This is the best and most comprehensive introduction to tensors available on UA-cam. 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
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.
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.
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
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.
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
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!
@@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.
Great video, now I understood what is the use of tensor and why does this introduced. I have taken an advanced course in tensor in iit Roorkee I did not understood the basics of tensor even.
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.
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.
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
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.
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?
I wish people didn’t talk about the coordinates OF vectors and tensors. I think a lot of confusion results from people thinking that vectors and tensors really do HAVE coordinates. I think it would be clearer to say something like “a coordinate system is a set of basis vectors/tensors that can be used to map other vectors/tensors into a matrix representation.”
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.
The breakdown components with respect to the basis is not invariant but the overall structure is, that is the aggregate bases*respective contravariant component.
@@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.
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.
@@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!
@@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.
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
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.
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
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. ua-cam.com/video/BbQmTmSzUCI/v-deo.html
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.
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?
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.
What's a good textbook for mathematical vector and tensor analysis at the undergraduate level? I've been trying to find one, but usually I only find vector calculus textbooks (i.e late part of calc 3.) or a book written mostly for physicists and engineers.
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.
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.
@@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.
@@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
I tried to understand tensors and threw the book aside. I think what I was missing was a motivation. Scalars and vectors are good examples of tensors, but I perhaps need one more layer of physical example, What was the person who created tensors thinking about? What *specific* problem was he trying to address? For beginners (like me), a physics problem is the *only* way to introduce tensors. Why should the path for inventing them and the path for studying them be any different?
I have a video that covers some basic motivation from physics, which is "part -1". Unfortunately a lot of my approach is based on pure math, although I do want to eventually do some videos on relativity.
I'm just an engineer. In this days, tensor calculus and tensor "construction" are fundamental for simulation of complex, multivariable proccesses like weather forecast, geodesic analysis, nuclear explosions, complex buildings, etc. Using tensor calculus and theme-related math, you can model an hurricane using an N-Rank tensor containing 3D data about: humidity, pressure, wind speed, temperature, spatial location, etc. And all of this being contained into a single tensorial entity. OR, you can use lower rank tensor, separating variables by classes and apply the math of the science being applied. If, by Fourier times, tensor math would have existed, he probably would have used them for his "heat theory" because he would have needed to express the 3D reality of the soil parameters with more components than a matrix (bidimensional problem). I believe that tensor theory arose as a solution to this kind of problem: to model reality beyond an unidimensional of bidimensional problem. Maxwell, with his theory of electromagnetism expressed in quaternions, was closer to the truth than anybody else, mixing scalar and vectorial components into his equations. Was Heaviside who, pushing vectorial algebra, gave rid of scalar components of Maxwell quaternions, giving his equations the current form. It has been suspected for more than 110 years that Nikola Tesla was very aware of the significance of the scalar components of the theory, wich could allow the flow of LONGITUDINAL Dielectric Fields (like in a thunderbolt). Tensor calculus is NON LINEAR, as also it is a trajectory and composition of a lightning, and there is not any LINEAR calculus that allow us to model this. Things would be very different with electromagnetism in a tensorial expression. This is my humble opinion. Also, the problem is that current EM theory works and tensor calculus is very difficult. I'm convinced that the military labs, working with directed energy, use a quite different set of equations for EM and, probably tensor math.
My mistake, as english is not my native language. When I wrote thunderbolt (Tesla part), I was talking about lightning. Everyone close to the subject observed the discharges of the Tesla Coil and Tesla Capacitor. Those are longitudinal dielectric displacements, highly random, and susceptible of being modeled with tensor theory and fractals theory.
*Why should the path for inventing them and the path for studying them be any different?* Because mathematics exist. It is not enough to have an intuition for what kind of problem you want to solve. You need to have a rigorous formal theory for those objects that will solve your problems and make sure that these objects are well-defined and the intuition you are looking for can be made coherent. Otherwise, how do you expect to be able to do anything, let alone solve your problem? Besides, you are wrong. To a beginner who is learning about vector spaces in an abstract context and has no interest in physics, physics is a useless method for explaining what tensors are. This is why the abstract definition exists and is necessary. And in fact, the abstract definition is what ensures that the physics that you do with tensors are actually valid.
I didn’t understand what’s exactly invariant about tensors in definition 2. Length? But it definitely can change after a coordinate change, “units” can change. Direction? But it’s defined in terms of a coordinate system, and all coordinate systems are “equally important”. If there’s an invariant, we should be able to calculate it before and after transform and check that it doesn’t change.
It's the combination of basis vectors and components that are invariant. Length is calculated using something called the "metric" (see video #9), and it will stay the same in all coordinates. The metric will change along with the coordinates so that the length of the vector never changes.
"a tensor is a crazy bababoey"
This is the best and most comprehensive introduction to tensors available on UA-cam. 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.
Absolutely the best instruction for tensor available on the internet
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
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 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.
This is very pedagogical, straightforward insight into the subject. This helps me a lot, thank you.
To my taste, this is the best introduction of tensors on the Net. Gracias, senior!
Thanks!
Brilliant video Chris. You explained without overwhelming or undermining the listener. 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
THANKS for making this topic so much more understandable! The best explanation I have found so far.
"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
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.
I went through several videos to have the intuition of Tensors. But this is the beast..!
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 😅
You're pretty much good because you actually have the passion of quantum physics and tensors well much of quantum chromodynamics
I have finally got it! This is the best introduction to what are tensors out there!
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
Best set of lectures on youtube!
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!
Ohhh mean. That's soo helpful, you don't know how much!! Thank you a lot!
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.
Simple, and to the point. Thanks
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.
best tensor explanation on youtube so far
this is actually the best explanation for a high school student who doesnt rly know the university level calculus, thank you :)
Great video, now I understood what is the use of tensor and why does this introduced. I have taken an advanced course in tensor in iit Roorkee I did not understood the basics of tensor even.
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
This answered all the questions for me, thank you so much
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 is extremely important content. Thank you!
Content, pictures, and visualization of equations are very nice. Can the volume be louder, please. Try to increase the lectures, looking for more!
i'll be going through your course in my free time, i look forwards to learning with you!
This guy really made avideo about freaking tensors and got half a million deserved views
excelent video, as always. Had a mini "mind blown" moment.
Thanks for this great video. It really sweeps me away.
Excellent presentation!!!! Thank you.
Excellent! Please make more videos like this
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
Good summarising of the analysis of the vector and tensor notations in algebra and geometry of the linearly combination with operations
came here because my high school teacher lied to me and didn't tell me about the inertia tensor.
thats good self promotion
Thanks a lot. You have explained in the easy way.
Shoulda called this series, "Tensors for Begensors"! Real missed opportunity with this one
finally someone who can explain "tensor" without using the word......... "tensor."
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
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.
اكثررر من رااائع اخيييرا فهمت شكراااا كوومايااااات 🎉❤😂
very easy to grasp explanation please provide more topics! Subscribed ^^
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 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.
Best video on tensor👍🏻
Many thanks for all this work, best so far !
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?
How big is the auditorium please ?
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....
I wish people didn’t talk about the coordinates OF vectors and tensors. I think a lot of confusion results from people thinking that vectors and tensors really do HAVE coordinates. I think it would be clearer to say something like “a coordinate system is a set of basis vectors/tensors that can be used to map other vectors/tensors into a matrix representation.”
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.
would you plz suggest me a good book for tensors?
I really like your videos.
this is awesome, thank you!
In this lockdown a.....good explanation
The breakdown components with respect to the basis is not invariant but the overall structure is, that is the aggregate bases*respective contravariant component.
This channel has seen some remarkable growth in the past month. Any idea how come?
Someone posted a reddit thread about me on /r/math, which got me 1000 subscribers in roughly 24 hours.
@@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.
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.
@@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!
@@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.
Simply the best explanation on UA-cam
thank you Sir for this series of lessons
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
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.
straussen
Vague waffle!
Can you answer me about some tensor questions, please
Nice. A big shot.
Great video! Thank you!
The length of the pencil *does* depend on the coordinate system. (I'm thinking of Special Relativity)
That's true, but it's spacetime interval/proper length (the true "length" concept in spacetime) is invariant.
best series!
THANK YOU SO MUCH for making tensor calculus crystal clear !!!
I love this. Thank you!
phenomenal content, thank you
This was amazing! Thank you, loads!
This was amazing
Thank you
Really good explanation
Hi Chris! great video but i'm more of a book kind of guy which book would you suggest for learning tensors?
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
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.
ua-cam.com/video/BbQmTmSzUCI/v-deo.html
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.
5:19 Wow, that door is way high up than I thought. I thought it was far but it's really high and to the right.
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?
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.
What's a good textbook for mathematical vector and tensor analysis at the undergraduate level? I've been trying to find one, but usually I only find vector calculus textbooks (i.e late part of calc 3.) or a book written mostly for physicists and engineers.
Exceptionally helpful
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.
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.
Thank you.
I am looking forward to viewing your videos on tensor calculus.
Take a look at math is beautiful, another good series and you will learn all about that but it is pretty heavy sailing.
@@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.
@@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
Beyond awesome!
You are the best man!
very helpful, thanks!
I get it! The speed of light is invariant regardless of reference frame.
I tried to understand tensors and threw the book aside. I think what I was missing was a motivation. Scalars and vectors are good examples of tensors, but I perhaps need one more layer of physical example, What was the person who created tensors thinking about? What *specific* problem was he trying to address? For beginners (like me), a physics problem is the *only* way to introduce tensors. Why should the path for inventing them and the path for studying them be any different?
I have a video that covers some basic motivation from physics, which is "part -1". Unfortunately a lot of my approach is based on pure math, although I do want to eventually do some videos on relativity.
Exactly.
I'm just an engineer. In this days, tensor calculus and tensor "construction" are fundamental for simulation of complex, multivariable proccesses like weather forecast, geodesic analysis, nuclear explosions, complex buildings, etc.
Using tensor calculus and theme-related math, you can model an hurricane using an N-Rank tensor containing 3D data about: humidity, pressure, wind speed, temperature, spatial location, etc. And all of this being contained into a single tensorial entity. OR, you can use lower rank tensor, separating variables by classes and apply the math of the science being applied.
If, by Fourier times, tensor math would have existed, he probably would have used them for his "heat theory" because he would have needed to express the 3D reality of the soil parameters with more components than a matrix (bidimensional problem).
I believe that tensor theory arose as a solution to this kind of problem: to model reality beyond an unidimensional of bidimensional problem.
Maxwell, with his theory of electromagnetism expressed in quaternions, was closer to the truth than anybody else, mixing scalar and vectorial components into his equations. Was Heaviside who, pushing vectorial algebra, gave rid of scalar components of Maxwell quaternions, giving his equations the current form.
It has been suspected for more than 110 years that Nikola Tesla was very aware of the significance of the scalar components of the theory, wich could allow the flow of LONGITUDINAL Dielectric Fields (like in a thunderbolt).
Tensor calculus is NON LINEAR, as also it is a trajectory and composition of a lightning, and there is not any LINEAR calculus that allow us to model this. Things would be very different with electromagnetism in a tensorial expression.
This is my humble opinion. Also, the problem is that current EM theory works and tensor calculus is very difficult. I'm convinced that the military labs, working with directed energy, use a quite different set of equations for EM and, probably tensor math.
My mistake, as english is not my native language. When I wrote thunderbolt (Tesla part), I was talking about lightning. Everyone close to the subject observed the discharges of the Tesla Coil and Tesla Capacitor. Those are longitudinal dielectric displacements, highly random, and susceptible of being modeled with tensor theory and fractals theory.
*Why should the path for inventing them and the path for studying them be any different?*
Because mathematics exist. It is not enough to have an intuition for what kind of problem you want to solve. You need to have a rigorous formal theory for those objects that will solve your problems and make sure that these objects are well-defined and the intuition you are looking for can be made coherent. Otherwise, how do you expect to be able to do anything, let alone solve your problem?
Besides, you are wrong. To a beginner who is learning about vector spaces in an abstract context and has no interest in physics, physics is a useless method for explaining what tensors are. This is why the abstract definition exists and is necessary. And in fact, the abstract definition is what ensures that the physics that you do with tensors are actually valid.
Thank you very much sir.
I didn’t understand what’s exactly invariant about tensors in definition 2. Length? But it definitely can change after a coordinate change, “units” can change. Direction? But it’s defined in terms of a coordinate system, and all coordinate systems are “equally important”. If there’s an invariant, we should be able to calculate it before and after transform and check that it doesn’t change.
It's the combination of basis vectors and components that are invariant. Length is calculated using something called the "metric" (see video #9), and it will stay the same in all coordinates. The metric will change along with the coordinates so that the length of the vector never changes.
Awesome. Thank you
Great video!
Thank you.
Thank you!
Thank you very very Sir
Really helpful ..
Awesome content. Thank you for it, But the sound quality... my ears cry of this narrowed high freq band.