This is the best and most comprehensive introduction to tensors available on CZcams. 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 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
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.
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
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.
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.
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
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!
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.
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.
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.
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
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.
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.
@@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!!!
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
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.
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.
Im my continuum mechanics class, my professor gave the following definitions: “A rank 2 tensor is a linear function from vectors to vectors.” “A rank 4 tensor is a linear function from rank 2 tensors to rank 2 tensors.” I was wondering how accurate these definitions are in general, because I think they were genuinely the most helpful definition I got in terms of understanding all the other properties of tensors, like how a covector is a function that maps vectors to scalars.
What you have said is true. But a rank 2 tensor can also be viewed as a linear function from a pair of vectors to a scalar. And a rank-4 tensor can map 3 vectors to a vector, or map 1 vector to a rank-3 tensor, and so on. There are multiple ways to interpret a tensor of a given rank. You just need to make sure the total ranks of the input and output tensors correspond to the rank of the "tensor map" you're using. I talk about this in one of the later videos in this series. Maybe 13 or so.
@@eigenchris I guess that usually where the distinction is drawn between a rank (2,2) vs a rank (1,3) or (4,0) tensor? We called all of them 4th “order” rather than “rank” and all was done in orthonormal coordinate systems, where the bases and dual bases are equivalent.
Those pairs of numbers (the tensor type), determine whether or not the inputs will be vectors or covectors. This doesn't depend on the basis. But even with a specific type, there are still multiple ways to use the tensor. Basically we can choose the number of inputs we put in. Populating all the input slots means the output would be a scalar. But the fewer inputs we give, the larger the output tensor will be.
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.
The breakdown components with respect to the basis is not invariant but the overall structure is, that is the aggregate bases*respective contravariant component.
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?
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.
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.
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,
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.
@@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.
@@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.
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.
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
"a tensor is a crazy bababoey"
A tensor is a fit ting innit
This is the best and most comprehensive introduction to tensors available on CZcams. 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
Glad you like them. I do plan on doing the basics of GR at some point in the next 6 months.
This is the best explanations on tensor I have ever seen.
@@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
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.
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
Absolutely the best instruction for tensor available on the internet
Brilliant video Chris. You explained without overwhelming or undermining the listener. Thank you
THANKS for making this topic so much more understandable! The best explanation I have found so far.
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.
"A tensor is something that transforms like a tensor"
Lol yeah, how can the definition of a tensor be that it's a collection of vectors? Aren't vectors tensors?
@@c.l.368 tensors are colections of tensors inside colectins of tensors, them u hav a fractal
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.
@@canyadigit6274 Can be a nice definition but in the moment i dont have the necessary iq to make any math with them ;-;
@@canyadigit6274 nice definition. Thanks
This is very pedagogical, straightforward insight into the subject. This helps me a lot, thank you.
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
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!
Thank you. This is the only video I could find on this that actually makes sense.
@BLAIR M Schirmer That one is crap lmao
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.
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.
@@crehenge2386 Or perhaps, more simply, it's a clearer video to me than the other ones I've found.
oh you are the vanilla tweaks guy
I have finally got it! This is the best introduction to what are tensors out there!
Simple, and to the point. Thanks
This is extremely important content. Thank you!
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.
Best set of lectures on youtube!
Excellent presentation!!!! Thank you.
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 😅
Thanks for this great video. It really sweeps me away.
This answered all the questions for me, thank you so much
I went through several videos to have the intuition of Tensors. But this is the beast..!
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
To my taste, this is the best introduction of tensors on the Net. Gracias, senior!
Thanks!
Thanks a lot. You have explained in the easy way.
i'll be going through your course in my free time, i look forwards to learning with you!
Ohhh mean. That's soo helpful, you don't know how much!! Thank you a lot!
Excellent! Please make more videos like this
Many thanks for all this work, best so far !
You're pretty much good because you actually have the passion of quantum physics and tensors well much of quantum chromodynamics
excelent video, as always. Had a mini "mind blown" moment.
very easy to grasp explanation please provide more topics! Subscribed ^^
Best video on tensor👍🏻
this is awesome, thank you!
Good summarising of the analysis of the vector and tensor notations in algebra and geometry of the linearly combination with operations
Content, pictures, and visualization of equations are very nice. Can the volume be louder, please. Try to increase the lectures, looking for more!
Exceptionally helpful
best tensor explanation on youtube so far
THANK YOU SO MUCH for making tensor calculus crystal clear !!!
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.
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.
This was amazing! Thank you, loads!
After my post graduation, first time I understood the meaning of tensors. Thank you!
How the hell did you graduate without understanding tensors
Yeh India hai yaha kuch bhi ho Sakta hai
@@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!!!
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
@@jptuser Tensor were in course but only few university has very good teachers, so those who are not lucky simple skip the part.
phenomenal content, thank you
thank you Sir for this series of lessons
I really like your videos.
best series!
This was amazing
Thank you
I love this. Thank you!
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..
You are the best man!
Nice. A big shot.
Simply the best explanation on CZcams
Great video! Thank you!
this is actually the best explanation for a high school student who doesnt rly know the university level calculus, thank you :)
Beyond awesome!
Nice explanation.
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.
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.
Thank you.
Thank you Chris for making such good videos that even a mere 9th grader like me is able to understand it :)
I’m in the same boat as you.
I am also on the same boat as you!
Hopefully we don't drown
Great video!
Really good explanation
Parfaitement clair! En plus je comprends parfaitement votre Anglais. Merci
De rien. :)
@@eigenchris Tres clair, en effet. Bravo!
Et bravo pour ne PAS saboter votre explication avec de la musique de fond....
Awesome. Thank you
Mind blowing
very helpful, thanks!
Im my continuum mechanics class, my professor gave the following definitions:
“A rank 2 tensor is a linear function from vectors to vectors.”
“A rank 4 tensor is a linear function from rank 2 tensors to rank 2 tensors.”
I was wondering how accurate these definitions are in general, because I think they were genuinely the most helpful definition I got in terms of understanding all the other properties of tensors, like how a covector is a function that maps vectors to scalars.
What you have said is true. But a rank 2 tensor can also be viewed as a linear function from a pair of vectors to a scalar. And a rank-4 tensor can map 3 vectors to a vector, or map 1 vector to a rank-3 tensor, and so on. There are multiple ways to interpret a tensor of a given rank. You just need to make sure the total ranks of the input and output tensors correspond to the rank of the "tensor map" you're using. I talk about this in one of the later videos in this series. Maybe 13 or so.
@@eigenchris I guess that usually where the distinction is drawn between a rank (2,2) vs a rank (1,3) or (4,0) tensor? We called all of them 4th “order” rather than “rank” and all was done in orthonormal coordinate systems, where the bases and dual bases are equivalent.
Those pairs of numbers (the tensor type), determine whether or not the inputs will be vectors or covectors. This doesn't depend on the basis. But even with a specific type, there are still multiple ways to use the tensor. Basically we can choose the number of inputs we put in. Populating all the input slots means the output would be a scalar. But the fewer inputs we give, the larger the output tensor will be.
Good. Keep going
In this lockdown a.....good explanation
اكثررر من رااائع اخيييرا فهمت شكراااا كوومايااااات 🎉❤😂
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.
Glad it's been helpful. I've actually never heard of Casually Explained. I'll have to check his channel out.
This guy really made avideo about freaking tensors and got half a million deserved views
Really helpful ..
Hi Chris! great video but i'm more of a book kind of guy which book would you suggest for learning tensors?
Thank you!
Thank you very much sir.
Really nice thanks
Very good video
Awesome 👍
Thank you!!
The breakdown components with respect to the basis is not invariant but the overall structure is, that is the aggregate bases*respective contravariant component.
Bravo!
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?
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.
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.
No, the video is correct, because you are assuming that components have the same length, which is not the case
would you plz suggest me a good book for tensors?
Thank you very very Sir
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,
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.
@@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.
@@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.
sweet! thanks!
Subbed! thanks.
thanks a lot!
thank you.
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.
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.
Based. Thank you.
Thanks eigenchris for your attention and quick respond.
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
Thank you
You're awesome ^^
very good
Thank you so much
thank you