"I decided to invest a large amount of time into producing a video series on advanced mathematics that's easy to digest for anyone because I'm passionate about it". Nothing but respect for you.
I beg you not to drop this, I know this kind of content doesn't get many views but you'll be forever recognized by those 1000 people whom you teached thoroughly and loved the content and we will be forever grateful :)
PLEASE ALSO MAKE A SERIES ON GEOMETRIC CALCULUS (it was all in caps to show my enthusiasm towards GA and GC) also please don't stop making videos for this series before you complete it, because this series will help thousands of people, maybe even hundreds of thousands, maybe even millions
yes!!! since a while, the fundamental theorem of geometric calculus is my new favorite equation. who cares about that e-to-the-i-pi stuff? :-P unfortunately, i do not yet understand it as well as i would like, so more material on geometric calculus would indeed be very welcome!
@@MusicEngineeer who cares about the e^(i*pi) stuff? Well, Euler's formula gives us an elegant way to represent the geometric product of vectors. For any two vectors u and v, let 'I' be the unit bivector with the same orientation as the outer product of u and v (so I^2 = -1), and let theta be the angle between u and v. Then the geometric product of u and v (i.e. the sum of the inner and outer products of u and v) is just the product of their lengths times e^(I*theta). Sure, that's only for the geometric product of vectors, but the introduction of the geometric product naturally extends an n-dimensional vector space into the 2^n-dimensional multivector system we call the geometric algebra of that vector space. You could argue that it's a bit overhyped, but my appreciation for Euler's formula increased significantly while learning GA. That being said, I agree with you that the multivector form of the Fundamental Theorem of Calculus is absolutely beautiful
I'm looking forward to this series. I saw another video on RU-vid where geometric algebra was applied to reformulate Maxwell's equations into a singular equation, and it looked so elegant and natural. Once you're done with this series, I would highly recommend making a geometric calculus series since that also greatly simplifies topics like differential geometry and differential forms, and would be very useful in physics. Maybe someday in the future geometric algebra and geometric calculus could set the foundation for finally combining quantum mechanics and general relativity into the holy grail of physics: the theory of everything.
@@sudgylacmoe LOL I just realized that you were the one that uploaded the video with Maxwell's equation. Well, I'm looking forward to your upcoming videos.
@Paul Wolf That's a fascinating subject that I can't go into fully in a comment. Something interesting is that the spatial unit vectors in relativity are actually not the same as the normal spatial unit vectors, and the outer product of the time unit vector with a spatial unit vector produces a normal spatial unit vector (which is a bivector in this case)! What I just said probably confused you though. I am planning on making a video on this topic at some point.
@Paul Wolf Time itself is not a 'vector' all on its own, instead, it's a specific dimension in a larger 'space' called 'spacetime'. But it's not exactly identical to the other 3 known spatial dimensions we're all familiar with, which is why it seems to be a 'separate' thing to us humans who live in a very much 'not even close to the speed of light' speed experience of the world. The main difference with the time dimension, from a mathematical point of view is that it interacts with the other dimensions not in the more usual human-experience 'Euclidean' geometry that we are all used to, where parallel lines never converge or diverge, and lines and planes seem 'flat'. Instead, time and space interact in what's called a 'hyperbolic' geometry, which is a bit weird and 'curvy'. Most of the time, this kind of interaction is only noticeable when things are moving at very high speeds (closer to the speed of light), or when very massive things (like the Sun, or stars, or black holes) change the 'curviness' of spacetime around them. Typically, when working within relativity, a vector wouldn't solely be in the 'time' dimension. Rather, the vector would have components in each of the 4 dimensions, 3 of space, and 1 of time. So, it would be called a spacetime vector, rather than a time vector or a space vector. So, while we are all moving at some velocity through space (with respect to each other and to other objects in the universe), we are *also* moving at some velocity through *time.* Or, to put it another way, our velocity is not through space or time separately, but through spacetime as a whole.
This is a hefty goal, but it’s exciting to see someone on RU-vid undertake it! I love to see the math community and math enthusiasts pursuing this interesting topic there are few resources on now. This is how educational revolutions happen!
My man. I’m paying thousands for Math classes in college that repeats already known information in a boring way while you just provided less seen information to the public…for free?!! Wow! This is just wisdom. Beautiful concepts that describe the natural world and how to engineer parts of it. Great foundation. You need more subs
This sounds like it will be a very ambitious project. But, from the quality of your previous video on the topic I can only assume this will end up being a great series. Looking forward to it.
Really looking forward to this! You should share this in the 3b1b discord for the summer of math exposition, this is exactly the kind of content the event is meant to proliferate.
I love that you started uploading videos more. You have very good explanations and your videos fit perfectly between highly complex and more simple. Keep up the good work
I am quite excited for this series, RU-vid is lacking good geometric algebra videos for sure and you seem to be a great person for the job of doing them! I can't wait to see the series
The world so needs an introductory Geometric Algebra course. Especially one with exercises.. good call! Your earlier intro to GA was an instant classic. Looking forward to seeing what you create.
Awesome! Can't wait! Subscribed! May I suggest, if you haven't heard of him already, checking out retired professor Norman Wildberger's channel called Insights Into Mathematics? He has covered many topics in the past which resonate with Geometric Algebra, without quite (yet) going 'all the way' into Geometric Algebra. You may be interested in his development of Chromogeometry, and Universal Hyperbolic Geometry, as being quite closely related to Geometric Algebra. He covers many other topics as well, including Foundations of Mathematics, Linear Algebra, etc., and his most recent work on Algebraic Calculus, as well as his older-but-still-very-relevant work on Rational Trigonometry. He also has a variety of other kinds somewhat more 'casual' or 'indirect' math-interest/math-hobbyist types of videos such as his Math History lectures (from the University of New South Wales), Sociology of Pure Mathematics, Famous Math Problems, and on and on. I will recommend this video series to him once you've got a couple more videos going, as I'm sure he'd at least be curious/interested, and hopefully it might get him to start incorporating Geometric Algebra ideas directly into his own videos as well. Also, I'm sure he'd be open to dialog/correspondence/whatever with you on whatever topics as you seem to have a very refreshing take on things! Cheers!
I’ve noticed that GA seems to have a heavy emphasis on using an orthonormal basis, whereas something like tensor algebra is all about using arbitrary bases. Is there any conflict there for describing something like GR?
Orthonormal basis provide a convenient rule for most spaces. For example, in Euclidean space, only the Kronecker's delta is needed. When dealing with general basis, the metric tensor can be used in the dot product / contraction. But, since GA allows to express algebraically relationships between geometric elements, GA can describe most of its content without referring to any coordinate system. For instance, given a non null vector a, the reflection of any vector v in a is a^-1*v*a. The formula doesn't rely on any basis. In conclusion, one of the greatest power of GA is the express geometric concepts entirely algebraically thus skipping lots of bookkeeping needed with "coordinate heavy" systems.
@@user-hh5bx8xe5o you don't need coordinates with the metric tensor too. It's a bilinear map. I guess to deal with arbitrary basis modify the geometric/clifford product such that v^2 = g(v,v) for a vector v.
I have noticed this too and wondered (also for non-relativistic applications). On the one hand, Geometric Algebra magnificently unifies many topics in mathematics and physics. On the other hand, Tensor Calculus allows for the explicit use of components in arbitrary curvilinear coordinate systems (which is needed for practical calculations using computers). This is why, to date, I believe the utmost mathematical language would be a combination of (1) Geometric Algebra as presented here and (2) Tensor Calculus as presented by Professor Pavel Grinfeld in the awesome RU-vid channel MathTheBeautiful (as presented by him (!), as he heavily emphasizes geometry, prioritizes intuition, and aims at general cases with no background presumed coordinate systems whatsoever, such as an orthonormal one). I am currently working on combining these two as a personal/professional project.
You can use a reciprocal basis for this, see ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-iv5G956UGfs.html for example (well, the only example on RU-vid that I'm aware of). Also there's no need to refer to any basis in most cases in the first place just like with tensor equations. GR can be formulated in GA, either the usual formulation with curved spacetime, or another one called Gauge Theory Gravity that introduces gauge fields and stays in flat spacetime.
Thank You for your valuable information & time for making this great mathematics videos series. If it's possible from your side can you also cover the use of Geometric Algebra in Artificial intelligence/Machine learning ? Thanks 👍
So looking forward to this series. I think I watched your last GA video 3 times already. Somehow I blindly believe this is one area which has something much deeper than I know.
Is there any book about the formalisation of GA? I really cannot wait to see your video on the topic, but since I am considering writing my thesis about it (I'm a Math major) I would love to have the reference to some textbooks or other sources. Thank you a lot!
Can you teach geometric calculus? The papers Hestenes wrote are profound but the notation is too confusing for me to grasp easily. Also, can you eventually get to projective and conformal GA?
If you haven't seen it yet, I found raw.githubusercontent.com/pygae/galgebra/master/doc/books/bookGA.pdf much easier to read than Hestenes' articles on Geometric Calculus (still not as good as it could be though, missing visualizations and so on).
There is one question that people forget to answer when doing these kind of lectures and it is "Why?" "How you calculated the cross product of two vectors." is often answered. "Why do we calculate the cross product or two vectors" is skimmed over, ignored completely or assumed to be obvious. The why really helps me learn. Matrices were a boring, dry subject till I learned there important in computer graphics. 😅 Please put lost of "Why" in the course. Looking forward to it. Thanks
Just saw this as the first in the series! I’m extremely interested in this because you hint at a potential unification of vector calculus, complex numbers and quaternions. I think there could be something very insightful in that. Although I heard your argument about making a video series (at least first), I’d like to encourage you make a book (out of the video series?) because although a video is likely the best way to teach a concept, being able to really study a book (that doesn’t move on until you are really ready) is IMO a better medium to really learn the nitty gritty from. These complement each other. Plus, maybe you can be THE go to reference for a new course in college math, which would be quite an achievement these days ;-). Looking forward to watching the all!
Ever since I saw Your first video on Geometric algebra a couple of weeks ago, I've been dying to get a deeper dive on it. I cannot express how overwhelmingly excited I am for this!!! Keep up the great work :)
Message to people who are planning to watch this.. This is a good helpful materials but if you're planning to watch this know, it may be helpful to learn a little bit trignometric unit circle and complex numbers just a little prerequisite.
Talk about timely. I have been searching for an into to Geometric Algeria for a couple of weeks now having only just learnt that it is a thing. I only have high school algebra but have had to learn more math for game development. I have had a feeling like there is a certain set of approaches that make more sense than others, but have not been able to put my finger on it till recently. For example when I started learning 3d math for computer graphics I could not figure out why the dot product of two vectors was soo important. Eventually someone explained the geometric interpretation and it all made sense. (it gives the angle between the two vectors) Cannot wait for this. Please follow through. Have you got a Patrion or something?
this is an amazing video with an amazing motivation, such beautiful information and concepts in a series like 3b1b has, but more in depth. also learning about geometric algebra in high school will make mathematical concepts further down the line much more simple without the need to understand so many different types of math. thanks for the video and i subscibed to you
I recently watched your other video on GA from about a year ago. It has been the first time I've been excited about mathematics since university 10 years ago...
I have deep respect and admiration for you, your work, and your efforts. Thank you very much, Sir! As someone who is currently an undergrad in Physics, this knowledge may open new paths of exploration for me, and of course for those who will come after us. I am standing on the shoulders of giants, and soon they will also stand on ours as well. To think that, after decades and centuries of progress after the Industrial revolution (the humble beginnings of which was when Electromagnetism was to be solidified) it really is mind-boggling awe-inspiring, and wondrous to see such a underrated Mathematical tool. Who knows. Maybe the applications of Geometric Algebra may be one of the missing insights we need to finally construct the theory of everything.
Really looking forward for the series. If I could give a suggestion, I'd like to see some of the examples to be connected with physics problems. It would be especially cool to see how to use the geometric algebra maxwell equation, ∇F=J, to actually solve a problem and to see how it makes this more simple than vector calculus.
My username is pretty mostly made up. The "Sudgy" part is a name I have used since I was a kid, and the "Lacmoe" part is something my sister and I came up with years ago.
I'll put my money where my mouth is and support you on patreon so that we can help you complete the video series. I think it is important, especially because universities aren't giving it the attention it deserves
@@sudgylacmoe Wow... I didn't see your FAQ... Thank you!. Please continue producing more videos on geometric algebra and its applications in physics. ..I think that in geometric algebra there is knowledge that is lost or ignored
I'm a former Nasa Principal Investigator and PGM and PROJ. Dir. of IMU. WE ARE GOING TO USE THIS MATH TO FACILITATE OUR R&D in IMU -lets keep in touch and should add you to our team and potentially complement your project (source for funding).
I may have written this already on a comment on one of your other videos in this series, but I'll gladly repeat it. In one long evening of watching your videos on Geometric Algebra, I learnt and understood more mathematics than I did in the years trying to understand first year mathematics (L.A. and calculus) in 1987-1990 at the Mathematics Department. I'm not in a position to sponsor anything, but I hope you will find the time and means to finish this.
You are a really good teacher I think. It’s honestly my pleasure to meet you. During high school, math makes so much sense, as they always adds up eventually. However in higher educations, things started to get weird. It’s not like it does not work, but the geometric meanings are getting abit off in my opinion. I’m studying and diving deep into this series of math, and see if it helps with my understanding of higher level maths. Thanks for making all this videos for free. Sincerely 🙏🏻🙏🏻
i watched your longer introduction video and you managed to give me a new favorite field of math in just that video. every new section i was blown away, from 2d multivectors being complex numbers, to 3d ones being quaternions, and then simplifying ALL of Maxwell's Equations into a single one
I am so glad I subscribed to you. I develop computer graphics software. Computer graphics almost exclusively uses gibbs' vector formulations rather than clifford's vectors. This stuff is sooooo awesome
