he still doesn't know what Tensors are. He is smart. He understands really complicated math. He studies cybernetics and applies for the job in the best it firm of our country. But he didn't get the Tensors. This terrified me.
@@РайанКупер-э4о no need to be scared, just like how the video described Geometric Algebra as a ‘special rule’ for multiplying (multi)vectors, tensors are a ‘special rule’ for transforming (nested)matrices. It’s convenient result that these transformation rules can also be written as the nested matrices that they transform
Best set of videos on the algebras I have thus far viewed. congrats. Very easy to understand and presented clearly. Viewer like me feel rewarded after viewing and very grateful
this feels like a video that would be unimaginably useful in the future, but now I just lost it around the 20 minute mark. It feels as if I will return to this in the future, and find it really helpful.
Edit: I remembered this video again now 5 months later, and I feel like I understand most of it! I've massively improved in maths in the past few months, and I'm even starting a bachelor's degree in just a few weeks.
How's the bachelor's going :)
@@julesk1088 nice! I got 89 on linear algebra and I'm right now doing set theory
@@smorcrux426 good job :)
@@eldattackkrossa9886 thanks! My semester is starting in just a few weeks and this time I'm taking some computer science courses and real analysis
@@smorcrux426 hey dude just here to say best of luck for you courses!
I.... My mind is effin' blown. I'm trying to finish my PhD in mechanical engineering this semester, at MIT. This is my 26th semester at MIT, counting previous degrees. And somehow, I had never even heard of geometric algebra until this week. Suddenly, the math I was stuck on for the last two weeks seems trivial. THANK YOU
"Charge density is a current moving through time"
Damn
LMAO
That will definitely take me some time to process it in all its implications
no, no, he’s got a point
I've been looking at the Spacetime Algebra formulation for electromagnetics and it gets even crazier. When you add a separate basis vector for time, the charge-current is no longer a multivector with a vector part and scalar part. The charge-current in Spacetime Algebra is a pure vector, with charge being nothing more than the timelike component.
Similarly, the electric field gains a timelike factor, turning the electromagnetic field into a single bivector with 6 components in 4D spacetime. Even with these modifications to the formulation, Maxwell's Equation still remains ∇F = J. Because WTF is up with this wizardry‽
@@angeldude101 cool facts! thanks for sharing
To quote William Kingdon Clifford from his founding article on geometric algebra: „ I may, perhaps, therefore be permitted to express my profound admiration of that extraordinary work, and my conviction that its principles will exercise a vast influence upon the future of mathematical science“ Clifford was refering to Graßmann, I would like to extend the compliment to your great video!!!
This video genuinely feels life changing to me. This is like you have finally given access to all the colors as an artist. This is absolutely gorgeous... I'm stunned
Thanks for the assist... I've been working on this for awhile, because I knew it had to exist! Incredible, we can ALL breath now and refocus, the chains are off now friends...
37:00 it actually IS the traditional gradient if you use relativistic coordinates (ct,x,y,z)
plus the identification of scalars and vectors pointing in the time direction
the d'lambertian, right?
I so wish I'd had explanations like this when studying physics 40 years ago! The intuitions are rendered apparent and the "just because" statements eliminated.
Your teaching style carried me a novice through your presentation, didn't understand hardly anything but somehow had enough grasp to eagerly finish without boredom enjoying every word. Such a powerful tool.
Thank you for clarity of concepts, 79 yrs senior, non math major.
Easily the most helpful algebra (mathematics) explanation video I have ever seen on RUclips! I will definitely watch this channel whenever I have a math problem to solve.
Nice presentation.
Some remarks as support:
1. Note that multiplication rules for orthonormal vectors (at 14:40) are the same as the rules for the Pauli matrices, which gives a direct connection to quantum mechanics. The Pauli matrices are just a representation of Cl3 (geometric algebra of 3D Euclidean vector space).
2. We can express classical mechanics in Cl3, calculations are much faster (up to 10 times), the Kepler problem can be reduced to a harmonic oscillator problem, we do not need tensors (like inertia tensor), etc. Then, we can use the same mathematics (Cl3) for EM theory (as in this video), the special theory of relativity (we do not need the Minkowski 4D spaces), quantum mechanics (without the imaginary unit), Dirac theory, even the general theory of relativity. Briefly, all main theories of physics with the same language, without imaginary numbers, matrices, and tensors.
3. As for the possibility of generalization and unification, just two examples. a) Once you write down a rotor formula in 3D, you can apply it immediately in, say, 121D. b) In geometric algebra, all integral theorems of the vector calculus (including those from complex analysis) are united in just one theorem (the fundamental theorem of calculus).
4. Geometric algebra likes programming; we almost do not need "if-then-else".
5. Without matrices, but with oriented numbers (k-vectors), geometric algebra gives us a geometric clarity at each step of calculations. This is very important, because it employs the tremendous visualization power that human beings have.
6. Geometric algebra sheds new light on the concept of numbers (oriented numbers). Note that Cl3 contains real numbers, complex numbers (but without the imaginary unit), hyper-complex numbers (say, vectors), dual numbers, quaternions (the even part of Cl3), spinors (they are just quaternions here), etc.
See also
www.springer.com/gp/book/9783030017552 (a book recommended by Hestenes),
www.researchgate.net/publication/343382199_Dirac_theory_in_Euclidean_3D_Geometric_algebra_Cl3
Hello, thank you for the two references.
Do you have any reference for in depth explanation of point 2: expressing Classical Mechanics in _Cl_ 3? Thank you in advance, or thanks to anyone contributing to the answer.
@@ChaineYTXF Hestenes: New Foundations for Classical Mechanics
@@miroslavjosipovic5014 thank you very much. Hestenes seems at the heart of the debate when discussing geometric algebra. I'll look it up.
@@ChaineYTXF Hestenes revived GA.
I love the fact that you used Tau instead of Pi... these are part of the essence of a revolution in math, where we need to make it shine in its intuitiveness.
This seems like the future of understanding physics. I'm hoping you make more videos about geometric algebra 🤞. Your ability to explain the subject matter has been the best I've found so far. And your use of manim is top notch.
I studied math & electrical engineering in the late 70s. This approach to the curriculum would have been a godsend. Dang.
This is the best math video in RUclips. Every single "wait a second" had me feeling shock and joy. Everything just falls out so naturally!
I can't even express with words how mindblown this video left me. Amazing work, the concepts made sense so easily.
This video blew my mind, again and again and again. Thank you so much for this beautiful presentation
Thank you for the corrections in the description. As someone who's into abstract algebra, some things in the video struck me as strange but intuitive, so I let them slide so that I could get an intuitive sense of what you were discussing. But after the intuitive comes the specifics - and pointing out where intuition goes wrong. And this is exactly what you did in the description! Thank you for the very well-taught broad and intuitive overview of geometric algebra, it seems very fascinating. And for the specificity and footnotes in the description!
This video caused me to go on an unhinged rant to my non-math friends, which is always a fun time. This genuinely blew my mind over and over, and was fascinating and informative. Thank you so much for making this, and I pray to God you make more videos like it
It should be noted that Pascal’s Triangle seems to tell how many of each component there are; in 3D it has 1-3-3-1 scalar-vector-bivector-trivector terms. I find this interesting
It's easy to prove this connection. Have a go at it !
Yea the link is combinatorics. E.g we have 3 choose 2 ways to get our bivectors in 3D.
It's the same pattern when multiplying polynomial equations of the same order.
I am a nerd,I love science and math, I love learning about things even when I don't need to. I've seen a lot of math videos because of this. I have never felt so invested into a price of media in my entire life. This video has single handed changed my view of geometry. This is a masterpiece
Subscribed.
Middle school teacher: you can’t add scalar and vectors
Mathematicians: why the hell not
Middle-school teachers aren't the only ones who have criticized the "addition" of scalars and vectors -- it's also been criticized by physicists. David Hestenes has given good explanations (summarized below) of what's involved, but he's also given flippant responses that cause trouble for us GA proponents. I wish the uploader hadn't indicated that we can "just add" scalars and bivectors.
Hestenes' "good" explanation is that Grassman and Clifford (the 19th-Century originators of GA) identified and defined a certain operation by means of which the information expressed individually by scalars and bivectors can be combined in a very useful symbolic form. Grassman and Clifford chose to call that operation "addition" (what else?), and to represent it via the symbol "+". As Hestenes explains in _New Foundations for Classical Mechanics_ (2nd edition, pp. 28-29), Grassman himself had balked at the idea of "adding" scalars and bivectors, but finally recognized (too late in life) that ...
_"The absurdity [of "adding" scalars and bivectors] disappears when it is realized that [that operation] can be justified in the abstract "Grassmanian" fashion which [originated with Grassman, and] has become standard mathematical procedure today. All that mathematics really requires is that the indicated relations and operations be well defined and consistently employed."_
Similar comments apply to the "addition" of bivectors. To answer the criticisms directed at such ideas, I made the video *Answering Two Common Objections to Geometric Algebra* , and wrote a document entitled *Making Sense of Bivector Addition* (available at vixra). The video is part of my playlist "Geometric Algebra of Clifford, Grassman, and Hestenes". Such topics are also under discussion currently in the LinkedIn group "Pre-University Geometric Algebra".
For a mathematician addition is just a binary operation(which is commutative most of the time).In that sense in any given set you can add things.
Mathematicians regularly add things that one might feel should not be added. One can form what is called "formal linear combinations".
My whole comment here is to justify this and give some examples.
This happens in the constructions of objects similar to that of geometric algebra: the exterior algebra of a vector space gotten by formal linear combinations of exterior products of vectors:
en.wikipedia.org/wiki/Exterior_algebra
Note that k-form is by definition an alternating multilinear map that takes k vectors and returns a scalar. This means that adding a k-form and a j-form does not really make sense, but it does algebraically (that is, abstractly) if you form direct sums of the vector spaces of all k-forms and all j-forms.
As the comment that Jim mentions, this abstract construction has nice properties (it is not done just for abstraction sake) because the exterior algebra formed by all such linear combinations of k-forms for different values of k is interesting.
You can also do something similar (adding things that do not seem should be added) when forming the tensor space:
en.wikipedia.org/wiki/Tensor_algebra
The tensor product of k vectors in V is technically a vector in the k-fold tensor product of V. However, since there is a natural way to multiply tensor products of different sizes, it is natural to put them all together into a large space. This is what is done above in the exterior algebra.
This construction with tensor products is actually useful in physics (if the Grassmannian theory is complicated and seemingly unmotivated).
If you have a Hilbert spaces H (think roughly: C^n an n dimensional complex vector space, where the unit vectors give you a state of a particle in quantum mechanics). Then you can form the k-fold tensor product of H with itself k times to get the space characterizing k independent particles.
Now, if you want to consider the space of all states where you might not know the number of particles, if the particles are allowed to change, or something than you can consider the common construction in mathematical physics: Fock space which is the tensor algebra for H.
Here is a video about it:
ruclips.net/video/jAw9WMkcCj0/видео.html
Formal Linear combinations / Sums:
math.stackexchange.com/questions/1029851/understanding-the-meaning-of-formal-linear-combination-and-tensor-product
math.stackexchange.com/questions/996556/why-formal-linear-combination
Often the term "formal" is used in math. This means that usually the math written has some meaning, but you are to ignore the technicalities and focus on some other aspect of it.
A famous example of this is the dirac delta "function", which is an example of physicists using a formal expression for an integral.
ruclips.net/video/SQwyLjVQwF8/видео.html
ruclips.net/video/SxNVcCVj-3c/видео.html
The mathematical theory of Distributions was developed to address "functions" like this.
Physicists are currently doing something similar with what is called the Feynman path integral, but we are not currently aware if there is a logical justification of it using mathematics... interesting unsolved question.
In knot theory, one sometimes does add things that one "probably should not" add. This is a way of turning combinatorial objects into algebraic objects where invariants are sometimes more easily introduced.
For instance, knot theorists form linear combinations c_1v_1 + c_2v_2 where c_1, c_2 belong to Z/2Z. That is, c_1 and c_2 are either 0 or 1 and arithmetic is done modulo 2, i.e. bit arithmetic like in a computer.
This is done in such a way that you can form linear combinations of literally anything:
1 square + 1 pentagon.
If you don't want to worry about a fraction number of cubes (but have multiplicity), then you form linear combinations with scalars in Z (integers). This is done in de Rham cohomology (the d is the exterior derivative of differential forms, related to the exterior product, and the differential \partial gives you the boundary of the region you are integrating over with some signs attached based on orientation) and the extension of Stokes' Theorem for higher dimensions, as generalizations of The Fundamental Theorem of Calculus in1 dimension, Divergence Theorem in 3 dimensions , Green's Theorem in 2 dimensions, Stokes' Theorem in 3 dimensions.
Introduction to Knot Theory:
ruclips.net/video/zNffZ3UcARs/видео.html
Another example of Z-linear combinations is at 57:23 of
ruclips.net/video/aY18jGeim38/видео.html
If you don't want to worry about negative numbers, you can form linear combinations over Z/2Z as I introduced it above. You can see
ruclips.net/video/RArAHA3Oe7M/видео.html
If you start at 42:20 and go to the end of the presentation and keep an eye out for sums of things that do not have a natural sum, then you can see that it fundamental to the construction. He starts with making some paths into an algebra: so you can multiple by adjoining trees and can add formally. Then he does a lot of algebra involving these trees. He later takes an infinite sum of these graphs wen defining "the differential"
At 45:23 the lecturer says "My algebra is all over Z/2Z out of some horrible laziness" referring to the idea that algebra involving scalars in Z/2Z is much, much simpler than over Z since arithmetic only involves 2 numbers, as discussed above.
You can't add scalars and vectors. In fact, you can't even add vectors and vectors!
Because original definition of addition - is "scalar+scalar", and that's it.
Every other "addition" - is in reality a different function.
This whole deal always revolved around how we write down stuff.
@@blinded6502 You can't add two vectors, addition is an operation between scalars and scalars, what you should do with vectors is addition. On that hand you can add vectors with vectors, but you can't add two scalars.
Clearing that misunderstanding up :)
This helps you see in pictures. You are able to connect different parts of physics with ease.
Master's in electrical engineering here. Absolutely floored. This is a great distillation of a topic that I have been always wary about touching. I am eager to step through this reformulation of electromagnetics to see what I might gain in the new viewpoint. Excellent!
At 22 minutes in I've seen every single formula before and had no idea they were so related. You are a teaching wizard.
38:00 a current that is moving through time instead if space
Yes! This intuition is really missing from so many textbooks and it really makes relativity so much more obvious. Density is a flow in time!
Density is negative mass
Man, for some reason I shed a tear at the end.
What a great video, thanks a LOT for sharing. As an undergrad in physics, this just adds even more motivation to pursue more knowledge.
Such a beautiful topic, I will definitely read more on it.
This is by far the most impactful math video I've seen all year. I bought a book for game engine development and it touched on Clifford Algebra, but this video brings everything together. Thank you so much!
Am I right in suggesting that this also perfectly explains why the cross product with Nabla/Del gives us a measure of Curl? It should actually be an outer product which would naturally results in bivectors, that are rotation objects! If I’ve got it right it’s absolutely gorgeous!!
Yep!
I am totally amazed by this. It was during quantum mechanics when I was thinking to myself that physics just gets too complicated and abstract for my brain to keep up, but in 45 minutes you've shown that with the right language, all natural phenomenon can be expressed in a shockingly simple way. When you pulled out the pauli spin matrices, I was caught totally off guard.
This might be the best math video I've seen on RUclips. Suddenly everything about vectors and complex numbers just clicks and makes perfect sense. I always had a hard time conceptually understanding complex numbers, cross products etc., I was always taught about them in a disjointed way. Now I see it's all a beautiful interconnected web.
I did a postdoc in geometric algebra applications to 5 axis machine milling. We used a version of geometric algebra which no one other than two people in the world use.
I am finishing a dual math/physics masters and I can't believe I've never seen this. My head exploded when the bivector/imaginary number connection was made. I love physics, but math never ceases to amaze me.
You're an incredible teacher. The presentation of this is brilliant and so well laid out! It somehow managed to keep my attention for the entire 45 minutes. I'm a physicist and never had to learn this formalism, but now that I know it it is so useful. Thank you!
You are a GOD amongst men !!! i appreciate your time and your mathematical vizualisation capabilities and your explanation of pure maths !! god bless you my brother !!!!
I have read up on these topics for years and never really understood them fully. This was the most informative thing I've ever encountered on any of these topics! WELL DONE!
My mind is blown. Time to extend what I learned with this.
Edit: why is this not taught in my physics program? This is so important!
please consider making part 2 with physics
As a Physics student my mind was blown again and again during this video...
Knowing a mathematical object that connects so many physical concepts (pseudo scalar, cross product, Pauli matrices, spinor...) that seem to be unrelated is just amazing
Brilliant. Having ventured into Manim myself, the effort in this shows, thank you.
i major physics, and recently been wanting to study some math. My math nerd friend suggested me to start from geomtric algebra, and i came across this video. Almost had heart attack when the scalar adding vector shows up, but after that, every his "wait a minute" blows my mind. This is such a great video! and I could never view Maxwell equation the same way again.
This video was incredible; truly a service to humanity. I hope someday we start teaching these concepts to much younger students so they can absorb the unity of geometry, algebra, and physics while their minds are still plastic. It really makes all obscure concepts that students struggle with so much more intuitive! You've changed the way I think with this video, I can't thank you enough. I wish you nothing but good things :)
Looks like a proper formalism to describe both, field theories and quantum mechanics. And it's so easy and illustrative that it even could be taught at high school.
This video is an instant classic. I'll definitely revisit this later and I hope you create more mathematics/physics related content. Got a new sub :)
kac yasindasin
Ehh
He man checked your playlist on mathematics really awesome ✨✨✨
@@ramansb8924 I didn't think anyone would see that :) Glad you liked it!
@@mami42g 🙌✨
why, why wasn't i taught this in my physics courses, this is eye opening, everithing pops up so naturally
This is the best thing ever happened. I literally watched entire video with goose bumps. I desperately needed some strong unification of electrodynamics under vector calculus and this has just made my life probably. Thank you so much 💓
All my physics classes and struggles flashed before my eyes lamenting how simple could it be if a knew this elegant and beautiful representation of geometric algebra.
I have been looking at Geometric Algebra for a couple of days now and this is by far the best first introduction I have seen so far. Thanks!
Some videos manage to blow your mind. A small subset even does so twice. But this video is unique: it blows your mind at LEAST five times.
Thank you sir for teaching a mildly aging engineer/associate professor an INCREDIBLE lesson ❤
Thanks for this. I have a bachelors in electrical engineering so I was familiar with some of this, but this video brings it all together beautifully in a way that makes me wonder why it isn't taught in college this way.
this video has inspired me so much, i think the point math existing is that it can simplify the real world problems into numbers, looking at the complex calculations of vectors that the school taught, and plenty of functions, this is a much elegant way of expressing
Absolutely mind blowing, this has fundamentally altered my perspective of physics math and how they relate. Please keep doing what your doing, maybe even delve into some of these topics like the Pauli matrices ect.
Hands down, this has been perhaps the best youtube video I have seen in the past 20 years. So many Ahhha moments! Beautifully explaining the original of many relationships that I took for granted or didn't understand where they come from.
Absolute banger of a video my guy
Definitely unified many concepts I’ve studied in my physics degree and in a super concise/simple/EFFICIENT way!
My god. I remember watching this video 2 years ago and thinking "hey thats pretty cool". Now I'm doing a masters degree in mathematics, trying to wrap my head around differential geometry, and seeing the way you've explained the wedge product has blown my mind, and helped me through this absolute slog of a degree 😂. Many thanks for this amazing video
Ayoo bro, you just made sense to my 20 years of education in just one video. WTF! IDK how tf everything makes sense now
First time learning geometric algebra and I love it. everytime you say "wait a minute" it makes me smile knowing that it's going to blow my mind.
I am in High school when you explained bivector are equal to imaginary number, I screamed in mixed emotions of horror and excitement and joy
The explanation of imaginary numbers being pseudo-scalars was a clear Eureka moment, this video was a genius explanation!
Thank you SOOOOO much for this video! I'm currently taking a differential geometry course and this video gave me insanely valuable intuitions for working with differential forms!
You blew my mind when you connected this to complex numbers at ~20 minutes in.
Finding connections in mathematics is one of my favorite things!
This is so amazing!!! I have a more or less basic knowledge on vectors, matrices, complex numbers and quaternions and never heard about geometric algebra, this is beautiful!
Hi there noob, go learn some splitcomplex numbers, biquaternions, tessarines and lorentz groups. You'll have fun
I watched this video for over 1 hour, because my mind get getting blown and I needed a few minutes to appreciate the simplicity in this mathematical approach.
This is a damn good video! I had to pause, go back 20 seconds, and rewatch several times, and each time I would end up understanding what was going on. I'll be revisiting this video for sure; you did an amazing job.
This is way too much information all at once and I love it. Geometric Algebra seems to provide highly useful results for facilitating conceptual understanding in physics and it seems like one of the most profound connection networks for multiple crucial fields of mathematics. Amazing video, I'll be looking forward to diving head first into the intricacies of this field in the future!
You've sent me on an absolutely wild goose chase. I've found lectures by David Hestenes and worked through his books (or at least tried).
Thank you for introducing this!
for so many videos in RUclips about Clifford algebra I watched so far, this one is the most comprehensive one without starting the algebra space based on coordinate and basis algorithm,
This approach shows the nutshell clearly of the Clifford algebra and it’s powerful influence in both maths and physics
I like to think of blades geometrically by extending the geometric description of vectors. We identify a vector *v* by its magnitude and direction. That is, the geometric properties | *v* |, Span{ *v* }, and a choice of orientation (+/-) determine *v* . In this view, a vector is an oriented quantity associated with a particular linear subspace. This is readily extended to all k-blades B = *v1* ^ ... ^ *vk* as an oriented quantity ±|B| associated with a linear subspace Span{B} = Span{ *v1* , ..., *vk* }. I feel like this perspective is more conceptually flexible, as blades are used to represent a variety of geometric and physical relationships, such as areas/volumes/etc., rotations, torques, and even linear subspaces themselves. Thinking of blades generally as quantities associated with a subspace, rather than something more specific like an oriented area/volume/parallelepiped/etc., helps unify these seemingly disparate ideas.
@@hyperduality2838 stop spamming this duality junk everywhere, it isn’t actually insightful. There are things about dualities that can be said and which are insightful, but none of the things I’ve seen you say about (alleged) dualities are at all insightful, and the ones that aren’t trivial are generally wrong.
If you want to say some somewhat more interesting things about dualities, learn some category theory at least to the point where you understand the precise sense in which products and coproducts are dual concepts.
There are valuable things to say about dualities, but the stuff you have been saying just sounds like someone who maybe did some recreational drugs and mistakenly concluded that they gained some great insight.
Thank you so much for this video!! This makes my physics journey so much more fun and opens up a whole new for me
I finally "Grokked" Geometric Algebra thanks to your visual presentation. I don't even think 3Blue1Brown could enlighten Geometric Algebra this well. Many thanks!
Lol i literally thought that this was a 3blue1brown video until I found this comment (I don’t recognize voices very well)
@@depression_isnt_real Don't get me wrong: 3b1b videos are amazing, but what's even better is that he made the tool he created for creating those videos open source. I can't wait to see how many great math videos in the 3b1b style like this one will be out there in a few years created by people like you, inspired by 3b1b. I believe that this is 3b1b's biggest gift, even bigger than his videos.
Yeah, I had an idea about it but this really sold it for me
@@depression_isnt_real And few years ago it was "Sal Khan's black backdrop with hand written notes" type of videos.
no need to compare
i literally cried After watching this video and i NEVER cry, this was a masterpiece and geometric algebra Is One of the most beautiful things i've ever seen, you did a fantastic job in explaining It and you probably changed my Life, I'm about to start a physics university and this Will surely help me SO MUCH, thank you!!!
I am a physics major and I sincerely thank you for your great work on this video. You solve my confusion lasting for many years when I encounter with multi-dimensional calculus. Although sometimes we just get used to the "rules" we maneuver the terms, I somehow feel uncomfortable to the interpretation of the equations. The concept of geometric algebra you introduced to me just take off the frame which trapping my mind for so long. I wish I could see this video earlier.
I have seen a _lot_ of math videos. This one is, without a doubt, the most mindblowing one I've ever seen
I'd like very much to see treating the Navier-Stokes full equation set (mass, momentum and energy) by this tool. The tensor treatment of those equations is too tedious.
Ricardo
good idea but start with the Euler equations (zero vicosity)
At 38:00 when you said charge is current moving in time, my brain imploded with the spacetime diagram and worldlines on it. Thank you.
Wow. It actually looks a lot more intuitive than the complex numbers and quaternions. Great explanation!
You are the math teacher I wish I had 25 years ago. Thank you.
It is just the missing lesson in the university, which I wanted to know for years, connecting linear algebra to higher level math concepts and dozens of physics theories.
Bloody brilliant
Im someone with just lin alg knowledge and you manage to make this video completly understandable