Tensors for Beginners 0: Tensor Definition

Less of a focus on physics in this one. More of a focus on the mathematical definition.

[@korwi7373]

"a tensor is a crazy bababoey"

[@user-fc8xw4fi5v]

A tensor is a fit ting innit

[@prateeksin]

This is the best and most comprehensive introduction to tensors available on KZbin. Thanks for your hard work Chris. I have been trying to get the fundamentals of General Relativity and trying to gather courage :-) This helped me a lot. Many thanks

[@eigenchris]

Glad you like them. I do plan on doing the basics of GR at some point in the next 6 months.

[@9846lcs]

This is the best explanations on tensor I have ever seen.

[@frankdimeglio8216]

@@eigenchris Consider what is TIME. Consider what is E=MC2. Consider what is physics/physical experience as it is seen, felt, AND touched. Consider what is THE EARTH/ground !!! Importantly, gravity is an interaction that cannot be shielded (or blocked) ON BALANCE. TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE, AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE). WHAT IS E=MC2 is dimensionally consistent. c squared CLEARLY represents a dimension of SPACE ON BALANCE. I have proven the fourth dimension. E=MC2 AS F=MA CLEARLY PROVES (ON BALANCE) WHY AND HOW THE PROPER AND FULL UNDERSTANDING OF TIME (AND TIME DILATION) UNIVERSALLY ESTABLISHES THE FACT THAT ELECTROMAGNETISM/ENERGY IS GRAVITY: A PHOTON may be placed at the center of what is THE SUN (as A POINT, of course), AS the reduction of SPACE is offset by (or BALANCED with) the speed of light; AS E=mc2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity. Indeed, the stars AND PLANETS are POINTS in the night sky. E=mc2 IS F=ma. Gravity IS ELECTROMAGNETISM/energy. Time DILATION ULTIMATELY proves ON BALANCE that ELECTROMAGNETISM/energy is GRAVITY, AS E=mc2 IS F=ma. Indeed, TIME is NECESSARILY possible/potential AND actual IN BALANCE; AS ELECTROMAGNETISM/ENERGY IS GRAVITY; AS E=MC2 IS F=MA. Great. "Mass"/ENERGY IS GRAVITY. ELECTROMAGNETISM/ENERGY IS GRAVITY. E=mc2 IS F=ma. (Very importantly, outer "space" involves full inertia; AND it is fully invisible AND black.) BALANCE and completeness go hand in hand. It ALL CLEARLY makes perfect sense. I have mathematically unified physics/physical experience, as I have CLEARLY proven that WHAT IS E=MC2 IS F=ma in what is a truly universal and BALANCED fashion. Consider TIME AND time dilation ON BALANCE. c squared CLEARLY (AND NECESSARILY) represents a dimension of SPACE ON BALANCE, AS gravity/acceleration involves BALANCED inertia/INERTIAL RESISTANCE; AS ELECTROMAGNETISM/energy is CLEARLY (AND NECESSARILY) proven to be gravity (ON/IN BALANCE); AS the rotation of WHAT IS THE MOON matches the revolution; AS “mass”/ENERGY involves BALANCED inertia/INERTIAL RESISTANCE consistent WITH/as what is BALANCED electromagnetic/gravitational force/ENERGY; AS TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE. ELECTROMAGNETISM/energy is CLEARLY (AND NECESSARILY) proven to be gravity (ON/IN BALANCE). GRAVITATIONAL force/ENERGY IS proportional to (or BALANCED with/as) inertia/INERTIAL RESISTANCE, AS WHAT IS E=MC2 is taken directly from F=ma; AS gravity/acceleration involves BALANCED inertia/INERTIAL RESISTANCE; AS TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE. Indeed, consider WHAT IS the fully illuminated (AND setting/WHITE) MOON. Consider what is THE EYE ON BALANCE. Consider what is the TRANSLUCENT AND BLUE sky ON BALANCE !!! Consider what is the orange (AND setting) Sun ON BALANCE. Consider what is THE EARTH/ground ON BALANCE !!! Again, gravity is an interaction that cannot be shielded (or blocked) ON BALANCE. c squared CLEARLY represents a dimension of SPACE ON BALANCE, AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE); AS TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE. Consider TIME AND time dilation ON BALANCE. Again, consider, ON BALANCE, what is the fully illuminated (AND setting/WHITE) MOON. WHAT IS E=MC2 is taken directly from F=ma, AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE); AS TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE. I have mathematically proven and CLEARLY explained (ON BALANCE) why AND how the rotation of WHAT IS THE MOON matches the revolution; AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE); AS TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE. I have mathematically unified physics. I have CLEARLY proven what is the fourth dimension. By Frank Martin DiMeglio

[@neil6477]

It has been a long time, Chris, around 50 years, since I've actually been excited to learn something about maths. So credit to you. Thanks a lot - I'm really looking forwards to delving into this stuff. Just hope my brain hasn't atrophied too much! Thanks again.

[@spspurdue9280]

Absolutely the best instruction for tensor available on the internet

[@BRUSCKORB]

I usually never like, comment and subscribe to videos I watch, but man you are insane. You are the first person I've seen that actually explain what a tensor is instead of throwing a bunch of mathematical stuff. Ty

[@iagojacob3785]

"A tensor is something that transforms like a tensor"

[@c.l.368]

Lol yeah, how can the definition of a tensor be that it's a collection of vectors? Aren't vectors tensors?

[@iagojacob3785]

@@c.l.368 tensors are colections of tensors inside colectins of tensors, them u hav a fractal

[@canyadigit6274]

C. L. The definition Chris used was actually one of the most perfect definitions I’ve seen. You can say it’s a “collection of vectors and covectors combined with the tensor product” because the definition of a vector is not a tensor. You can define a vector as a rank 1 tensor but that’s not the only definition. A vector is a mathematical object that can be scaled, rotated, not divided, and can be added and subtracted while being invariant under a change of coordinates.

[@iagojacob3785]

@@canyadigit6274 Can be a nice definition but in the moment i dont have the necessary iq to make any math with them ;-;

[@ibrahimyakupaydn]

@@canyadigit6274 nice definition. Thanks

[@GrantGryczan]

Thank you. This is the only video I could find on this that actually makes sense.

[@sebastianelytron8450]

@BLAIR M Schirmer That one is crap lmao

[@RAMBOTNPC]

Sebastian Elytron is very good in fact, for me that I’m a student that this kind of things haven’t been teached. He explained the basics of vectors, what I want to learn. The definition of tensors was unclear, the tensors part was made for an experience guy in that area.

[@crehenge2386]

That means you don't understand it then. He's very sloppy and often wrong. If you don't get tensors it's probably because you don't understand multivariable calculus and linear algebra enough.

[@GrantGryczan]

@@crehenge2386 Or perhaps, more simply, it's a clearer video to me than the other ones I've found.

[@distendedmist5840]

oh you are the vanilla tweaks guy

[@BiffBifford]

THANKS for making this topic so much more understandable! The best explanation I have found so far.

[@uncouthyouth2433]

Brilliant video Chris. You explained without overwhelming or undermining the listener. Thank you

[@chrisladd5855]

I love the way you break it down. The pencil pointing to door being invariant with the coordinate systems being the varying components was so easy to understand. Nice.

[@MarkHewis]

I have finally got it! This is the best introduction to what are tensors out there!

[@drwhackadoodle360]

Thank you, I have been reading about relativity and stuff on my own and it is super helpful to have everything distilled in such an easily digestible way. This is really good!

[@fgvcosmic6752]

I've just completed my Vectors and Matrices course, where we very briefly mentioned Tensors (in the form of the Levi-Cevita alternating symbol) and I got curious as to how they actually work. This was a great introduction and I'm absolutely going to binge the rest of this series

[@ourfalli]

This is very pedagogical, straightforward insight into the subject. This helps me a lot, thank you.

[@motmot2694]

Best set of lectures on youtube!

[@vincenttortora6330]

This is extremely important content. Thank you!

[@rojinhasan7922]

Ohhh mean. That's soo helpful, you don't know how much!! Thank you a lot!

[@mohammedkhan4990]

Excellent presentation!!!! Thank you.

[@larryhernandez536]

Simple, and to the point. Thanks

[@adrianloma]

excelent video, as always. Had a mini "mind blown" moment.

[@vavikapeter8843]

To my taste, this is the best introduction of tensors on the Net. Gracias, senior!

[@eigenchris]

Thanks!

[@peterrobinherbert]

A really good inroduction. I went through the series and the follow up series on tensor analysis. You don't really need to understand every single thing in all the videos to get a general idea of tensors. However I decided to go through from the very beginning and not to progress until I had thoroughly everything in each video.

[@patrickcollins8048]

Mega thanks for this explanation! I’ve looked at dozens of textual definitions, and i really didn’t understand any of them before this video 😅

[@AlI-xy9jx]

This answered all the questions for me, thank you so much

[@arunnavinjoseph9262]

Content, pictures, and visualization of equations are very nice. Can the volume be louder, please. Try to increase the lectures, looking for more!

[@alitareq1586]

Thanks a lot. You have explained in the easy way.

[@phuocnguyenbui71]

Thanks for this great video. It really sweeps me away.

[@michaelcheung6290]

Excellent! Please make more videos like this

[@Sleir_]

i'll be going through your course in my free time, i look forwards to learning with you!

[@PhysicswithRoky]

I went through several videos to have the intuition of Tensors. But this is the beast..!

[@thebcl1547]

You mentioned that the metric tensor gvu represents lines and rows of a grid, or later we understand it as components. Being a person with not much math bkgd, when I see u and v I wonder if they are only placeholders, that is their values depends on the components in the change of coordinates in a coordinate system. Btw, your videos are the best I can find on GR, etc for a person who doesn't know about math or physics.

[@sonamt4872]

Best video on tensor👍🏻

[@AniSepherd972]

very easy to grasp explanation please provide more topics! Subscribed ^^

[@carlosagorreta5276]

best tensor explanation on youtube so far

[@sunphysics]

After my post graduation, first time I understood the meaning of tensors. Thank you!

[@BangMaster96]

How the hell did you graduate without understanding tensors

[@harishsinghbhoj9692]

Yeh India hai yaha kuch bhi ho Sakta hai

[@sunphysics]

@@BangMaster96 In graduation our University didn't teach this. In MSc also it was not the part of course. But, while finding some foreign author books, I came across!!! Where you have studied in maths or physics!!!

[@jptuser]

In msc you didn't encounter tensor ..what.. tensors are everywhere in physics. From electrodynamics to quantum field to astronomy to cosmology .. ever branch has it

[@inquiringhuman2582]

@@jptuser Tensor were in course but only few university has very good teachers, so those who are not lucky simple skip the part.

[@physicspoint]

Exceptionally helpful

[@7177YT]

this is awesome, thank you!

[@Philmad]

Many thanks for all this work, best so far !

[@g3452sgp]

Concerning tensor definition, I like the geometrical definition of tensor you presented here , since this is the most intuitive way of defining tensor to me. So here , I put down my definition. " Tensor is the multi-dimensional matrix which incorporates geometrical structures in it." What do you think? And I want to see precisely what the geometrical structure in tensor is.

[@Andres186000]

A definition that may be more rigorous without being too much less accessible might be "a tensor is a mathematical object that can be expressed as a multi-dimensional array (or matrix) that has some features that don't change under some mathematical operations." To be fair, this definition sounded a bit more accessible before I wrote it.

[@mustafizurrahman5699]

Crazy video on tensor. Such a lucid explanation with geometric interpretation. Hats off. High school students who know little about geometry or vector can fathom the intuition of tensor. Love ❤you sir

[@thegamingsquad5885]

You're pretty much good because you actually have the passion of quantum physics and tensors well much of quantum chromodynamics

[@Jacob011]

A very important point here, that should be emphasized more to really hammer it in is the distinction between a vector and its components. This isn't at all obvious, because often we hear that a vector is determined by its components, in which case, then, how can vector remain the same when its components change? I love the series btw, by far the best explanations around and very helpful for someone like me who have always been curious about this stuff, but didn't have time for it at the uni.

[@ninabhuva2879]

This was amazing! Thank you, loads!

[@yousufnazir8141]

Good summarising of the analysis of the vector and tensor notations in algebra and geometry of the linearly combination with operations

[@aaravkansal9251]

Thank you Chris for making such good videos that even a mere 9th grader like me is able to understand it :)

[@darkfox8101]

I’m in the same boat as you.

[@abhayjaiswal7622]

I am also on the same boat as you! Hopefully we don't drown

[@thanasis2561]

thank you Sir for this series of lessons

[@josephmoore4764]

You have been the best resource I've seen for this, thanks. Also, you wouldn't happen to also be the guy who does Casually Explained? You sound exactly the same.

[@eigenchris]

Glad it's been helpful. I've actually never heard of Casually Explained. I'll have to check his channel out.

[@huraahaljabery814]

This was amazing Thank you

[@AnanyaChadha]

phenomenal content, thank you

[@Ivan_1791]

You are the best man!

[@WilEngl]

THANK YOU SO MUCH for making tensor calculus crystal clear !!!

[@narfwhals7843]

Hi, Chris! Are you familiar with the Dirac notation? If so, have you considered using it for tensor maths? It seems a very convenient way to write things down without index notation. |v> is a general vector,

[@eigenchris]

Yeah, I've mostly seen it used in quantum mechanics. For the tensor product, I've seen people either use the traditional ⊗ symbol, or just write the basis vectors side-by-side.

[@narfwhals7843]

@@eigenchris yeah I'm watching a clourse on qm(by professor m does science) and often when they explain bra ket and operator relations I keep thinking "wait, this is just tensor algebra". Which makes sense because it is basis independent linear algebra. I was just wondering if the notation was useful in other contexts.

[@eigenchris]

@@narfwhals7843 I guess it's a matter of preference. The "bra" notation is nice because it makes it very clear that the "bra" (covector) is supposed to act on a "ket" (vector) to produce a scalar. You could re-write all of SR/GR using bras/kets if you want. But I've never seen a textbook do that.

[@jeanpaulziegler587]

Parfaitement clair! En plus je comprends parfaitement votre Anglais. Merci

[@eigenchris]

De rien. :)

[@jamesdonalfaulkner]

@@eigenchris Tres clair, en effet. Bravo!

[@jamesdonalfaulkner]

Et bravo pour ne PAS saboter votre explication avec de la musique de fond....

[@bishalbanjara2891]

That was my confusion.....different people define tensors in different forms as you said.....that made me confuse....it's fine now...very nice video..

[@antonioesposito8071]

Hi Chris! great video but i'm more of a book kind of guy which book would you suggest for learning tensors?

[@myousefzadeh]

Dear Eingchris I would appreciate if you let me know the name of software or program you used for creating these fantastic lectures. I mean that fart of lecture that you write texts, math symbols and geometric figures, not video and audio parts of the lecture. The reason I’m asking this question is, I’m going to take notes from your lectures and writing with hand would be very time consuming. Thanks for these fantastic, professional and concise lectures for those love mathematics. Regards.

[@eigenchris]

I made the slides in Powerpoint. I can upload the slides to a online share in the next hour or two. I'll let you know when I've done that.

[@yogibrijkumar]

Nice explanation.

[@mahendrapratapyadav4684]

Thank you.

[@deconfinedQPT]

Can you suggest a book with exercises so that we can accompany your lecture series with it. Upon completing your lectures I would like to try myself with actual exercises too. However there are a lot of books around and I do not have any idea which to use. Thank you for your efforts and sharing your knowledge.

[@eigenchris]

I'm sorry but I don't have any recommendations. You'll have to try googling or asking someone else.

[@eigenchris]

I used a lot of online PDFs and course slides that I have lost track of. I did read the first few chapters of a book called Gravitation by Misner, Thorne and Wheeler to learn tensors and tensor calculus, but I'm not sure it's best for beginners.

[@bis9kashdd2as60]

would you plz suggest me a good book for tensors?

[@RalphDratman]

I love this. Thank you!

[@user-th5ge7ly8p]

this is actually the best explanation for a high school student who doesnt rly know the university level calculus, thank you :)

[@pradeepkumaryadav2643]

tensor= an object(object size and orientation) that is invariant under a change of coordinates (you mean coordinate systems ?), and has components that change in a special , predictable way under a change of coordinates (i think its coordinate systems) now i am trying to understand that sir, is there any objects also present of which the size and orientation changes with the change in coordinate system , OR is there any kind of conditions present in which the object's size and orientation changes with the change in coordinate system ? second question that what about the state of stress developed at a point of member under influence of external forces , that stress is also the tensors, isn't it ? but sir stresses are not the objects , isn't it ? and other things in your videos is excellent sir , but i am confused in above questions please clear my problem sir. THANK YOU

[@eigenchris]

Yes, I do mean "coordinate systems". When I say "object", I am being somewhat vague. An "object" doesn't have to be something you can draw on paper. It can be any idea you can come up with, including the stress tensor. The stress tensor is sometimes visualized as an ellipsoid (you can google "stress ellipsoid" to see this), so if you want to think of the stress tensor as a geometric object with size and orientation that doesn't depend on a coordinate systems, you can think of it as an ellipsoid that we measure with different coordinate systems. The numbers inside the stress tensor matrix will change depending on the coordinate system, but the ellipsoid will have the same size and orientation in all coordinate systems.

[@davidkoleckar4337]

straussen

[@DennisBetts1]

Vague waffle!

[@user-ov7lq7gh7i]

Can you answer me about some tensor questions, please

[@gordonchan4801]

I really like your videos.

[@tshiamooriel5113]

best series!

[@g3452sgp]

You said "tensors as partial derivatives and gradients that transform with the Jacobian matrix". I am very much interested to see what you mean by that.

[@eigenchris]

The current plan is for the non-calculus videos to go up to about 17 videos. After that I'm going to start a new series on tensor calculus where I go into detail about that.

[@g3452sgp]

Thank you. I am looking forward to viewing your videos on tensor calculus.

[@philandthai]

Take a look at math is beautiful, another good series and you will learn all about that but it is pretty heavy sailing.

[@angelmendez-rivera351]

@@stellamn No. The Jacobian and the gradient are not the same thing. You said it yourself: one is a matrix, the other one is a vector. Also, the Jacobian is a transformation between two vector functions in a vector space. The gradient is a linear operator on a scalar function. So they are not even remotely the same concept. Please, go and repeat your vector calculus course.

[@stellamn]

@@angelmendez-rivera351 please go away. Bc here is no space for condescending people who forbid others to question concepts and forbid to express themselves. What do you think is the purpose of a comment section of an educational video channel. Go read a book

[@user-qq5to6ve1b]

There's a question bothering me: Is it because it is mathematically invariant under basis transformation so it is a tensor (in other words: the covariant parts and the contravariant parts is balanced), or because we want it to be invariant so we make it tensor? It might be a little bit silly to ask this kind of question (since invariant and tensors are logically "if and only if"), but I am a bit curious about how people think about it. for example: when I think about vectors, should I not imagining it but just write down v = (v^i)(e_i) or should I think about arrows (geometric object which are independent of the coordinate) first. this question bothers me for I don't know that is it because (gradient f) is invariant because "we want it to be invariant" or because "we find it to be invariant". (or everythings is just because the chain rule altomatically make it to be true.)

[@quaereverum3871]

This channel has seen some remarkable growth in the past month. Any idea how come?

[@eigenchris]

Someone posted a reddit thread about me on /r/math, which got me 1000 subscribers in roughly 24 hours.

[@quaereverum3871]

@@eigenchris Sweet, well deserved. I thought for a moment the world suddenly started to take great interest in differential geometry, but alas. If I can ask, at what point during undergraduate/graduate school do you think these concepts are moslty taught? I myself am in my last year as an undergraduate mathematics student, and I have never had tensors introduced to me in class.

[@eigenchris]

I never learned tensors in my undergrad physics degree, also never learned differential geometry in school. I would expect you'd be in 3rd year at the very least before you had a class that used this stuff, since you need linear algebra and multivariable calculus under your belt. More likely you'd see it 4th year or even grad school since that is usually when General Relativity is taught.

[@quaereverum3871]

@@eigenchris I see, I suppose that makes sense. I will look forward to taking those courses later on, then, and enjoy your series as preparation in the meantime. Thanks!

[@electric_sand]

@@quaereverum3871 hey hey, can we connect on discord? From your comments, you seem to be an interesting person. I'm currently bailing out the waters of math which seems to be flooding my mind's boat.

[@michaellewis7861]

The breakdown components with respect to the basis is not invariant but the overall structure is, that is the aggregate bases*respective contravariant component.

[@abstractnonsense3253]

Really good explanation

[@albretchmueller3637]

I don’t know if you are a teacher. Your intro to tensors, with a little bit of more step-by-step guidance, I would even use to teach high school students. Questions: 1) any books you would suggest for avid high school students?; 2) could you point out to examples of profitably using tensors for natural language text processing/corpora research?

[@amalm007]

Great video!

[@parigim]

Best videos on Tensors that I have seen so far. So I hate to point out an error.. In the coordinates definition example, the magnitude should remain invariant. So should the dot product. So, to is not right. to will work.

[@eigenchris]

If you watch Tensors for Beginners video 9 on the metric tensor, you'll learn how the metric tensor components are involved in computing the dot product. This isn't obvious in cartesian basis, but it becomes more important for non-cartesian basis.

[@nobunagaoda4840]

No, the video is correct, because you are assuming that components have the same length, which is not the case

[@AJ-et3vf]

Great video! Thank you!

[@martinhall7636]

Hi Chris - can you recommend a good text book on linear algebra please? I'm not a mathematician by background but have an 'A' level in maths (16-18) and a degree in Chemistry from years ago and would love to spend some of my spare time learning about tensors with a view to moving on to the basics of GR. Any suggestions would be very much appreciated.

[@eigenchris]

Sorry, I don't have any recommended books on tensors specifically. I have another playlist on GR. I also have an "eigenchris recommendations video" that contains some texts/ sources for GR.

[@mamathakumari9911]

In this lockdown a.....good explanation

[@michaelvitalo3235]

Beyond awesome!

[@kaustuvregmi1469]

Sir I've studied somewhere that tensor are the quantities with magnitude and multiple directions. Here yo gave an example of pencil which has only one direction. Is there any restrictions that a vector you took for tensor must have multiple direction?

[@eigenchris]

A vector/pencil is a "rank 1" tensor because it is one-dimensional. There are other tensors like rank 2, 3, 4, etc. Linear maps and the metric tensor are both rank 2 tensors. I cover them in videos 7-9 of this series.

[@kaustuvregmi1469]

eigenchris thank you sir

[@haneenjawarneh2317]

اكثررر من رااائع اخيييرا فهمت شكراااا كوومايااااات 🎉❤😂

[@gbpferrao]

This guy really made avideo about freaking tensors and got half a million deserved views

[@amritanshraghav3793]

Can you give an example of objects that do vary with change of co-ordinate system changes? Might be a dumb question but I am trying to figure out something that would *not* be a tensor

[@eigenchris]

Not a dumb question at all. Non-tensors are harder to find than tensors, from what I have seen. An example of non-tensor is the Christoffel symbol, which is used in the definition of the covariant derivative. It's a 3D array which doesn't follow the ordinary forward/backward transformation rules. This is a more advanced topic that requires calculus to understand.

[@AJOEHlFY]

Just like a tensor is an object that is invariant under change in perspective you can define objects that do change with perspective. A simple example might be an object that is always facing your direction, if you change where you stand this object will have to change to always face in your direction. Another example is the resulting vector of a cross product. This vector is invariant under most coordinate transformations except orientation. Such object that is invariant to most coordinate transformations but not all can be called a pseudo tensor. So it's like a scale with at the one end we have invariance under all coordinates 'tensors' and at the other end we have objects that is fully dependant on which coordinates we pick.

[@kevon217]

very helpful, thanks!

[@75hilmar]

Nice. A big shot.

[@g3452sgp]

I have two question about tensor definition. 1. If a vector is given, there is always a co-vector associated with the vector given and the co-vector is unique. Is this understanding correct? 2. Just like there is a co-vector when a vector is given, there is also a co-matrix when a matrix is given and the co-matrix is unique. And this hold true with any Nth-dimensional matrix. Is this understanding correct?

[@professoreisenoxid9882]

In my Calculus-Class was the covector of a vector v introduced as his "dual-vector" v* and because every vectorspace (with finite dimension) is isomorph to his dual-vectorspace there is unique relationship between a vector and a dual-vector/covektor. We never talked precisely about dual-matrices or camatrices but matriced are also vectors or yield a vectorspace. So, my first explanation can be applied to matrices too.

[@davidjones5319]

Awesome. Thank you

[@kevin7314]

What is "the pencil"? Is it the length of the pencil? Is it the length of pencil and its orientation? Is it the pencil length and its orientation related to the door and constraints? Something else?

[@eigenchris]

The pencil is supposed to be a vector. In this case yes, the pencil is completely defined by its length and orientation. If you want a formal definition of a vector, you can try watching video 7 where I give three different possible definitions: (1) a list of numbers, (2) an picture of an arrow, (3) a member of a vector space. The 3rd definition (member of a vector space), just means vectors are things that we can scale and add together. It's a bit abstract, so if you prefer you can just think of a vector as an arrow.

[@Samcanplaymusic]

Mind blowing

[@n4nyhwwnv327]

Simply the best explanation on KZbin

[@akshaysehgal]

Subbed! thanks.

[@thevegg3275]

Could you please explain mathematically how the g sub mn matrix if formed using ds^2=dr^2 + r^2 d (theta)^2 ? What was presented was g sub mn = [1 0, 0 r^2]. Note [1 0, 0 r^2] is a two by two matrix

[@eigenchris]

I cover this metric in my "Tensor Caclulus 11" video (link below). The "1" for dr^2 because the e_r basis vector is 1 unit long. The "r^2" for dθ^2 means the e_θ basis vector is "r" units long. The zeros mean the r and θ directions are orthogonal, so e_r · e_θ = 0. kzbin.info/www/bejne/eJO0noejiN-Ieas

[@edwardlulofs444]

Good. Keep going

[@abcdefgabcdefg3017]

Thank you very much sir.

[@myousefzadeh]

Thanks eigenchris for your attention and quick respond.

[@eigenchris]

The slides are here. I tried to correct any mistakes, but there might still be some problems. drive.google.com/drive/folders/12erLlD6MbFdPAm6VsneeMTDlURv3oOax?usp=sharing

[@djamilahad9179]

Really nice thanks

[@a.007]

Really helpful ..

[@pendalink]

Very good video

[@chenvinc6776]

thanks a lot!

[@CAL1440]

thank you.

[@HR-yd5ib]

sweet! thanks!

[@ramishrashid8935]

Awesome 👍