Is there a video where above problems are solved using Tensors?

the best lecture series out there for people starting to learn linear algebra. great concept building for a solid foundation to understand more complex problems. way better than the vast collection of calculation tutorials that just throw algorithm after algorithm at you. great work.

Thanks a lot dear professor , I just watched this lecture and I am struggling with sth I couldn't come up with, How can we prove that the RCT is related to the intrinsic properties of the surface?

Helpful lecture

Well done Pavel

Thanks!

Earlier in the series you mentioned that some of the geometry problems "become a joke" when we start solving solving them with vectors. Can we expect examples of that in the near future?

I've finished my MSc in Computer Science, and I watch these lectures out of interest, and they make me feel like a student again :D

Very few may understand the depth of these lectures..

Hey, a bit off topic, but I visited your website and saw that you reworked your approach to tensors, which I will try to learn after this, because you told me this is the prequel and not the other way ;) Do you have any plans on making them available in another format ? As a book or maybe a PDF ? Its not super important, but I like having different formats and purchasing it would also show my appreciation. Additionally I wanted to ask you if you had any recommendations on exercises for differential geometry to go along with this course.

Do what we can and see where it takes us. That’s a lot of normal.

Maybe a dumb question, but why did you use arrows for the vectors v and e when determining alpha, but not for any of the others ? ~ 3:00

he's back!!

Very clear and comprehensive lecture

absolutely amazing

It always felt wrong to me when doing vector calc exercises, that some coordinates were implicitly expected. Like "prove this relation" and I just thought "why in cartesian and how would that work in general" when x,y,z were used in the solution.

Awesome

I’m assuming these are all active rotations since the basis doesn’t change. A passive transformation would represent the same point in space but just with different coordinates?

you're the best <3. im so thankful we have your lectures on youtube.

❤

what have you just done right here. I have never been so captivated about a concept that exists so abstractly in my mind

beautiful explanation, thank you.

mind pothondi raa rei..........su sir

When I first read about the theorem in my textbook about polar decomposition my first thoughts were "is this really true? If you can write any matrix A as Q and S (let's use the same letters as in these videos), you can also write S as XDX^T and get A=QXDX^T which is just one rotation matrix multiplied by a diagonal matrix, multiplied by another rotation matrix. That can't be right. We can't express every linear transformation as rotating and reflecting, then scaling and then again rotating and reflecting, right?" So, my intuition was wrong then, this is apparently possible to do with every matrix, if I understood the videos correctly.

I really like the way you teach this. I thought I knew linear algebra, but this takes things to another level.

idk why the quality on the lemma website is so bad. can't read much that you have written on the board.

This is not the end yet, is it ? Nevermind: You just uploaded again :)

Sir I abousulutly love your teaching may you always be well and teach maths to all of maths lover

I have two questions about surfaces. Is there a way to find the inverse of a surface? The other question is we have parametrization by arc lengh, but is there a way to define a parametrization by surface area?

Even though differentiation of functions having vectors as arguments is weird to think about you can still make an intuitive approach to it geometrically and see where the errors first occur.

Thank you for your amazing videos.Speak of the rotation stuff, could you make a video details how the quarternions works. I have watched so many videos still could not fully understand the concepts .Thanks for your reply in advance

When something is so universal like this I start to wonder if it can be somehow generalized to non-linear transformations like diffeomorphisms

Can we restrict the form of A using this decomposition if we know that A is nil-potent with index 2?

At time 43:00 you write the equation for the surface MT (4 values) each of them as a function of the ambience MT (9 values). You make a contraction and now each value of the surface MT is expressed as a function of a PORTION of the ambience MT i.e. only 6 values because beta will never be = 3. Is it correct ?

I saw the little messy 😂😂😂😂

At first look I wanted to calculate coefficients of Legendre polynomials but for numerical methods we don't need them We need nodes (roots) and weights which can be calculated from eigen problem with symmetric tridiagonal matrix b_{k,k+1} =b_{k+1,k}= k/sqrt((2k-1)(2k+1)) For nodes we need eigenvalues and for weights we need first entry of eigenvector corresponding with eigenvalue There is other method based on Newton's method and asymptotic approximations but i dont know the details Maybe video with code written from scratch

6:34

1:20 how do you prove that the two empty spaces are the same, like the size of the empty space does not change if you rearrange the pieces?

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

than

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