0:40 I swear to god this is pretty much the explanation my teachers gave me. "Tensors are things that be like they do, but only sometimes. anyway here's your homework, you little shits. enjoy "
This is by FAR the best series I have ever found explaining tensors in a very clear and concise way that isn't only the math but the intuitive explanation behind the maths. Subbed and liked!
Thank you again for giving the clearest explanation on this topic that I have every heard. Finally, when it comes to working with these things, I'm not as...tense!
This is some of the best handwriting I've seen from a lecturer. Some of mine had very messy handwriting, or didn't fully erase previous boards (just lightly rubbed out), so I spent a lot of the time trying to work out what was written on the board!
It makes sense once you learn the maths behind it lol that definition isn't really meant for people who are learning it for the first time, just like this definition of a vector "A vector is an element of a vector space" it makes sense once you know what a vector space is.
You deserve more subscribers, I really hope that your channel grows and get known, your explanation is really good and you cover university topics that are fairly hard to find and whats more, that are explained that well. Tensors were a total nightmare for me to understand and you really saved me from drowning in that topic. Thanks a lot!
The best video explaining clearly about the behind the scenes that happen during a transformation of a tensor. Thanks for this great video . Also could you please refer me to any book where I can learn more about Tensors
That's actually not the true definition of a tensor. Given a vector space V and its dual space V*, then a tensor T is just a multilinear map such that T: (V ⊗ V*) --> ℝ All properties can be deduced from that. It's a much more elegant and rigorous way of defining it. But of course, it's less intuitive and imposible to understand if you're not familiar with linear algebra.
@@MsAlfred1996 No it is not... It might be "elegant" with regards to writing or abstract formalism, but surely not for understanding. It is exactly this crap that kept me busy for years. Also if you try to understand General Relativity, it would be much better to use Geometric Algebra, then you even don't need tensor stuff to solve some things. It is a problem of modern science to get more and more abstract instead of relating things to a simple understand, WHICH IS STILL POSSIBLE! If definitinos lead to the same things, and principles, then there is no TRUE definition. There are always several definitions possible.
Oh, well a coordinate transformation is a sort of something that transforms the coordinate system to a new one, may it be scale or angle or both, so the inverse of that transformation will return the transformed object to its original state, when we say that the basis vectors will transform according to L-1, that means once the first transformation will take place, the second one will only apply to the basis vectors, thus making the basis vectors invariant to the first transformation.
So tensors are mathematical ways to describe reality. The reality doesn't change but the numbers describing the reality changes when you have a change of co-ordinate system
This is an extremely coordinate-centric point of view of tensors. I don't think much can be learned about them until you understand the basic, abstract definition of a tensor product. Otherwise, the change of coordinate rules (which aren't even actually explained here) will seem mysterious and unmotivated, and hard to remember (so many coordinates!). Tensor products, abstractly, are *really easy*. A general element of the tensor product, let's say of vector spaces V1, V2, ..., Vn, is just a linear sum of terms (v1,v2,....,vn), where each vi is in the vector space Vi (in the video, each Vi is R^3, which is of dimension 3). So a general element of the tensor just looks like a.(a1,a2,...,an) + b.(b1,b2,...,bn) + ... + z.(z1,z2,....,zn) where the a,b,...,z are just scalars and, again, the ai, bi, ..., zi are in the vector space Vi for each i. This is subject to some rules: Firstly, you have distributivity: R1: (x+y,v2,v3,...,vn) = (x,v2,v3,...,vn) + (y,v2,v3,...,vn) and similarly if you split in coordinates other than the first one. In other words, you can combine two tensors which disagree in one coordinate by combining the answer in that coordinate. Secondly, you have scalar multiplication: R2: c(v1,v2,...,vn) = (c.v1,v2,...,vn) and similarly if you choose a different coordinate. In other words, you can bring multiplication by a scalar into (any) one of the coordinates. We also implicitly assume that (c1+c2)(v1,...,vn) = c1.(v1,...,vn) + c2.(v1,...,vn), for scalars c1 and c2. Now, if V1 has a basis of vectors b1, b2,..., bd (so V1 has dimension d), then we all know that a general element v1 in V1 can be written c1.b1 + c2.b2 + ....+ cd.bd. It follows that a general component like (v1,v2,...,vn), where each vi is in Vi, in the tensor product can be rewritten c1.(b1,v2,...,vn) + c2.(b2,v2,...,vn) + ... + cd.(bd,v2,...,vn). That is, you can split your v1 as a sum with respect to the basis for V1 and distribute in the tensor. You can now split each of these terms in the V2 component given a basis for V2, each of those can then be split for V3,... and so on. That means that you can always express an element of the tensor product as a linear sum of elements (e1,e2,...,en), where each ei is a basis element of Vi. These are linearly independent, so one sees that the tensor product of V1,V2,...,Vn is of dimension dim(V1)*dim(V2)*...*dim(Vn). This is opposed to the dimension of the standard product, whose dimension is the _sum_ of the dimensions. Anyway, that was longer than expected! But the point is: 1) the rules for the tensor product are really easy if you just give the abstract definition. There are just two rules R1 and R2 you need to know really. 2) you can choose a basis for the tensor product by choosing bases for each vector space you're tensoring, and a basis element for the tensor then corresponds to choosing one basis element for _each_ vector space (in the video, each Vi has the same dimension m, so the tensor has dimension m^n) 3) anything else, such as changes of coordinates, can (and in my opinion *should*) be easily derivable from this basic definition. You can start with the coordinate-wise definition, but I think that this approach encourages people to think of tensors as just high dimensional arrays of numbers, which is _not_ a good intuition.
After reading you're explanation on tensors and comparing it to the video, I now realize that this video relies way too much on the geometric point of view without focusing more on the linear algebra aspects of the tensor product.
@@evanurena8868 and your opinion is exactly why many people hate math, not every newbie is genius, they need some easier way to understand abstract things like math, things like abstract concept can just be studied later after they understand roughly what it mean.
@@That_One_Guy... Not every newbie may find the geometric explanation much easier then the algebraic one because those people have more trouble with spatial reasoning. What you've stated was completely irrelevant to what I said because you're making dangerous implications that don't really connect with my initial stance.
@Corona Kuro What do you mean by sloppy? Not everyone who isn't a traditional student watches these videos because they want to feel smart but maybe other reasons such as a genuine interest or an alternative explanation for a certain concept. The mind is a complex entity of the human body and math is an extremely broad area, so people are bound to have varying logical interpretations in how math works for them. That same person who doesn't comprehend this video on tensors is probably better adept at understanding another topic in math like differential equations or doesn't understand tensors from a geometric point of view . Whether one is in or out of school, i guess all of us in some way are students trying to learn something new one way or another.
@@evanurena8868 just ignore Corona Kuro, obviously a complete douche. I am an 'actual student' and found this video and others like a really excellent, superior in many ways to my degree text books. I used to watch things like long before I started my degree and your quite right that one doesn't need to be an 'actual student' to grapple with these topics. Universities don't hold a monopoly on learning!
Along this series resource, TH-cam and internet really evolves on how people learn. Great Videos on tensor you made. I often think tensor and matrix are actually same thing. But it is not. One question I have, when you notate tensor from the cube surface one by one, you write them down in matrix form from the first row to the third row. Why not write down these surface breakdown components in column ? From the left first column to the right.
So what if the transformation law is flipped? ie. The magnitude of the components change according to L inverse, but the basis vectors change according to L? Is this a different type of tensor or does the transformation have the same effect?
I have a question. Component of tensor will transform according to L, does component mean magnitude? If yes, then if magnitude is transforming according to L and base vector(direction) Transforming according to L-1. How are they cancelling each others effect. One is magnitude the other is direction.
What does it mean for a vector to change (or remain the same)? That was not explained. And why do the components transform according to L and the bases according to L^-1?
If L transforms a component to another coordinate system, why does L^{-1} transform the basis vectors? That fact doesn't seem obvious. Also, if a tensor component looks like p_ij e^i e^j and we want to transform that into the prime coordinate system, it seems like the calculation would be p'_ij = L p_ij * L^{-1} e^i * L^{-1}e^j so the L's do not cancel. Or does p_ij get a copy of L for each index? Are these things discussed in the upcoming videos in the playlist? Thank you!
Please just confirm if these dimensions, of the terms in the field equation, found on the internet, are correct or wrong. [G(miu,niu)]=M.L^-1.T^-2; [R]=[Lamda]=L^-2; [g(miu,niu)]=M.L.T^-2; [T(miu,niu)]=M^2.T^-4. And if the dimensional homogeneity of the relativistic gravity equation is checked with these dimensions of the tensors and curves in the field equation?
Hi, thanks for your videos!!! I think they are absolutely amazing. Could I ask you something? Could you please numerate them? I think it's kind of confusing getting to know which one is the next. Thanks. Again. Amazing videos!
Appreciate the kind words! For the numeration, it's a good idea, but I'll wait to do that until I finish the playlist; that way I can write (1/n) next to the videos, where n is the total number of videos. For now, here's a link to the playlist so you can watch everything in order: th-cam.com/play/PLdgVBOaXkb9D6zw47gsrtE5XqLeRPh27_.html
It seems obvious that when you change the reference (the basis) then the description (the coordinates) will also change, but reality will remain the same. It's like using a different unit for temperature or money, or like describing an object in different languages, the words will change, not the object itself, be it a thing, a vector, a tensor, or anything. That said, I still don't know what a tensor is, I just know the way it's written depends on the reference used, as anything else. I certainely missed something important in this video, but what ?
I'm with you. Could it be that a tensor is an array of numbers that express some vector space values and that each point contains vectors that give the rules for correct each point values to the new coordinate system? That's probably not right.
Regarding your first definition, Neuenschwander made a joke about that very thing in his book in his opening chapter... Seriously, It must be some kind of hazing process by mathematicians to define tensors based on how they transform. LOL. One of those "if you don't understand it don't expect me to be obliged to explain it to you."
my guess (im not fully sure) is you can say that basis vectors transform under L, but then the components wld transform under -1... so perhaps its a matter of convention
@@tanchienhao You guessed correct! It's just a matter of convention; I was using it to intuitively describe how tensors would transform. The overall point is that basis vectors and components would transform in an inverse manner.
''A tensor is an object that transforms like a tensor'', What does this mean? Does the second ''tensor'' mean an physics object which can transform something like a rubiks cube?
I think it depends on what sources you look at; I've always learned it as collections/arrays of numbers (and wikipedia + a number of other sources agree), but your version is also correct (and one that wolfram mathworld happens to agree with)!
From the Google dictionary: a rectangular array of quantities or expressions in rows and columns that is treated as a single entity and manipulated according to particular rules. But you don't *have* to manipulate it. Just sitting there it is the first part of the definition, only.
@@robertbrandywine I don't rely on online dictionary definitions, but on my university classes. The term appeared in 2 domains, math and computer sciences. In none of them where matrices just collections of numbers. A random collection of numbers is just an array or a multi-dimensional array. In algebra, we were presented matrices as representing transformations. As with many notions in math, once you know that and the rules to use it, you can abstract from the fact that they represent transformations, and just apply the algebraic rules to manipulate them. Now you can *choose* to forget what they essentially are and just *consider* them as collections of number (just like you can represent a vector as just a collection of numbers), but *you* doing so does not essentially changes what a matrice is. It's like looking at a text in chinese, and saying it's just a collection of symbols. Someone would tell you it's not just that, it's a language, it has meaning and purpose. But you can choose to see it as just a series of random symbols. *Edit* : if you want to convince yourself that matrices are not simple arrays, try to find *why* their algebraic operation rules are what they are. Especially, why the multiplication of matrices is the complicated formula it is, and you'll find that it is directly derived from the composition of functions (the f°g operation). The relation of matrices with functions operations (addition, substraction, composition) will also tell you what the numbers in a matrice actually represent.
I totally appreciate the fundamental approach to teaching tensor types. However, one thing I would suggest is to use a more practical example of the beam. Affix one side of the beam to a surface, and place the O stress tensor at the other. You can then apply the concept directly, rather than leaving it up to the reader to interpret the concept in some arbitrary way. Thanks! ☺
The way this guy explains tensors is the exact reason why you go through college without understanding tensors. All those bad books and pdfs start explaining tensors from some application point of view. So they tell you about tensors and what tensors do without telling you what a tensor is. This video is more about the stress tensor than a general tensor. If you want to have even a basic understand of tensors, start with a modest understanding of linear algebra, then came back and focus on basis vectors, then learn forward and backwards transformations from one basis to another. Do tons of examples till you understand forward and backward transformation from one basis to another well, then go study covectors, again do practice examples till your fingers are blistered. Then go study the tensor product. Then you will have the correct intuition and working knowledge of tensors. Do this and you will thank me later. Don't waste your years with the stress tensor, or moments of inertia, or the metric tensor etc. Come back to all these high level applications after you have a working knowledge of tensors. In fact the first application you should study after you follow my recommendation is the metric tensor. You probably could learn right now how to write the metric tensor for any surface by just deriving the first fundamental form for that surface, extracting all the components of the metric tensor and then writing out the metric tensor for the surface or space. However, that will not give you even a basic working knowledge of tensors. If you want to go higher after following my guidance, then you can start studying tensor calculus.
the "invariant under change of coordinates" meme is more confusing that useful. What varies with coordinates? Matrices? But matrix changes to a similar matrix when you change coordinates. This sort of description borders on meaninglessness to me. In mathematics we have tensor products of vector spaces. That's a coordinate free definition. If you want coordinates you choose a basis on the underlying vp and the tensors change accordingly when you change the base. I genuinely don't understand this video and I do understand tensors.
My way of defining tensors relates more to the physical argument as opposed to the mathematical one: take the stress tensor for example. The stress tensor is a physical quantity that describes how a beam or solid object is affected by different forces (e.g. shear, stretch) on it. If I change the coordinate system from Cartesian to spherical, those forces don't change and affect the beam in the same way in both coordinate systems. That is, even if the way I represent the components of that stress tensor changes, how the beam is affected isn't going to change with a simple change of coordinates. I guess your frame of reference is more mathematical whereas mine is Physics-based: I feel like teaching with the latter frame of reference in mind has broader appeal, but I recognize that not everyone fits that boat. In any case, I appreciate the discussion your comments generated here!
@@FacultyofKhan thanks for the reply, I haven't realized this is a definition from the point of view of physics. I still don't like it, but I get it now. This approach puts the equivariancy of the data in the center. I appreciate now this is a different point of view, it seems somewhat cumbersome to me, but I guess physicists have a good reason to use it like this. I've never had to think about stress, strain, viscosity or elasticity and the algebraic multilinear approach might be cumbersome in those applications. The main immediate problem was that it's not obvious what kind of object tensor is in this definition. Like what's its type. Now I understand it's simply a collection of numbers (with an additional property).
Indeed.... Is the most important fact of a tensor. I think that this sentence means that a tensor transforms in a special way, that is, this way is unique and only the tensors transforms it this way
0:40 I swear to god this is pretty much the explanation my teachers gave me.
"Tensors are things that be like they do, but only sometimes. anyway here's your homework, you little shits. enjoy "
General of Relativity: a thing that behave just like the general of relativity
Well, no. But actually, yes.
This is by FAR the best series I have ever found explaining tensors in a very clear and concise way that isn't only the math but the intuitive explanation behind the maths. Subbed and liked!
Wow the only video that actually knows what it’s talking about. Amazing thank you!
Glad it was helpful!
Thank you again for giving the clearest explanation on this topic that I have every heard. Finally, when it comes to working with these things, I'm not as...tense!
"A tensor is an object that transforms like a tensor"
I've never seen a finer example of circular reasoning in my life, bravo Academia!
A duck is an animal that looks and acts like a duck
This is some of the best handwriting I've seen from a lecturer. Some of mine had very messy handwriting, or didn't fully erase previous boards (just lightly rubbed out), so I spent a lot of the time trying to work out what was written on the board!
Wow, thank you!
Hey, man! You have very clear explanations on a very subtle topic. Keep it going!
0:40
I feel targeted, hurt, and generally unsettled, all of which are made up for by the rest of this video.
I'm glad that you share my feelings towards that definition.
I would like but ur comment has 46 likes
It makes sense once you learn the maths behind it lol that definition isn't really meant for people who are learning it for the first time, just like this definition of a vector "A vector is an element of a vector space" it makes sense once you know what a vector space is.
This will be a good Saturday evening :)
You deserve more subscribers, I really hope that your channel grows and get known, your explanation is really good and you cover university topics that are fairly hard to find and whats more, that are explained that well.
Tensors were a total nightmare for me to understand and you really saved me from drowning in that topic.
Thanks a lot!
Thank you! Share my videos with your friends to help me become more known!
These are one of the finest ed videos I have ever seen. I'm glad that the topic is discussed so finely. Great job!
this is perhaps the clearest guide to anything on youtube - thanks :) (applicable to part one as well!!)
These videos are honestly the best resource I've found explaining what tensors are. Thank you!
Amazing videos as always. Please keep it up with upper-year undergraduate physics and math courses!!!! Love it
The best video explaining clearly about the behind the scenes that happen during a transformation of a tensor. Thanks for this great video . Also could you please refer me to any book where I can learn more about Tensors
Schaum's Outline is one good option for beginners. Good luck with studying!
Coordinate systems are used to describe reality. A change of system changes the description, but not reality!
Clap clap clap. Nicely said
@@_DD_15 Hahaha ;-)
You did a marvelous job... Greetings from Nigeria..🤝
Anthony Zee: A tensor is something that transforms like a tensors...
黄瀚 “noshit,Sherlock”
That's actually not the true definition of a tensor.
Given a vector space V and its dual space V*, then a tensor T is just a multilinear map such that
T: (V ⊗ V*) --> ℝ
All properties can be deduced from that. It's a much more elegant and rigorous way of defining it. But of course, it's less intuitive and imposible to understand if you're not familiar with linear algebra.
@@MsAlfred1996 No it is not... It might be "elegant" with regards to writing or abstract formalism, but surely not for understanding. It is exactly this crap that kept me busy for years. Also if you try to understand General Relativity, it would be much better to use Geometric Algebra, then you even don't need tensor stuff to solve some things. It is a problem of modern science to get more and more abstract instead of relating things to a simple understand, WHICH IS STILL POSSIBLE!
If definitinos lead to the same things, and principles, then there is no TRUE definition. There are always several definitions possible.
You and 3blue1brown are making my move from Electronic Engineering to Mechanical Engineering a pleasant walk in a wood. Thanks man!
What is foo? Foo is an object that transforms like a foo. OK...
2:50 the displacement does not change only the components do
wow, it just clicked
I know what tensors are now, thanks!
excellent video. very clear conception of a tensor and a lucid explanation
0:40
Every 60 seconds in Africa, a minute passes
Clocks tick faster at the top of Mount Kilimanjaro. You can fit 1 million Kilimanjaro seconds into 1 sea level minute
best ever video for tensor ，thank you sir
5:54 the basis vectors will transform according to L inverse
I don't get what you mean. Please always use examples
Oh, well a coordinate transformation is a sort of something that transforms the coordinate system to a new one, may it be scale or angle or both, so the inverse of that transformation will return the transformed object to its original state, when we say that the basis vectors will transform according to L-1, that means once the first transformation will take place, the second one will only apply to the basis vectors, thus making the basis vectors invariant to the first transformation.
love to hear u ...it clears my all concepts about tensor
Awesome video sir. Your efforts are really appreciated. I love your channel so much
So tensors are mathematical ways to describe reality. The reality doesn't change but the numbers describing the reality changes when you have a change of co-ordinate system
This is an extremely coordinate-centric point of view of tensors. I don't think much can be learned about them until you understand the basic, abstract definition of a tensor product. Otherwise, the change of coordinate rules (which aren't even actually explained here) will seem mysterious and unmotivated, and hard to remember (so many coordinates!).
Tensor products, abstractly, are *really easy*. A general element of the tensor product, let's say of vector spaces V1, V2, ..., Vn, is just a linear sum of terms
(v1,v2,....,vn),
where each vi is in the vector space Vi (in the video, each Vi is R^3, which is of dimension 3). So a general element of the tensor just looks like
a.(a1,a2,...,an) + b.(b1,b2,...,bn) + ... + z.(z1,z2,....,zn)
where the a,b,...,z are just scalars and, again, the ai, bi, ..., zi are in the vector space Vi for each i.
This is subject to some rules:
Firstly, you have distributivity:
R1: (x+y,v2,v3,...,vn) = (x,v2,v3,...,vn) + (y,v2,v3,...,vn)
and similarly if you split in coordinates other than the first one. In other words, you can combine two tensors which disagree in one coordinate by combining the answer in that coordinate.
Secondly, you have scalar multiplication:
R2: c(v1,v2,...,vn) = (c.v1,v2,...,vn)
and similarly if you choose a different coordinate. In other words, you can bring multiplication by a scalar into (any) one of the coordinates. We also implicitly assume that (c1+c2)(v1,...,vn) = c1.(v1,...,vn) + c2.(v1,...,vn), for scalars c1 and c2.
Now, if V1 has a basis of vectors b1, b2,..., bd (so V1 has dimension d), then we all know that a general element v1 in V1 can be written
c1.b1 + c2.b2 + ....+ cd.bd.
It follows that a general component like (v1,v2,...,vn), where each vi is in Vi, in the tensor product can be rewritten
c1.(b1,v2,...,vn) + c2.(b2,v2,...,vn) + ... + cd.(bd,v2,...,vn).
That is, you can split your v1 as a sum with respect to the basis for V1 and distribute in the tensor. You can now split each of these terms in the V2 component given a basis for V2, each of those can then be split for V3,... and so on.
That means that you can always express an element of the tensor product as a linear sum of elements (e1,e2,...,en), where each ei is a basis element of Vi. These are linearly independent, so one sees that the tensor product of V1,V2,...,Vn is of dimension dim(V1)*dim(V2)*...*dim(Vn). This is opposed to the dimension of the standard product, whose dimension is the _sum_ of the dimensions.
Anyway, that was longer than expected! But the point is:
1) the rules for the tensor product are really easy if you just give the abstract definition. There are just two rules R1 and R2 you need to know really.
2) you can choose a basis for the tensor product by choosing bases for each vector space you're tensoring, and a basis element for the tensor then corresponds to choosing one basis element for _each_ vector space (in the video, each Vi has the same dimension m, so the tensor has dimension m^n)
3) anything else, such as changes of coordinates, can (and in my opinion *should*) be easily derivable from this basic definition. You can start with the coordinate-wise definition, but I think that this approach encourages people to think of tensors as just high dimensional arrays of numbers, which is _not_ a good intuition.
After reading you're explanation on tensors and comparing it to the video, I now realize that this video relies way too much on the geometric point of view without focusing more on the linear algebra aspects of the tensor product.
@@evanurena8868 and your opinion is exactly why many people hate math, not every newbie is genius, they need some easier way to understand abstract things like math, things like abstract concept can just be studied later after they understand roughly what it mean.
@@That_One_Guy... Not every newbie may find the geometric explanation much easier then the algebraic one because those people have more trouble with spatial reasoning. What you've stated was completely irrelevant to what I said because you're making dangerous implications that don't really connect with my initial stance.
@Corona Kuro What do you mean by sloppy? Not everyone who isn't a traditional student watches these videos because they want to feel smart but maybe other reasons such as a genuine interest or an alternative explanation for a certain concept. The mind is a complex entity of the human body and math is an extremely broad area, so people are bound to have varying logical interpretations in how math works for them. That same person who doesn't comprehend this video on tensors is probably better adept at understanding another topic in math like differential equations or doesn't understand tensors from a geometric point of view . Whether one is in or out of school, i guess all of us in some way are students trying to learn something new one way or another.
@@evanurena8868 just ignore Corona Kuro, obviously a complete douche. I am an 'actual student' and found this video and others like a really excellent, superior in many ways to my degree text books. I used to watch things like long before I started my degree and your quite right that one doesn't need to be an 'actual student' to grapple with these topics. Universities don't hold a monopoly on learning!
0:40 "Ah yes the floor here is made out of floor"
Along this series resource, TH-cam and internet really evolves on how people learn. Great Videos on tensor you made. I often think tensor and matrix are actually same thing. But it is not. One question I have, when you notate tensor from the cube surface one by one, you write them down in matrix form from the first row to the third row. Why not write down these surface breakdown components in column ? From the left first column to the right.
Thank you! Now I know more about tensors!
props my dude, you are ridiculously good at explaining shit
Great video series! subscribed
Very useful video for understanding!!)
So what if the transformation law is flipped? ie. The magnitude of the components change according to L inverse, but the basis vectors change according to L? Is this a different type of tensor or does the transformation have the same effect?
I have a question. Component of tensor will transform according to L, does component mean magnitude? If yes, then if magnitude is transforming according to L and base vector(direction) Transforming according to L-1. How are they cancelling each others effect. One is magnitude the other is direction.
Thanks a lot. Hope u channel got at least 1M subscribers :)
Hi thx for the explanation. Can you explain why the basis vectors will transform according to L-1?
What does it mean for a vector to change (or remain the same)? That was not explained. And why do the components transform according to L and the bases according to L^-1?
thank you, love khan
If L transforms a component to another coordinate system, why does L^{-1} transform the basis vectors? That fact doesn't seem obvious.
Also, if a tensor component looks like p_ij e^i e^j and we want to transform that into the prime coordinate system, it seems like the calculation would be p'_ij = L p_ij * L^{-1} e^i * L^{-1}e^j so the L's do not cancel. Or does p_ij get a copy of L for each index?
Are these things discussed in the upcoming videos in the playlist? Thank you!
When you changed the coordinates for the stress tensor. Did you change the cross sections as well?
Yes, as far as I understand
Thanks for you
Goteeeeeem!
cool tut!!!
is changing betwee cartesian and polar coordinates a tensor transformation?
Please just confirm if these dimensions, of the terms in the field equation, found on the internet, are correct or wrong.
[G(miu,niu)]=M.L^-1.T^-2; [R]=[Lamda]=L^-2; [g(miu,niu)]=M.L.T^-2; [T(miu,niu)]=M^2.T^-4. And if the dimensional homogeneity of the relativistic gravity equation is checked with these dimensions of the tensors and curves in the field equation?
Hi, thanks for your videos!!! I think they are absolutely amazing. Could I ask you something? Could you please numerate them? I think it's kind of confusing getting to know which one is the next. Thanks. Again. Amazing videos!
Appreciate the kind words!
For the numeration, it's a good idea, but I'll wait to do that until I finish the playlist; that way I can write (1/n) next to the videos, where n is the total number of videos. For now, here's a link to the playlist so you can watch everything in order:
th-cam.com/play/PLdgVBOaXkb9D6zw47gsrtE5XqLeRPh27_.html
It seems obvious that when you change the reference (the basis) then the description (the coordinates) will also change, but reality will remain the same. It's like using a different unit for temperature or money, or like describing an object in different languages, the words will change, not the object itself, be it a thing, a vector, a tensor, or anything. That said, I still don't know what a tensor is, I just know the way it's written depends on the reference used, as anything else. I certainely missed something important in this video, but what ?
I'm with you. Could it be that a tensor is an array of numbers that express some vector space values and that each point contains vectors that give the rules for correct each point values to the new coordinate system? That's probably not right.
But, looking at the components, how di touvkniw that the tensor is the same is the components change with the transformation?
Regarding your first definition, Neuenschwander made a joke about that very thing in his book in his opening chapter...
Seriously, It must be some kind of hazing process by mathematicians to define tensors based on how they transform. LOL. One of those "if you don't understand it don't expect me to be obliged to explain it to you."
Try to make a video on Tensor Product
In the future, I will!
bruh how'd he write so straight without lines
As a senior in high school, this video raped my mind, yet was extremely helpful. Overall 9.9/10
Can someone explain, what would be a reason you would change the coordinate system on a stress tensor?
0:40 that exactly what my Math teacher says. LMAO
At 6:20, why do the basis vectors transform according to L-1 and not L ?
my guess (im not fully sure) is you can say that basis vectors transform under L, but then the components wld transform under -1... so perhaps its a matter of convention
@@tanchienhao You guessed correct! It's just a matter of convention; I was using it to intuitively describe how tensors would transform. The overall point is that basis vectors and components would transform in an inverse manner.
@@FacultyofKhan Does inverse here mean "counter-acting"?
Can you give an example of a matrix that has physical significance but is not a tensor?
[23, 27, 33] where these numbers are the ages of some three people at a given point in time.
''A tensor is an object that transforms like a tensor'', What does this mean? Does the second ''tensor'' mean an physics object which can transform something like a rubiks cube?
Thanks for the video 👍. Just one remark, matrices are not just collections of numbers, they are linear transformations.
I think it depends on what sources you look at; I've always learned it as collections/arrays of numbers (and wikipedia + a number of other sources agree), but your version is also correct (and one that wolfram mathworld happens to agree with)!
From the Google dictionary: a rectangular array of quantities or expressions in rows and columns that is treated as a single entity and manipulated according to particular rules. But you don't *have* to manipulate it. Just sitting there it is the first part of the definition, only.
@@robertbrandywine I don't rely on online dictionary definitions, but on my university classes. The term appeared in 2 domains, math and computer sciences. In none of them where matrices just collections of numbers. A random collection of numbers is just an array or a multi-dimensional array. In algebra, we were presented matrices as representing transformations. As with many notions in math, once you know that and the rules to use it, you can abstract from the fact that they represent transformations, and just apply the algebraic rules to manipulate them. Now you can *choose* to forget what they essentially are and just *consider* them as collections of number (just like you can represent a vector as just a collection of numbers), but *you* doing so does not essentially changes what a matrice is. It's like looking at a text in chinese, and saying it's just a collection of symbols. Someone would tell you it's not just that, it's a language, it has meaning and purpose. But you can choose to see it as just a series of random symbols.
*Edit* : if you want to convince yourself that matrices are not simple arrays, try to find *why* their algebraic operation rules are what they are. Especially, why the multiplication of matrices is the complicated formula it is, and you'll find that it is directly derived from the composition of functions (the f°g operation). The relation of matrices with functions operations (addition, substraction, composition) will also tell you what the numbers in a matrice actually represent.
I have to understand tensor to implement a research paper, damn its getting a bit too much
I totally appreciate the fundamental approach to teaching tensor types. However, one thing I would suggest is to use a more practical example of the beam. Affix one side of the beam to a surface, and place the O stress tensor at the other. You can then apply the concept directly, rather than leaving it up to the reader to interpret the concept in some arbitrary way.
Thanks! ☺
HAAAAllelujah. HAAAllelujah. HALLELUJAH, HALLELUJAH, HALLELOOOoooOOOJAH!
Honestly I could weep.
How do you speak such long sentences
A tensor is something that transforms like a tensor
Who is Marcin Maciejewski,
Kto to jest Marcin Maciejewski?
Czy zna on Mateusza Macielewskiego?
Great
The way this guy explains tensors is the exact reason why you go through college without understanding tensors. All those bad books and pdfs start explaining tensors from some application point of view. So they tell you about tensors and what tensors do without telling you what a tensor is. This video is more about the stress tensor than a general tensor. If you want to have even a basic understand of tensors, start with a modest understanding of linear algebra, then came back and focus on basis vectors, then learn forward and backwards transformations from one basis to another. Do tons of examples till you understand forward and backward transformation from one basis to another well, then go study covectors, again do practice examples till your fingers are blistered. Then go study the tensor product. Then you will have the correct intuition and working knowledge of tensors. Do this and you will thank me later. Don't waste your years with the stress tensor, or moments of inertia, or the metric tensor etc. Come back to all these high level applications after you have a working knowledge of tensors. In fact the first application you should study after you follow my recommendation is the metric tensor. You probably could learn right now how to write the metric tensor for any surface by just deriving the first fundamental form for that surface, extracting all the components of the metric tensor and then writing out the metric tensor for the surface or space. However, that will not give you even a basic working knowledge of tensors. If you want to go higher after following my guidance, then you can start studying tensor calculus.
Thanks for that buddy
could you plsss share this lecture in Pdf. if you are reading this comment, i would say i love you ❤.
What is a Saturday? A Saturday is a day that feels like a Saturday. A Saturday that doesnt feel like a Saturday is n o t a Saturday.
Tensor apt regression_likely
Actually a matrix is a (1,1) tensor.
the "invariant under change of coordinates" meme is more confusing that useful. What varies with coordinates? Matrices? But matrix changes to a similar matrix when you change coordinates. This sort of description borders on meaninglessness to me. In mathematics we have tensor products of vector spaces. That's a coordinate free definition. If you want coordinates you choose a basis on the underlying vp and the tensors change accordingly when you change the base. I genuinely don't understand this video and I do understand tensors.
My way of defining tensors relates more to the physical argument as opposed to the mathematical one: take the stress tensor for example. The stress tensor is a physical quantity that describes how a beam or solid object is affected by different forces (e.g. shear, stretch) on it. If I change the coordinate system from Cartesian to spherical, those forces don't change and affect the beam in the same way in both coordinate systems. That is, even if the way I represent the components of that stress tensor changes, how the beam is affected isn't going to change with a simple change of coordinates.
I guess your frame of reference is more mathematical whereas mine is Physics-based: I feel like teaching with the latter frame of reference in mind has broader appeal, but I recognize that not everyone fits that boat. In any case, I appreciate the discussion your comments generated here!
@@FacultyofKhan thanks for the reply, I haven't realized this is a definition from the point of view of physics. I still don't like it, but I get it now. This approach puts the equivariancy of the data in the center. I appreciate now this is a different point of view, it seems somewhat cumbersome to me, but I guess physicists have a good reason to use it like this. I've never had to think about stress, strain, viscosity or elasticity and the algebraic multilinear approach might be cumbersome in those applications.
The main immediate problem was that it's not obvious what kind of object tensor is in this definition. Like what's its type. Now I understand it's simply a collection of numbers (with an additional property).
what is the purpose of change of coordinates?
Some problems becomes easy after changing the coordinates
In a few words a tensor is Aijk i©j©k. Where © is the tensorial product 😂 while the matrix representing the coordinates of the tensor is only represented by the coordinates Aijk. (Pretend the c inside the circle © is an x) 😂 That's the only thing I found on my phone.
**A tensor is a mathematical object that transforms like a tensor.**
Hmmm yes the floor here is made out of floor
Indeed.... Is the most important fact of a tensor. I think that this sentence means that a tensor transforms in a special way, that is, this way is unique and only the tensors transforms it this way