Tensor Calculus For Physics Majors 002 | Vector Transformations and Vectors as Matrices

Part II of the preliminary vector stuff section of this series on Tensor Calculus. We go over transformations through rotation, space-time interval invariance, transformation coefficients as partial derivatives, vectors as Matrices (Bra-Ket Notation), outer products, completeness, calculating matrix elements, and change of basis.
Link to Tensor Calculus for Physics Book:
www.amazon.com...

[@michaeljburt]

Just throwing this in here, but dude, this is a masterclass series on tensor calculus. Your derivations are concise and extremely valuable. Too many professors like to gloss over details of something "they once derived". To be able to do it on the fly and off the top of your head shows a truly deep understanding. Well done

[@luisgeniole369]

it's far more elegant to factor out the dx² and dy², so you drop the sin² + cos² = 1

[@mikevaldez7684]

Michael Burt, "On the fly" ? "Off the top" of his head? Are you a kiss ass or just an idiot, maybe both? He stumbled through a very elementary introduction😂

[@michaeljburt]

@@mikevaldez7684 yeah, I guess it is really too bad that your mother didn't love you

[@user-ki6pt2zg1h]

Watch this video examination in life never give up-kzbin.info/www/bejne/oWXaZaOXoJtpjJI

[@pneptun]

@@mikevaldez7684 this video is pure gold. if you don't see it, no problem, but let others enjoy it. what i always hated in college (which was pre-google and pre-youtube when you couldn't just "look it up online") was "i leave this derivation to the reader" or "you can easily derive this at home" - well i DIDN'T WANT to spend a whole afternoon (or more) deriving this stuff at home, or having to go to the library and look it up in a book. just show me the damn thing. exactly this attitude is what makes maths/physics more inaccessible than it needs to be. lot of the text books just skim over derivations and proofs, skipping 5 steps at a time, assuming you'll "fill in the blanks in your own time", where they could easily have spelled everything out and spoon fed it to the reader. that's why i'm glad today's kids have it much easier with videos like these. i totally remember hating that the professors forced us to do half the work ourselves, totally unnecessarily.

[@gabriellove4361]

you're such a great teacher. you're way of explaining linear algebra is so intuitive, its actually been super helpful with my first year math course even though it doesn't come close to covering tensors. really appreciate your work.

[@AndrewDotsonvideos]

Thanks a lot!

[@mritunjayaparashar7996]

Really man this was the best your way of explaining was more effective as you covered the vectors first, I struggle when writing vectors in dirac notation and this video cleared some of my quantum mechanics problems too !.....keep up the good work and thank you 🙏

[@scienceexplainedsimply8115]

I'm a junior in high school and I'm using this series to learn about tensors, and I'm so happy that I'm able to understand this! I've always wondered what a tensor is.

[@luckygamer9197]

alright, so what's a tensor 😮

[@mohzmagdy]

I remember being around here in the channel 4 years ago while in highschool when you were posting this sort of content, and I was eagerly waiting to be able to understand it. Now I have an exam in a couple of days and was just searching up tensors on youtube when I found this video again lol

[@josephmoore4764]

If you rotate a camera by theta, the scene in the video appears rotated by -theta. Best way I can think of it in my head

[@user-ki6pt2zg1h]

Watch this video examination in life never give up-kzbin.info/www/bejne/oWXaZaOXoJtpjJI

[@kevivmodi7019]

@@user-ki6pt2zg1h bruh

[@callummilburn8204]

This lecture coarse supports my theory that you can learn anything if determined enough and you have the right learning opportunity and environment.

[@richardstone5096]

amazing teaching, you taught this so much better than my linear algebra classes, i actually understand the details without feeling like i'm just memorising rules

[@trumanburbank6899]

Another great video. In the rotated Cartesian coordinates part of your video: if we rotate the point, we have an active transformation and the angle of the point is changed from θ to θ + α. In a passive transformation, the angle from the new coordinate system to a fixed point is changed from θ to θ - α. In tensor analysis only passive transformations are used ("All is invariant"). Also, the equation of transformation between rotated coordinates can be perceived quickly, if you use the rule that the transformation coefficients are the 'direction cosines' between the various axis'. This is also true for rotations in 3D.

[@raunaksarada]

really dude what a nice explanation of basics for tensors

[@johnnyhackett199]

Dotson: All this is is the basic rules of matrix multiplication... Me: Damn it!

[@acatisfinetoo3018]

There is something terribly wrong with the education system when i can learn from a youtube video more than i ever could learn in university.

[@lezliebeans5639]

Dude, your lectures are gems! My Mathematical Physics professor is totally on crack or something, and all he does is waving hands for couple of minutes and saying bullshits and done... Your lectures are very detailed and they're probably gonna save my ass on the upcoming days....

[@raunaksarada]

really appreciate your work

[@apolloniuspergus9295]

Appreciate*

[@Mforader1792]

Thanks again dude makin lap 3 lol

[@allrounder2367]

You are a legend Sir! Alot of love from Pakistan. I was searching for a video on spherical coordinate and nothing cleared my concept but then I saw your video and I learned it.

[@pranay1546]

This is a sick tensor series, really love it

[@catherinegrimes2308]

Do you mean slick?

[@VerkehrtesteWelt]

Everything you explain, I can understand! I would prefer you over many of the professors I already had on my tecnical university.

[@muhammadasifzubair4923]

totally loved it man . the way you approached tensor . huge thumbs up bro

[@auroisflying]

As a first timer approaching tensors i really appreciated this

[@underfilho]

I finally understand why I studied Linear Algebra in the physics sense

[@ozgurnazli_]

by watching and studying with writing this series I learn so many things, even my handwriting is fixed

[@AndrewDotsonvideos]

😂

[@Mforader1792]

Dude you have NOOOOOO clue how much this helped!!!! Much appreciated my dude again🫡🤙

[@Mforader1792]

Like here we are about a half year later and well......its going awesome!

[@Mforader1792]

Except all the world crap going on. War el presudenté thinger. Mane its gettin real out there. Lol

[@ahsanrizvi2205]

It is amazing how you simplify these topics same as our toper friend a night before exam😍

[@bgjhgjghkjuhgbg]

I expect that in the next videos I'll start being confused by indices being on top or bottom of the letters. So I wonder if there's already a reason here why they are up or down, or if it is something I'm supposed to understand later in the course.

[@monkeyemperor1223]

it becomes clear when you start to do transformations and the indices change from top to bottom. If you want to express exponents, just put parentheses around the variable

[@dr_drw]

Thank you bro you make this is so simple and explain it so well!

[@apolloniuspergus9295]

As a nine year old, I love your explanations!

[@Mforader1792]

Alright back again have a muxh better grasp because derivations!!!! No joke it helped correlate everything....like kinda from the get go! Thanks again dude! Ill d3f be back!

[@AndrewDotsonvideos]

@@Mforader1792 really glad you’re finding the videos useful🙌🏻 just let me know if there’s anything that could use some clarification brotha

[@zakariazaki7513]

Thanks for lesson keep going 🤠 greeting from Morocco

[@Ok-np9rp]

Those are some clean sigmas my dude. Also, using this to look ahead of some classes for some summer research. Appreciate the vids!

[@seehaahok]

Hello and happy Andrew. Thanks for this great video. Your help has been invaluable to me. Very clear and very well explained. I watch your lessons with great pleasure and interest:

[@account1307]

Brilliant video!

[@judedavis92]

Amazing!

[@miguelaphan58]

clear and neat...two chops up andrew..!!!!...quep in this mode....you will be great !!!!

[@sir_charlie]

mate seriously, biggie up cheers

[@EpicMeerGaming]

Hi Andrew--I have been a fan of yours for quite some time, and recently I've started to go through your tensor calculus playlist to learn some more. I am a little confused about the last section. I understand that we are treating the change of basis from the i system to the alpha system as a transformation from A to A prime, but I am confused as to why you can equate the general projection equation (the summation of the bra-ket times the ith component of A) to the transformation of coordinates that was defined only in the special case of rotations (the summation of the jth component of A times its component derivatives). Surely the projection onto a basis cannot always be described by the rotational transformation of coordinates? Or am I missing something and does the coordinate transformation equation hold more generally and not just for rotations?

[@AndrewDotsonvideos]

Really good question and your suspicion is correct. I mention that these are two different representations of the same vector, but I don't think I made that very clear and I especially failed at clarifying that the transformations we were considering were the same. means I want the the component of A along the direction of alpha. And we know the components of A in a different basis (the basis summed over latin index i). You can absolutely consider one basis to just be a rotated version of the other for this example, but it doesn't matter what the transformation is. It just matters that the transf. are the same. So I'm really just demonstrating that the two operations (partial derivatives of one basis with respect to another and the sum over the inner product of the basis vectors in the 2 different bases) amount to answering same question: Given a vectors components in one basis and what the transformation is, what do the components look like in another basis? As an example of how these answer the same question. We wrote the partial derivative transformation rule from the rotation of cartesian coordinates: x' = x cosw +y sinw y' = -x sinw + y cos w We can instead just take the inner product of the new basis with the old, since it's already in terms of the old: x'.x = cos w x'.y = sin w ... etc But we already know cosw = partial x'/partial x, etc, so then you see they're the same. So we could just as well have written that transformation rule as x^i = x^alpha sum over alpha. The bra/ket expression is just also supposed to help people who had a little bit of quantum mechanics realize that they're more familiar with this concept than they think. I hope this helps, I had to rewatch the video to remember what my point was 🤣

[@EpicMeerGaming]

@@AndrewDotsonvideos Thank you so much for the thoughtful response! This definitely helps.

[@vwcanter]

These are great lectures. I was a little surprised when i cross i and j cross j were not zero, when you multiply them as a column then a row. You called it the out product, but i expected that to be identical to the cross product But then i remembered when you foil out to vectors, the cross product is everything except the diagonal. So the fact that the diagonal is all 1's tells me this is compatible with the old fashioned cross product. But i have to think about it some more

[@gautomdeka581]

At the very last as u compared the derivative of prime wrt un prime with the matrix , well i did not get that point how could u compare that since A'i ≠ Ai

[@chrisryan6464]

At 12:30 you multiply everything by 1/dt to get the velocities, but how does x/dt = v ? Shouldn't that 1/dt be d/dt?

[@rudrayangooptu9476]

Possibly a naive question but, the "transformations" we're using to define vectors, these do not imply a change of basis right? Like the resultant components are still going to be represented in the original basis?

[@khansaahmad2558]

The last thing you did didn't went clear, how you skipped summition over i in the last result

[@user-en5vj6vr2u]

It works out if you eliminate the subscripts on the Lambda and treat it as a square matrix. The Lambdas with subscripts are just the elements of that matrix

[@klong4128]

Good physic-dirac bra-ket maths-operator application .

[@dibyadarshandash2255]

why the chain rule not followed while differentiation at 6:40

[@monkeyemperor1223]

Hey Andrew, why do you multiply by 1/dt instead of taking the derivative and then call it velocity and acceleration? I thought vector calc still applies here

[@juanfa98]

Really nice, I just have one question. Is the outter product commutative? Becouse if it was it would fuck some things up, wouldn´t it?

[@rafaellisboa8493]

man i love your videos

[@criticalthinking575]

Dirac notation is tricky in the first look.. It's just for me Or everyone think so

[@MrKatana333]

Hey, thanks a lot for this!

[@sonoda7723]

Hi really great video, I have just one question: isn't the alpha an upper index in the lambda at 38:40?

[@baptistebauer99]

I was reading the comments and I found yours, which presents in my opinion a really interesting question. I'm no expert in tensors myself, so I went to the book Andrew's following to see if it was mentioned - and it was: quoting the book, "the Λ matrix elements are not tensor components themselves; subscripts are chosen so the notation will be consistent with tensor equations when distinctions eventually must be made between upper and lower indices". I hope it helped :)

[@SoowDeJu]

You actually really look like Sam from the youtube channel Node! You could easily pass for his brother!

[@peterclark5244]

So for clarification: 1. Tensors are equivalent to matrices as operations between vector spaces, where tensors have components specifically that follow the partial derivative transformation rule T_ij = dx'_i/dx_j? 2. A tensor is invariant iff |det(T)|=1?

[@jonasdaverio9369]

1. Tensors are more than matrices. You will probably discover in the next video. 2. By invariant, do you mean invariant under change of coordinate? If so, every tensor is invariant under change of coordinate system. Tensors are independant from coordinate system (and, in particular, vector are invariant). Only their components differ.

[@angelmendez-rivera351]

Peter Clark No. Tensors with two indices are equivalent to matrices in component space, but tensors can have any number of indeces, so this would not be true if the number of indeces was any different than 2. For instance, single-index tensors are vectors in component space, not matrices. Actually, I suppose all vectors are single-column matrices, and the true may be said for scalars, which are 1-by-1 matrices, but the same cannot be said for tensors of three-indeces, which are most definitely not matrices. Also, tensors represent objects in component space, but the object they represent is by definition always invariant, since they represent a physical object or quantity of a system.

[@ritamandi5196]

thank you

[@mohtasimtamjeed]

hey you went to the same school as Goku!

[@davidepascu3026]

Asyam Abyan he's wearing the Goku shirt

[@mohtasimtamjeed]

​@Asyam Abyan the Turtle Hermit school

[@mikevaldez7684]

Dotson

[@uniziakharchandy4452]

thank you for helping out noobs like myself

[@royalsgaming03]

why did you take theta in minus , can i do it with positive theta

[@jonatanpaschoal7362]

Fiquei com dúvida em 37:25. Mudança de base.

[@erickmubai9157]

The inner product in called contraction. what about the outer product?

[@anoshwasker4635]

At 16:20 you talk about x' = x_1 x^2 = y_1 x^2. Where are we getting that from? I'm stuck at where we get those from.

[@gamerdio2503]

He introduced this notation in the last video. x^1 is defined to be x, x^2 is defined to be y, and x^3 is defined to be a. Its important to note that these are NOT powers. We're not saying x squared is y, we're saying x with a superscript of 2 is the same as y

[@juliecrawford9428]

Andy, what is it that you might like to do with your education ? If you had to choose between math and physics, which, and Why?

[@handhdhd6522]

He chose physics, this math is needed for PhD physics research, it’s beyond difficult

[@Maxwell_Integral]

For the transformation to the A prime alpha frame of reference, shouldn't the equation be, "The sum over i, lambda alpha i, A"?

[@oni8337]

Yeah I got confused about that too, but I think he is probably just representing lambda as the full matrix describing all inner products between alpha and i basis vectors which would succinctly summarise the transformation of an arbitrary vector from one frame to another

[@Maxwell_Integral]

@@oni8337 thank you

[@jeromeperlstein7388]

There is some hand waving going on near the beginning of this video which should be pointed out. In deriving the transformation of coordinate axis (x,y) to (x',y'), he does so not by rotating the coordinate axis, but by rotating the position vector and keeping the coordinate axis fixed. The result is that (x',y') are the coordinates of a NEW position vector in the ORIGINAL reference frame. That this happens to be equivalent to rotating the reference frame in the opposite direction and keeping the position vector fixed is not mentioned.

[@spb1179]

Oh wow this was interesting. These first few videos are flying and I learned a lot. I doubt I can keep up in the next one considering I'm in highschool rn. Why did dS`^2 have to equal dS^2? Also at 23:04 what does the complex conjugate thing mean? Other than that the whole video made perfect sense until the last 3 minuets. I think I got tired and will try and watch that part again tomorrow.

[@aarnavchaturvedi2552]

Aye I'm in highschool too. So here's my shot at this 1)Essentially the whole idea is that regardless of your choice of cordinates, your vector "length" needs to be fixed, that's how the transformation is defined, one where length is preserved 2) Complex conjugate is basically when you have a complex number x = a +bi, conjugate of x = a -bi. So when you multiply a complex number by it's conjugate, you get a² + b²

[@spb1179]

Aarnav Chaturvedi I need to start over on this series lol I left it here and have missed a lot of it. I probably need to learn more linear algebra and the next videos would make more sense. Keep up your learning in this time of isolation! Best of luck

[@afifakimih8823]

Excellent lecture bro!!

[@meeharbin4205]

asalamu alykum

[@afifakimih8823]

Waa-alaikum Salam.

[@murad3460]

Dude i want to study Physics,but math seems too hard.Should i risk it?

[@AndrewDotsonvideos]

You're introduced to the more sophisticated math gradually. It's really not that bad.

[@ThePremez]

It's doable, got through the khan academy playlists meticulously. Do the linear algebra playlist and the single variable calculus playlist before your go to university. The differential equations playlist will help as well.

[@KhoaNguyen-os5zv]

No proof heavy so don’t stress out

[@XanderGouws]

As long as you stay on top of your classes you should be fine. Don't save the homework for last minute - especially for math. Also, practice as much as you can - if there are any extra practice questions, do them.

[@thehippievan1288]

Try Professor Leonard on KZbin, his lectures make wonders! :)

[@ElDiarioLudita]

Hi! What book do you use for Tensors? i am starting it and i am really interest. So much thanks for this course.

[@baptistebauer99]

He said it in the episode one of this series :) He's using Tensor Calculus for Physics, a concise guide, by Dwight E. Neuenschwander. He is actually following the book's content very closely.

[@ElDiarioLudita]

Hi Bauer. Sorry, but, you have the book for free? Can you spent to me the book? I will be so thanksfull to you enormly. My gmail is: [e-mail] too many thanks!!!

[@housamkak646]

use libgen.is for downloading any book u want

[@pacolibre5411]

37:30, what does the alpha refer to? I get that it has to do with “a new basis” but is alpha just the name of the basis? Is it an index? Is it a list of orthogonal basis vectors? If its just a name, how can you sum over it?

[@physicsfaith]

alpha is the symbol for the basis. Just like we used i to mean the normal set of basis vectors, he picked alpha for a different set. He then summed over the set of the alpha basis vectors. Its just a symbol to represent a new set of basis vectors.

[@volbla]

When he first writes he says it finds the alpha'th element of A. That makes it sound like it's a unit vector? In the top row i is the i'th unit vector that gets iterated over. But in this sum alpha doesn't depend on i. So is it a matrix, and selects the i'th basis vector of alpha? Quite confusing.

[@jack48382]

12:30 Mathematicians triggering xD

[@zenithalizesquads4873]

Anyone who saw this comment, please answer. How come that |A^'α ⟩=Λ_αi |A⟩? But really, I am confused that, how can he compare the term ⟨α│i⟩=(∂x^'α)/(∂x^j ), because it is from different summation, by the context different I mean, I do understand that if the function is the same and you can express it into 2 ways, you can compare the term, but A^i and A^'i isn't the same, A^'i is the transformed one. Please correct my understandings.

[@mariadaniela5813]

Hey, you just take the alpha as your new basis (so you can write it with the ') and i as your old basis, so if you look at equation (1.82) from Tensor Calculus for Physics by Dwight E. Neuenschwander, it's equivalent, because can be written in that previous notation as (∂x^'α)/(∂x^i), you just have to change the index according to this example :)

[@erfanmohagheghian707]

You should know there's an obvious difference when you rotate a point or the axes. Your derivation was for rotation of a point theta CCW and that's why you got the discrepancy initially.

[@angelmendez-rivera351]

Erfan mohagheghian Not really. For every rotation of a point, there exists an equivalent rotation of the axes, and vice versa. Furthermore, the rotations are inverses of one another. So, yeah, not quite. If anything, the explanation lies with the fact that he used inverse rotations. Hence the - sign. This is exactly what he said, though.

[@erfanmohagheghian707]

@@angelmendez-rivera351 He knew the answer but got it wrong initially, then corrected himself! If he had noticed that he should have rotated the axes he would have got the correct formula at the beginning. As simple!

[@angelmendez-rivera351]

Erfan mohagheghian So? He still explained the math correctly. In fact, the first set of equations he wrote was not even technically intended to be the same. The first was motivated by wanting a definition, the second by diagram. The diagram need not apply to the first set, so saying he obtained the wrong answer is dishonest and a misunderstanding of how pedagogy works.

[@erfanmohagheghian707]

Are you okay? He derived the equations for rotation of point not axes. He's so knowledgeable but he made a mistake, and my problem is not the mistake, it's that he doesn't prepare fully before recording.

[@angelmendez-rivera351]

Erfan mohagheghian Am I okay? I am perfectly fine. I think you are the one who is not okay here. I already said everything that needed to be said concerning this issue. His formula IS for the rotation of axes. Remember what I said: every point-rotation corresponds to an axis-rotation. Since the angle is arbitrary, he can just rename it. So, no, he did not make any wrong derivations. If you do not understand that basic mathematical concept that even a 7th grader can understand, then clearly it is you who is not fine here and needs help. But I cannot help you with your delusions if you have any. That is all that needs to be said. End of the conversation, and I hope you never talk to me again. Good bye.

[@hr1623]

The equation you are writing at 37:28 confused me a little,how did that come? The rest of the video was great👍

[@hydraslair4723]

Really late, but still: it comes from the above equation for |A〉 after multiplying on the left by 〈a|.

[@zendium1]

A^i=

[@samuelhawksworth1923]

At 14:10 I don’t understand how the derivatives equate to that, cos theta differentiates to -sin theta doesn’t it?

[@lunkel8108]

If you take the derivative with respect to θ. But we're taking the derivative with respect to x, something cos doesn't depend on, so it's just a constant for our purposes

[@timetraveler5128]

23:21 Why do we take the complex conjugate?

[@physicsfaith]

In Dirac notation (which you probably have never seen unless you've taken a few quantum mechanics courses or linear algebra with imaginary numbers), the vectors are represented by kets, that's the |i> symbol. The ket is a column vector. There is also a bra vector,

[@datsmydab-minecraft-and-mo5666]

34:30 so true :)

[@mikamikamusic7792]

16:42 where did the j index come from

[@AndrewDotsonvideos]

The j index is a dummy index which is being summed over. Just write out the sum explicitly term by term to see that the only real index that's "actually there" is the i index. This is how it should be since i is the only free index on the LHS.

[@phelipeantonie6638]

I love you

[@Mforader1792]

Annnnd we are baaaccck lol

[@ahmedkarrar25]

Hey guys, can anyone tell me how we get x', y' and z' in the beginning of the video?

[@zenithalizesquads4873]

From the transformation of the coordinate system, which transformation can be done by rotating the entire 2D space x,y around the z-axis by angle \theta, either clockwise or counter-clockwise. Then you can prove it by using the trigonometric functions.

[@gappythegoat5397]

Yeet boi

[@yuvraj7214]

Don't tell me that's how you write *lambda*

[@mritunjayaparashar7996]

Elvie Shane its uppercase lambda that's how you are supposed to write it.

[@alexismisselyn3916]

graduate physics and not know the 6 first letters of the alphabet, haha great video abc efg h

[@baptistebauer99]

I think it's because in the book he's following, the author (purposely?) explained it the exact same way, by "skipping" a letter :)

[@angelmendez-rivera351]

Alexis Misselyn Some letters are skipped because they would be confusing. For instance, e is a number, so also using e as a variable or index is extremely confusing, which is why it is never done. The same is true with using the letter O. In font, it is much easier to distinguish from 0, but when written by hand, it is just unnecessarily confusing, some conventions strictly ban the usage of the letter for any notation. It is not a matter of not knowing the alphabet, it is a matter of using convenient notation.

[@micayahritchie7158]

First?

[@zokalyx]

sorry I didn't see your comment. I mean it didn't appear when I was writing down my comment. You must be a fast boi

[@mileslyfe5239]

[@joaquin6719]

Physics=write fancy vectors symbols

[@bhaskar08]

The notation is disturbingly confusing.

[@mattraymond1497]

welcome to tensors lmao

[@angelmendez-rivera351]

Bhaskar Bhardwaj That is 75% of the difficulty of understanding tensors. They do not sell tensor notation textbooks for nothing.

[@jamesyeung3286]

physicists ran out of symbols and so they reused old notation lmao.

[@physicsfaith]

I think he is trying to motivate tensors by showing you that vectors transform the way he will show you tensors transform. He picked the example of rotation of basis vectors on an x-y axis. He then wanted to show people who have taken QM that Dirac notation is handling a lot of this transformation as well, but you probably didn't realize. I suspect a lot of viewers have not taken enough QM to understand his point with Dirac notation of vectors. In fact probably only the advanced undergrad or graduate viewers have ever seen Dirac notation, so just enjoy the ride. :-)

[@Panice111]

That's why I prefer maths major, its less calculation in this way haha

[@sciencestararvinsinghk]

As a thirteen year old, I love your explanations!

[@sciencestararvinsinghk]

Third!

[@micayahritchie7158]

Science With Arvin u understand all that?

[@sciencestararvinsinghk]

Yeah. I've been teaching myself calculus since seventh grade, two years ago

[@manishsingh-vk8if]

@Mathew Brigugliomost probably, just to look smart and brag about.

[@non-inertialobserver946]

@@manishsingh-vk8if or because he was curious and couldn't wait until 11/12th grade