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Yes this is exactly what I was looking for. You explained it great

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I could be wrong but I think the dot product came from multiplication of quaternions. If you multiply 2 quaternions q1 and q2 (a good exercise; giving them generic components a bi cj dk etc.) you will find that many of the terms cancel, leaving you essentially with a dot product term and a cross product term. They can be thought of as symmetric and antisymmetric parts. The history of it began with quaternions first. Then people like Heavyside and Gibbs were proponents of extracting these parts of the quaternionic product into a dot product and cross product and the rest is history. It's an interesting period of history where even some of the greats struggled with quaternions and also with Maxwell's laws. There was a lot of organization and consolidation that took place, but by forgetting the history we expose ourselves to risk of a lot of confusion in the sea of vector calculus and namblas and hodge star duals, clifford algebras, etc etc.

Can I switch between row and column operation while calculating elementary matrix?

Sublime lecture. Such illuminating examples. 🪩 I also smiled and appreciated your dislike of ∈. I'm not against it (it's e for element at least) but it's true that I always feel a bit unnecessarily pompous when I use it.

Why do you keep erasing the Tau term in favor or Pi? Do you not find Tau to be far more unifying?

thank you, amazing explanation, may God bless you

This feels like a Charlie Chaplin movie after mic died.

This is amazing, thank you so much for revealing the beauty of the mathematics. why not upload videos for Calc I. I am looking forward to your precious videos

great explanation, thanks!

ME,I THINK IT SHOULD READ <PARAMETRIZATION>.Cf. e.g. <THE OXFORD DICTIONARY OF MATHEMATICS>. ROLF NEVANLINNA HAS WRITEN A BOOK WITH THE TITLE <UNIFORMISIERUNG> [GERMAN] SPRINGER -VERLAG ,1967 2.EDITION.

ME,I THINK IT SHOULD READ <PARAMETRIZATION>. Cf. e.g. THE OXFORD DICTIONARY OF MATHEMATICS.

I didn't realize math could be so beautiful! :)

Very clear and impressive MASTERCLASS

What textbook do you use please?

Taylor's expansion or finding power series representations for functions is just taking derivatives to find simplest core of the function, such as straight line acceleration creating curve speed and then leading to quadratic location function. the Taylor's series are such representation with initial conditions at every level of differentiation. Seeing it in the perspectives of control system engineering or Dynamics , that is how Equations of Motion and Euler-Lagrange Equations work.

Mathematics is inherently and should be inherently beautiful and elegant. Its fundamental core is simple and unsophisticated, yet it possesses a magical beauty akin to art. Similar to art, mathematical concepts can be explained, appreciated, and embraced from various perspectives. This diversity evolves into the realm of philosophy, where not only one beautiful interpretation exists, but where limitless possibilities unfold. When all mathematical concepts converge and their interpretations harmonize into unity and none. then we will find the heaven, the kingdom of God, the Brahman. Thank you for teaching the math the way it is supposed to be taught !

I find it odd that you claim the ear is "one sensor", while with the eye, you count each rod and cone. There is more than one hair cell in the cochlea, you know....

[[x,0] [x] = [x^2] [0,x] [1] = [x]

Well done, I think I actually get it now! The presentation was wonderful, very enjoyable indeed. Cheers. SUBSCRIBED

Gorgeous stuff.

BOOM! This may very well be the best way to bring the metric tensor into introductory physics.

Pasha, you are more of an artist than a mathematician!. Marvelous the presentation.

Your video is like a mini-masterclass. So valuable!

oh did you mention to your students the identity matrix hidden in the metric? 'cuz you know

Ooooh Kayyyy Gram Matrix from now on

In the thumbnail, you can use LaTeX coding *\imath* and *\jmath* to get a dotless *i* and *j* respectively.

After decades of using linear algebra in professional work, I still find these treatments of the basics quite refreshing and sometimes even enlightening. There is an elegance to the part of the world that can be modeled and predicted by linear maps.

Thank you 🙏 a😭😭

Excellent as usual. How long is a line? Twice as long as a half of a line. Which one you choose as a reference length is arbitrary.

The radian is related to arclength of the circle. Hence, to do angle measurement in radians, one needs differential geometry.

if we allow ourselves to think with an extra dimension, is there some "arc length parametrization" for 3d surfaces? like a square grid of unit length? then the second degree of freedom (orientation) actually becomes continuous aswell (any angle of rotation). idk, just a thought EDIT: now that I think about it, adding an extra dimension might break other things... what even is R(s+h)? h would have to have orientation.. I dont want to think about this anymore :)

I can sortof see how the argument is different in that area is not quite like circumference/length - but the argument still reminds me of the (incorrect) proof that pi=4 by drawing a 4-perimeter square around a unit circle and repeatedly folding the corners of it inwards so the path of the folded square approaches the border of the circle

You might want to correct the spelling in your title.

I am myself a Maths teacher. And there are very few teachers out there , I really like to listen to. You are one of them. Your way of presenting things is just perfect

I love these videos! 😊

This one is an example of geometry leading the algebra 😊(ie we know it should be possible because geometrically it is possible)

There's a lot more insight to be gained by looking at the motion of a rigid rod than one might initially think.

I enjoy non-Cartesian logic. There's a Cartesian product (called conjunction), but there is also a non-Cartesian product (called fusion or smash product, or just "and") that has some advantages. I wonder about non-Cartesian coordinate systems. What makes it Cartesian?

I thought the parameterization of the unit circle does not involve angles and transrectal functions!

Amazing content

I was just looking at this today while studying kinematics!!! And I was wondering whether placing the velocity vector at the tip of the position vector is "wrong" because it should be at the origin and I got confused!!!😅🤣🤣🤣

When you negate the parameter, the orthogonal component of U' flips its direction, but not the parallel component.

I've only watched a few of your videos so far. This lesson was terrific in that it was full of more detailed insights than I ever learned long ago, e.g. higher order derivatives. Thanks.

this video is marvelous from start to finish, and the mute subtitles did really help

Love the smiley face 🙂

Thorold Gosset banished a second time to the intellectual netherworld, first by Burnside, now by Grinfeld ... tsk, tsk

It is an extremely poorly narrated video. Finding the coefficient of the operations you do in your mind is a part that is not really well explained.