Tensor Calculus 4a: The Tensor Notation

This course will eventually continue on Patreon at bit.ly/PavelPat...
Textbook: bit.ly/ITCYTNew
Errata: bit.ly/ITAErrata
McConnell's classic: bit.ly/MCTensors
Table of Contents of bit.ly/ITCYTNew
Rules of the Game
Coordinate Systems and the Role of Tensor Calculus
Change of Coordinates
The Tensor Description of Euclidean Spaces
The Tensor Property
Elements of Linear Algebra in Tensor Notation
Covariant Differentiation
Determinants and the Levi-Civita Symbol
The Tensor Description of Embedded Surfaces
The Covariant Surface Derivative
Curvature
Embedded Curves
Integration and Gauss’s Theorem
The Foundations of the Calculus of Moving Surfaces
Extension to Arbitrary Tensors
Applications of the Calculus of Moving Surfaces
Index:
Absolute tensor
Affine coordinates
Arc length
Beltrami operator
Bianchi identities
Binormal of a curve
Cartesian coordinates
Christoffel symbol
Codazzi equation
Contraction theorem
Contravaraint metric tensor
Contravariant basis
Contravariant components
Contravariant metric tensor
Coordinate basis
Covariant basis
Covariant derivative
Metrinilic property
Covariant metric tensor
Covariant tensor
Curl
Curvature normal
Curvature tensor
Cuvature of a curve
Cylindrical axis
Cylindrical coordinates
Delta systems
Differentiation of vector fields
Directional derivative
Dirichlet boundary condition
Divergence
Divergence theorem
Dummy index
Einstein summation convention
Einstein tensor
Equation of a geodesic
Euclidean space
Extrinsic curvature tensor
First groundform
Fluid film equations
Frenet formulas
Gauss’s theorem
Gauss’s Theorema Egregium
Gauss-Bonnet theorem
Gauss-Codazzi equation
Gaussian curvature
Genus of a closed surface
Geodesic
Gradient
Index juggling
Inner product matrix
Intrinsic derivative
Invariant
Invariant time derivative
Jolt of a particle
Kronecker symbol
Levi-Civita symbol
Mean curvature
Metric tensor
Metrics
Minimal surface
Normal derivative
Normal velocity
Orientation of a coordinate system
Orientation preserving coordinate change
Relative invariant
Relative tensor
Repeated index
Ricci tensor
Riemann space
Riemann-Christoffel tensor
Scalar
Scalar curvature
Second groundform
Shift tensor
Stokes’ theorem
Surface divergence
Surface Laplacian
Surge of a particle
Tangential coordinate velocity
Tensor property
Theorema Egregium
Third groundform
Thomas formula
Time evolution of integrals
Torsion of a curve
Total curvature
Variant
Vector
Parallelism along a curve
Permutation symbol
Polar coordinates
Position vector
Principal curvatures
Principal normal
Quotient theorem
Radius vector
Rayleigh quotient
Rectilinear coordinates
Vector curvature normal
Vector curvature tensor
Velocity of an interface
Volume element
Voss-Weyl formula
Weingarten’s formula
Applications: Differenital Geometry, Relativity

[@DanielLopez-ys5ji]

"my arm is getting tired, because I didn't use the tensor notation." lol Dr. Grinfield, I like your sense of humor.

[@reububble]

Or maybe it's because it was writing out so many Zs

[@scholar1972]

Dear professor, You make this stuff so easy to understand . I wish, I have this thirty-five years ago. I have no problems understanding notations, dot products, and the tensor notations. I have been working on unified field theory independently for more thirty years, however, explains helps to understand the works of Cartan,and Tullo better.

[@feynstein1004]

I envy you. You say it's easy to understand and yet I'm having a really hard time understanding it.

[@yashabyadav588]

@@feynstein1004 It's okay. Usually happens with me when I study something for first time. A good tip is to read some material before watching the lecture itself. You don't need to understand it. It will just give you a big picture of things. Or maybe you can watch lectures twice.

[@feynstein1004]

@@yashabyadav588 Hmm okay

[@superj1e2z6]

It feels so surreal that just 10 years ago, I was struggling with fractions and now watching Tensors video with close to no actual numbers used.

[@David-km2ie]

Lol, I was 7

[@l.3ok]

@@David-km2ie he too; probably

[@nemanjastankovic941]

It keeps getting more and more interesting. Thumb up for Dr. Grinfeld from Serbia :)

[@LucaIlarioCarbonini]

And from Italy too!

[@stephenhenrich2625]

This is exactly the pace and explicit instruction I need for this material

[@xXZeroSynDromeXx]

I tend to assume I'm the only one who would be interested in something like this. So when I see that this video has 58k views, it's both surprising and reassuring.

[@englemanart]

I eat this kind of stuff up. I just find it beautiful for its own sake, even though I know it is used for applied mathematics/science.

[@feynstein1004]

Well, idk about you but I'm learning this because I want to learn general relativity.

[@CstriderNNS]

for me math is the language that god wrote existence with, so by us learnig it we are learning about he/she/it/whatever GOD is.

[@hushaia8754]

I like his humility

[@MathTheBeautiful]

He's also very modest

[@yashabyadav588]

This is my third time watching this and I think I am starting to get it now.

[@AkamiChannel]

It was absolutely wonderful to have this spelled out in full detail because a lot of youtube videos, including susskind etc., sort of gloss over it due to time constraints. I don't know if I realized there were 2 sets of variables be indexed over. Your book is arriving today btw. Thanks.

[@SWiSHRoyal]

Great Lecture. In my courses they only told us that a a 1D tensor is a number, a 2D Tensor is a vector and a 3D Tensor is a matrix. They just used it. Finally I get what it means.

[@RanEncounter]

That kind of approach is quite usual in physics. But in mathematics and theoretical physics you need to know more about the properties.

[@bonbonpony]

Now the fun part is to find an answer to this question: what does the THIRD-rank tensor represents geometrically? :>

[@AAASweetandDaring]

I have been looking for understandable tensor explanations for relativistic electrodynamics and having found anything that really tells you what tensors are, this is really helpful!!! thank you so muchh

[@kedonsiemen]

Thank you! You're a brilliant teacher!

[@MathTheBeautiful]

Thank you, that's very kind of you.

[@aeroscience9834]

I find putting the primes on the index to be very confusing. It looks like i and i' are just different indexes (like i and j) for the same set of variables. Especially when you use both i and i' in the same equation where you want separate indexes like at 39:20 , It makes me think of contracting the i's. I think it's much more clear to use 2 different letters for the indexes (i and j) and put the prime on the z.

[@adarshkishore6666]

I was always thinking that there must be a better way to capture the chain rule other than using a myriad of partial derivatives. This is a very neat way and your explanation is also very neat.

[@vralev]

I just got to say kudos to this. Great job explaining it.

[@juborajroypavel8342]

I'm studying Economics & Computer Science. I don't know if there are any crucial applications of Tensor Calculus in any of these fields I'm studying, yet I felt so attracted to it just by knowing its beauty. I'm studying this with the hope that someday I could use it in my research. My concentration is Artificial Intelligence. I just want to say, THANK YOU SO MUCH, DEAR PROFESSOR. YOU ARE AN AMAZING TEACHER. Loves & Regards! (From Bangladesh)

[@MathTheBeautiful]

Thank you! That means a lot!

[@user-wj5in3ww6q]

So what, it helped somewhere?

[@Alex4LP]

Incredible lesson. Really awesome. Thank you very much for sharing.

[@rkpetry]

N.B. We generally say "partial F partial a" for ∂F/∂a, and "dee y dee x" for dy/dx, etc....

[@EddieVBlueIsland]

Nicely done. Thank you.

[@arunbharadwaj145]

Where can I find the solution to the second order derivative of the Jacobean function that was supposed to be the homework? I just need to see if my solution is right Oh, and +MathTheBeautiful, I love your videos. I'm an undergrad freshman, and I've pretty much self taught myself Calc 2&3 (through some books) and some of Linear Algebra and now Tensor Calculus, both of which would have been near impossible without your channel. Thank you very much!!

[@sohailmakhani5474]

Dear Professor, Can you please make video lectures on Differential Geometry? Need it badly after this playlist.

[@lucassanches8693]

Thank you very much for the video! It will be helpful in my Fluid Mechanics test later this week! (:

[@savinolupo2379]

Thank you very much for your crystal clear lectures!

[@fangjunkuang5061]

非常易懂，谢谢！

[@pedroaragao1754]

Thank you very much for the lecture!

[@taraspokalchuk7256]

22:43 So we won't be using a(m)= a1(m1, m2); a2(m1, m2); a3(m1, m2) ? Edit 25:59 - that was my question. I think i got it. We use Z^i for different identities, but if it is a list of variables or functions inside a function we surpress it?

[@Cfx45321]

Amazing lectures !! Thanks

[@feynstein1004]

This melted my brain. I guess I'm the only one who found it difficult to follow. I think I understood some of it but most of it went right over my head.

[@MathTheBeautiful]

Hey Feynstein 100! Check out the new Vector Calculus series which is a prequel to this series!

[@fhzsduras]

As a Pkmn Master I can't keep it only to myself: The time that we can name variables after Pokémon has come. MEWTWO!

[@jose-alberto-salazar-jimenez]

I really like his way of teaching.... with patience, asking, joking...

[@terryphi]

man, this is brilliant, but I wish it was partitioned into shorter segments

[@anugrahmathewprasad172]

I'm interested to know about what was skipped a 54:32. How does swapping horiztonally amount to taking transpose? Or in general how to convert from matrix notation to tensor notation. Can someone help me with this?

[@cartmansuperstar]

As far as i understand it, the first index will always be the row-index. Therefore, when switching from A^i_k to A_k^i and summing over k, it now (in the second case) means, that you will sum over the i-th column of A, while still summing over B^k_j (i.e. the j-th column of B), which is the same as, what you´d do, if you´d take the matrix-product of the transpose of A with B.

[@ethandrood]

I love this guy.

[@MathTheBeautiful]

How can you not!

[@cartmansuperstar]

1:06:52 "because if this was k, it would just be the product of A and B". Wouldn´t it rather be just the element in the i-th row and k-th coulumn of AB? Or do you regard it as the whole matrix-product, because i and k can be chosen freely?

[@CstriderNNS]

at 48:50 Did you change the position of the (i,j) indices to better represent that the i th index is a row, and the j th index is a column ? or for the fact that one is associated with contrivance, and the other is associated with covariance ?

[@UnforsakenXII]

52:50 lol what.

[@prabhatp654]

great lecture again, but somebody tell me please right after getting the identity of two coordinates or system, why do we differentiate them? Is it due to covarient basis but here we only parameterized it only, we didn't impose any coordinate system to the space.

[@adriangheorghe2327]

What are the physical dimensions of the tensors and curves in Einstein's field equation in L, M, T, or in m, Kg, s, also called the relativistic formula of gravity or the general relativity formula?

[@kevinz4396]

Second derivative of Jacobian is missing a squared on right hand side, no? 01:25:00 around.

[@connorfrankston4040]

Is there a way to deal with contraction over tensors of infinite size? For example, couldn't changing the order of contraction cause some issues?

[@eco8user]

It is almost impossible to watch this course using the android youtube app; there are a dozen of ads in each video interrupting the explanations when you need to be most concentrated in order to follow. On the desktop (in the web-browser) there are no ads though. Is it possible to make the android version add-free, so that I can also study the courses when travelling?

[@RanEncounter]

yeah this is a pain in the ass. And the adds are so loud too :(.

[@gqwang6277]

Geometrically and physically, does any one wonder why the line in blackboard gets lower and lower toward right ?

[@SalvatoreIndelicato]

please can you insert subtitles?

[@geographymathmaster]

Hello, Thank you very much for the content. I was wondering why at 50:00 you said that the A^i_j notation is horrible for linear algebra. I find the double subscript notation often very cumbersome and sometimes add in a comma between the i and j since it makes it easier to understand, especially when the i and j are expressions. Thank you.

[@MathTheBeautiful]

+geographymathmaster Generally, if you are interested in an operator as a whole object, rather than its individual entries, then the matrix notation is superior. On the other hand, when you need access to the individual entries, the tensor notation is superior. In elementary Linear Algebra, you more often encounter the former situation. In tensor calculus, the latter.

[@geographymathmaster]

+MathTheBeautiful Thank you. I still think that using the tensor notation to denote individual elements might be easier at many times.

[@walter9029]

When you are talking of theta and phi which one is the polar and which one the azimut angle ?

[@leonig100]

At 105.12 we are told that Delta i upper index and I lower index is three. This is based on the Einstein summation principle. But the summation principal refers to two different elements with the same letter. What is our justification to adding the three terms to get 3? Why don't we multiply them for instance.

[@lemmafundamentals4431]

It doesn't need to be a product of two different elements. It can be one element. In fact you can think of two elements in a product as forming one element which is the result of the product. Then the contraction takes place on that element.

[@sufyannaeem2121]

what are contravariant basis or vectors? could'n understand???? need physically visualization

[@JamalAHamad]

Thank you professor

[@ShubhamSingh-lq5bl]

Sir, can you provide me the ebook of the tensor calculus.

[@jamesmarlar4231]

Please move the camera closer to the board.

[@mathieu5332]

What is the name of the textbook you use ?

[@andrewstallard6927]

Aren't there 48 terms in the formula at 1.19.49? In the first part are there not four summations--i, j, alpha and beta--that would make the number of terms 3x3x2x2=36 and in the second part three summations--i, alpha and beta----making 3x2x2 or twelve terms for that part for a total of 48?

[@rob3c]

Andrew Stallard the equation itself still uses alpha and beta indices on the left side, so the 2x2 is implicit. So 12 terms for each of those 4. Now, indexing against x,y,z to get r,theta seems confusing for a change in coordinates unless there’s also a projection involved, but that’s another matter. He sometimes says r,theta,phi when suggesting extensions using x,y,z, but he seems to conflate 2d and 3d here.

[@leophysics]

Thank you.. so much prof

[@viaprenestina3894]

any exercise?

[@debendragurung3033]

41:00, isnt that expressions dZ^i' / dZ^i just the Jacobian of cordintae transformation for unprimed to primed.... and dZ^i/dZ^i' the inverse ...

[@bhargavmanohar507]

sir, my doubt is in the eq. J^i' i J^i i' won't it be considered as summation because we have an upper i and a lower i ?

[@MathTheBeautiful]

Correct!

[@carlosmehicano8052]

hi i was just confused as to why the jacobian was introduced for what i thought were multivariate functions, it may be a stupid question but is it normal to present the polar coordinate system in terms of vectors? is my problem that im associating the vector notation with the cartesian coordinate system?

[@MathTheBeautiful]

Hi Carlos, I'm not 100% sure what your question is. Can you paraphrase it? Pavel

[@Smoothcurveup52]

Thanks sir

[@cfriedalek]

Just finished lecture 3 on covariant basis and this starts with reference to inverse Jacobian which wasn't mentioned at all in the previous lecture. Did I miss something? Is there a lecture missing or perhaps I'm watching in the wrong order. I'm watching in playlist order.

[@suluclacrosnet61]

Yes, there's a lecture missing that talks about change of variables. It's actually been lost and Dr. Grinfeld intends to rerecord it this week (3/23/2014).

[@cfriedalek]

Suluclac Rosnet Thanks for the update

[@6AP6APblCKA]

Профессор, меня немного смущает место расположения штриха в записи Z^i^штрих. Это выглядит, как та же исходная система координат Z, которой назначен новый итератор i^штрих. Далее, можно было бы просто переименовать i^штрих в, например, j, и мы получим Z^i^штрих = Z^j. И т.о. мы полностью потеряем связь с новой координатной системой Z^штрих. Не будет ли верным, на самом деле, писать вместо Z^i^штрих другое обозначение: Z^штрих_i^штрих? В этой записи меня ничего не смущает - есть система координат Z^штрих и ей назначен итератор i^штрих.

[@6AP6APblCKA]

И, конечно же, большое спасибо за потрясающие лекции! В Вашем изложении мне удается понимать все с первого раза в такой сложной теме. Особо хотелось бы отметить Ваш подход, при котором всегда объясняется, зачем нужен тот или иной объект. Постановка цели и этого ключевого вопроса "зачем?" как мотивирует к вниманию и изучению дальнейшего материала, так и упрощает его. Сама исходная лекция №0 является огромным мотиватором ко всей серии - возникает желание "досмотреть сериал до конца", чтобы узнать, чем же все кончится, и как геометрия объединится с алгеброй.

[@MathTheBeautiful]

Sorry for the delay in responding. Yes, this notational choice (reportedly introduced by Ya.S. Dubnov) is surprising at first. But after some getting used to it proves to be very natural. It also helps the interpretation that a tensor (or variant) exists apart from any coordinates and simply finds its manifestation in a particular coordinate system. Given that it's been a month since you asked your question, perhaps you have already gotten used to this notation!

[@hennadiimadan6993]

Ну и чудо. Сижу и сосу, значит, барбариску, впервые наверно за 3 года. В этих краях это огромная редкость - кто-то откопал запасы одной русской студентки. Смотрю лекции по тензорному анализу, освежаю то что прогуливал лет 8 назад. И тут такой комментарий с таким ником. Ну и совпадение, однако, прям чертовщина!

[@foobargorch]

Why is it Z^i' and not Z'^i? In my mind the separation (and therefore the priming of the symbol) is between the groups of coordinate variables (different groups of Zs indexed by numbers, as opposed to different groups of indices). Since this would suggest Z'^i, I am assuming I must have gotten confused about something...

[@MathTheBeautiful]

foobargorch That's what any person would assume the first. Technically, the prime should go on the letter. But to put it on the index was a *brilliant* idea. I believe it is due to the Russian mathematician Dubnov. This idea kills several birds with one stone. 1. It indicates the alternative coordinate system. 2. It saves a lot of letters (i can be used twice: as i and as i'). 3. In Jacobian-type objects J^i'_i, it indicates what is being differentiated w.r.t. what.

[@foobargorch]

Ah, makes sense. Thanks!

[@bonbonpony]

09:12 But that rule was for top and bottom indices next to the SAME symbol, wasn't it? :P Here you have the SAME symbol, and the indices are at the SAME position: they're on TOP in both! I don't think Einstein's shorthand could work the same with the NUMERATOR and the DENOMINATOR being the "top" and "bottom" here :q

[@lawrencemwangi8846]

HI, how do I donate?..has really helped me and wanted to donate

[@Revan176]

this upper and lower indexes are confusing me a little bit. In my Tensor Calculus Lecture for Kontinuumsmechanik we just used lower index and the summation takes place when the same index appears two times in a term.

[@MathTheBeautiful]

In a very pragmatic sense, tensor calculus accomplishes, among others, two goals: 1. enables us to work with multidimensional arrays 2. enables us to work with coordinates other than Cartesian If you stick to lower indices only, you accomplish 1. but not 2.

[@Revan176]

ok

[@jefferywyss8740]

great!!!

[@yadercarrillo2261]

Good job!!. I'd like to know what's the title and author of the book that he used around the minute 1:21:00

[@MathTheBeautiful]

It's in the description.

[@bitesofmathematics4356]

FWIW I've worked out one way to demonstrate how to think of Z the independent variable vs. Z the function ie. what symbol means what ? Answer : put in some actual numbers and attempt to evaluate ! Consider say : R(X(R,T), Y(R,T)) = R then if this is true for all R and T it should be true say, for R = 3 and T = 4. The only logical substitution into that identity is : R(X(3, 4), Y(3, 4)) = 3 meaning that you evaluate the X and Y functions for the inputs 3 and 4, the values thus achieved from the functions X and Y by those arguments is then input to the R function to give an actual specific number ( here 3 ). Compare this with say : 3(X(R, 4), Y(R, 4)) = R which has no sensible semantics, unless you name functions using the symbols used to represent numbers ( silly ). I very much like Pavel's comment that we are "differentiating in the abstract" ie. what must always be true regardless of coordinate system choice, transforms etc ?

[@didles123]

Mike Hewson I was thinking it would be better to just use lowercase for the independent variables and capitals for the functions. Then we can say: R(X(r,t), Y(r,t)) = r T(X(r,t), Y(r,t)) = t Back when he took the derivative with respect to r, we would write: R_X * X_r + R_Y * Y_r = 1 T_X * X_r + T_Y * Y_r = 0

[@MathTheBeautiful]

+didles123 That's a good approach, too. I find that reusing the letters has some benefits, too. When "x" and "y" are reused, then the same quantity (regardless of its functional role) is denoted by the same letter. Secondly, I find that it invites care and forces me to talk myself through what I'm doing.

[@theflaggeddragon9472]

at 1:11:12 by writing ∂f/∂μ^α does the alpha refer to both mu's or just μ_1. Or is μ_1 the same as μ^α? is μ_2 the same as μ^β? does alpha and beta refer to one or the other?

[@MathTheBeautiful]

+The Flagged Dragon The index α and β refers EITHER to 1 or 2. In other words, that single expression captures two identities: one for when α=1 and one for when α=2. The second expression captures 4 identities: one for when α=1 and β=1, etc.

[@theflaggeddragon9472]

+MathTheBeautiful okay that's what i suspected. Thank you very much, I'm really enjoying these. I also really appreciate your diligence with answering questions. Keep up the good work

[@comprehensiveboycomprehens8786]

The camera was too far from the board. It's driving me nuts!

[@anantgairola3394]

@MathTheBeautiful, could you (or someone else reading this) please clarify what was skipped @54:32. Thank you for the wonderful lectures.

[@MathTheBeautiful]

Yes, I'll have to make a video about that.

[@anantgairola3394]

@@MathTheBeautiful Thank you for your prompt response, Professor Grinfeld. I look forward to that video.

[@leonadams6529]

Would it be possible to list the text being used in the class?

[@isaacnewton4625]

If I'm not mistaken, it's in the description: goo.gl/IaCQC2

[@Yoyimbo01]

Wait, when he discusses identity equal to Jacobian times its inverse, how is he allowed to use the same notation for the entire matrix of derivatives (Jacobian) and the contracted chain rule formula?

[@MathTheBeautiful]

Hi Erik, Which moment in the video are you referring to?

[@Yoyimbo01]

Thank you for replying sir! At 1:02:40, the tada moment :). It was ironically the only part which left me confused. Let me try to elaborate on why I am confused, we arrived at the derivate notation on the right by contracting sums while we arrived at the derivative notation for the jacobian by contracting matrix notation. I fail to see how they become the same automatically

[@Ykulvaarlck]

In what lecture did you talk about covariancy and contravariancy?

[@MathTheBeautiful]

Formally, kzbin.info/www/bejne/inScaX6cnqx0hc0

[@YoungColCol]

53:00 in this example [A,B]=0. Are there any instances where the entries in the matrices would be operations that don't commute?

[@MathTheBeautiful]

Why are you saying that [A,B]=0. I don't think that's implied!

[@YoungColCol]

sorry I edited the original comment, but I guess it didn't load. [Aik,Bkj]=0*

[@MathTheBeautiful]

+fred col I see what you mean and I'm glad you brought up this point of view. The statement [Aik,Bkj]=0 is technically correct, but it is not saying much. The beauty of this notation, which often gets lost because we are so used to the powerful matrix notation, is that Aik can be simultaneously seen as a collection of numbers (i.e. a matrix), if you mentally allow i and k to assume all possible valid values, or as a single number, if you assume that i is a specific number is k is a specific number. The interpretation depends on the context. So to state that [Aik,Bkj]=0 is to say that products of numbers commutes, e.g. [1.34 1.45] = 0.

[@tonk6812]

I think ur studnt told that there are for identities at 1.20 and term is 12 actually identites shd be 9 and term is 12 of course coz i j and i are dummy indices...i think he was looking at the book without realizing it what was taken

[@ibrahimsalat778]

I have difficulties with exercise 47 from the text book. can someone help?

[@andrewxc1335]

I imagine that: a) you don't need this anymore (sorry :p), and b) that you should expand the Jacobians into derivative notation.

[@Hillis360]

Can someone help me with the exercises of section 7.2 in the book? I'm getting cofused, maybe there are some mistakes ?

[@MathTheBeautiful]

Hey, author here. Which exercises in particular?

[@youteubakount4449]

You should have more faith in your students, professor. I got the second derivatives on my first try! I think you have made the thought process very clear from the beginning so don't worry about your students :) Thank you for this series!

[@Methuselah_]

The Prof. is communicating with ALL students/people - including viewers.

[@veronicanoordzee6440]

Why the use of Z's for all kinds of different objects? Didactically a pretty peculiar choice. I haven't seen it anywhere else. Don't think it will increase the sales of your book.

[@MathTheBeautiful]

All the objects have different indicial signatures which is sufficient for distinguishing among them. Using different letters would make the notational system more cumbersome.

[@foobargorch]

36:00 - the Kronecker delta only represents these derivates (∂zi/∂zj = δij) for an orthogonal coordinate system, right?

[@MathTheBeautiful]

foobargorch No, in all coordinate systems. It's a purely arithmetic fact!

[@foobargorch]

Isn't that just for the product of the jacboians? (I don't mean the general formula discussed in the lecture just the one in 36:00) Or am I wrong in thinking that e.g. under an affine coordinate system ∂zi/∂zj ≠ {0,1} for at least one combination of i≠j?

[@MathTheBeautiful]

+foobargorch ∂zi/∂zj is always δ^i_j. E.g.: ∂x/∂x = 1, ∂x/∂y = 0, ∂y/∂x = 0, and ∂y/∂y = 1. The difficult part here is to keep track of when "x" stands for a function and when for independent variable.

[@geographymathmaster]

+MathTheBeautiful Hello, thank you very much for the content. I was wondering why you said why \delta_i^j is technically different than \delta_j^i. I mean, they are always equal since: "i eq j" is equivalent to "j eq i", so I was wondering why you said at 36:00 that there is a technical distinction. I know that there is a technical distinction between \frac{\partial z^i}{\partial z^j} and \frac{\partial z^j}{\partial z^i}, so is this what you are referring to? Not the kronecker's delta?

[@MathTheBeautiful]

+geographymathmaster You're exactly right. I didn't mean "technically" - I meant "generally". For most other objects the order matters, but because the Kronecker δ is "symmetric" (which is a term that needs to be clarified for an object with an upper and a lower index) - in this particular case the order doesn't matter so we don't pay attention to it.

[@iyalovecky]

Pavel Grinfeld uses that r(x(r,o), y(r,o)) = r, ok, we know it's true in case of cartesian/polar coordinates, but Is it true in general?

[@MathTheBeautiful]

iyalovecky Yes. You should approach it in two steps: 1a. Two functions f and g (like x and 1/x, or ln(x) and e^x), are - by definition - the inverses of each other if f(g(x)) = x. 1b. Two sets of functions f1, f2 and g1, g2 are the inverses of each other if f1(g1(x, y), g2(x, y)) = x f2(g1(x, y), g2(x, y)) = y 1c. Two sets of n functions f^i and g^i and the inverses of each other if f^i ( g(x) ) = x^i (where the indices of all function arguments were suppressed. Good idea to write it out for n=3) 2. If you have two sets of coordinates, x^i and X^i, and two sets of functions f^i and g^i linking the different coordinates: x^i = f^i(X) X^i = g^i(x) then - by definition - the sets of functions f^i and g^i are the inverses of each other. So the answer to your question is YES.

[@iyalovecky]

MathTheBeautiful Thank you very, much. Also want to say, that i ordered your book yesterday, hope it will help me learn more deeply and faster!! Thanks again :)

[@brandontumeo4048]

MathTheBeautiful, I have taken up to Calculus 3 (Multivariable Calculus) but no Linear Algebra. what mathematics should I take in order to truly understand your lectures? (Note, I already have your book! :D )

[@MathTheBeautiful]

Certainly linear algebra. Tensor calculus does for curved things what linear algebra does for straight. And Calculus 3 and Vector Calculus

[@brandontumeo4048]

MathTheBeautiful Would elementary linear algebra suffice or would you recommend a higher level of linear algebra?

[@MathTheBeautiful]

Brandon Tumeo Linear algebra is good for everything. Just watch my or Professor Strang's lectures.

[@geographymathmaster]

+Brandon Tumeo There are also lots of linear algebra stuff on this channel. I'd recommend looking at it.

[@MathTheBeautiful]

+geographymathmaster Me too!

[@andrerossa8553]

nice, tks

[@parthsas8002]

1) Does anyone know how to speed up the video in the youtube from settings? 2) Does anyone has the answer to the last question? I just want to cross check my answer.

[@feynstein1004]

1) Click the cog icon on the lower right corner of the video. It will open up settings where you'll find speed : normal as default.

[@BlackEyedGhost0]

Probably should've used something besides "z i prime" for the second set. I feel like "w j" would've been a better choice to make it more readable.

[@MathTheBeautiful]

+BlackEyedGhost Notation is a personal choice, but the notation that I used is very carefully thought out and Z^(i') has many benefits - you'll see!

[@BlackEyedGhost0]

I've since watched more videos and now understand the benefits of using the same symbol since it implies a relationship that allows raising and lowering indices.

[@astroza_science]

Liked the joke from 48:26

[@johnheneghan3518]

Schaum's uses Bars instead of primes to indicate admissible coordinates. (Remember the differential is a linear map too.)

[@MathTheBeautiful]

It's a good suggestion. That said, "alternative coordinates" discussions (and, therefore, primes) don't come up once the technique is established.

[@prathamsaxena4801]

01:19:15 Lol!

[@IXSigmaXI]

well I sure am making mistakes at an alarming rate, so I got that going for me.

[@MathTheBeautiful]

same here!

[@Bel_Riose]

1:04:03 "I don't even see the code. All I see is blonde, bunette, redhead..."

[@terryphi]

also, I really find the content useful, but I find the delivery a bit slow. just fyi

[@MathTheBeautiful]

Many viewers watch these videos at 1.25 or 1.5 speed.

[@enricolucarelli816]

Terry Price I think speed is perfect. As you comment, one can always watch it at higher speed if his/her brain can keep up with it. Thank you very much for all these excellent lessons. People like you are the real heroes of modern time.

[@feynstein1004]

Slow? I tried watching it at 1.2x speed and I couldn't understand anything he said.

[@keshavl1089]

MathTheBeautiful we are not einstein 😂😂😂😂

[@zubairkhan-en6ze]

Very few may understand the depth of these lectures..

[@MathTheBeautiful]

I hope that's not true!

[@zubairkhan-en6ze]

@@MathTheBeautiful I meant that if someone has read history of geometry, linear algebra, calculus of variation and then watch your lectures he will appreciate the depth of these lectures.

[@sir.charless.7443]

He looks like a combination of frasier and john green

[@jackdkendall]

Haha, mu-2 = mewtwo

[@RanEncounter]

Also know as the pokemon variable. :D

[@feynstein1004]

To catch them is my real quest, to train them is my cause. Never more so than here.

[@BArdekani]

This would have been much clearer if instead of Z and Z' you simply used u and v.

[@MathTheBeautiful]

Valid thought, but you'll change your mind!

[@BArdekani]

@@MathTheBeautiful What makes it a little confusing is that the prime is sometimes applied to the variable as in Z' and sometimes applied to the index as in Z^i'. So if we are using the prime notation to denote a different coordinate system, I'd rather see Z^i and Z'^j and keep the prime next to the variable.

[@qwadratix]

Physicists do it in four dimensions. :D

[@hisxmark]

But it also works in Hilbert space, R(n), and it starts to get interesting in Minkowski space.

[@ChannelMath]

we all do it in 4-d

[@hisxmark]

ChannelMath >>"we all do it in 4-d" If string theory is correct, at least in 11-d!

[@Entanon]

Thats nothing. When I do it in 4 dimensions, I move at the speed of light in dimension 0. (The time dimension)

[@feynstein1004]

+hisxmark Minkowski space *is* 4-dimensional, three spatial and one temporal.

[@ProfeshenulRedneck]

omg index mind fuck... so many variables....

[@kapilk1644]

haha yes I hope I get used to seeing all the indices :\

[@MathTheBeautiful]

Well, I always quote Elie Cartan who didn't like "orgies of indices". I also quote Hermann Weyl who pointed out that the alternative is "orgies of formalism" with "endless profusion of name and symbols in addition to an intricate set of rules for carrying out calculations". The best choice is to not choose between the two camps but to see the benefits of both points of view.

[@thePrinceOfPurpose]

you are too far away from the board

[@SalvatoreIndelicato]

please can you insert subtitles?