Why are tensors EVERYWHERE? | Tensors for beginners
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- เผยแพร่เมื่อ 16 ส.ค. 2021
- A tensor transforms like a tensor. Why does this explanation keep circulating? Tensors as a mathematical object motivates deep appreciation to how physics and mathematics are related, but rarely is that properly explained to beginners.
The video explains the motivation for tensors, as well as an example calculation, and exploring electromagnitism in spacetime.
Music: Mark Tyner - Close To You
As a mathematician I always oppose thinking of geometric objects as some numbers with rules of transformations. But the video explains nicely the measurement-oriented point of view of a physicist, so you have my like!
Thank you very much!
because tensors are NOT geometric objects, they are algebraic objects, you don't need geometry whatsoever. However, they can become geometric objects if you attach them to a geometric structure. The way to properly describe them is using the language of principal and associated bundles, where the "transformation law" is the local projection of the group action onto the base manifold of the associated bundle. Sadly, this doesn't provide an intuitive explanation for the general public.
@@MagufoBoy In the context we have here, when people say tensors, they actually mean tensor fields on a manifolds: sections of a tensor product of the tangent bundle (and its dual). So of course they're geometric objects, just like vector fields, but with an added linear-algebraic flavor.
Fortunately, there are other ways to describe it properly.
The description of covariance and contravariance was explained so beautifully, it's criminal that you aren't a big science TH-cam channel.
One of the best videos that sums up the essence of what tensors really are. I am on my journey to learn them, and this video, I found to be very insightful, especially in terms of connecting some dots of understanding that I have gained from other sources. Really brilliant illustration!
This was exactly what I wanted to do! Thank you for the message!
Best concise explanation about tensors I ever came across. I wish I'd know this during my studies, 20 yrs ago.
Thank you very much for the kind message!
Just found this channel and it’s actually insane can’t beleive you don’t have so many more subscribers
Man this is an underrated video it covers the inspiration behind the Invention of tensor notation so well including all the important concepts related to tensor in such a short span of time. Its marvellous. 🎉
Thank you so much for your comment!
Please continue making videos. Best explanations I've seen so far for all the topics that always seemed just beyond my grasp.
Learning physics is a waterfall moment and it all makes sense at once. Glad I can help with yours :D
Super clear and easy to follow. You have a wonderful touch in explaining these phenomena. Thanks so much.
I wouldn't mind watching you working through some of these example equations start to finish. Showing the work.
I like to see how the sausage is made
Finally ,after months of searching I understand this fascinating physical object
I really like this video
as a physics student i have a problem with understanding, what tensors are.
After this video i feel i know them better, but still don't know what are they.
I realise after making the video, I should have made a simple 30 second segment outlining the actual maths.
I genuinely believe knowing the maths in physics does very little when it comes to understanding and intuition, but not including it at all was a mistake.
Eigenchris does a great video series on tensors, so watch that if you want a head start on university maths.
I loved this definition of Tensor. It is a better intuition
Very informative, hope your channels grows even larger!
Celsius and Fahrenheit are 2 buildings next to each other. The Celsius building has 5 floors for every 9 floors in the Fahrenheit building, and it's on a hill as tall as the Fahrenheit building's 32nd floor. They have a shared basement with numerous subterranean floors. You're on the 20th floor of the Celsius building, looking at the Fahrenheit building - which floor are you across from? (the 68th)
All Hail Lord Kelvin! Leader and prophet of the master race!
You are doing a great work..your explanation was awesome..!!
Really lovely. Thank you. Brilliant to start with the lamppost!
Took me a while to write the video. Glad every draft ended up in some shape in the final video :D
believe it or not , this was the first clear explanation I have come across , of what a contravariant and covariant tensor is ! Most other explanations just dive straight into partial differential equations .
This is one of the reasons I made this video. My lecturer is a great person, but my god they were confusing T-T
It's not that tensors are everywhere, it's that tensors describe something that is everywhere.
Thank you English tree for your grammatically correcting fruits
Bruh loved your explanation
This is an excellent description of Tensors!
Thank you very much!
You are a hero
Great work! Love the cartoon style. I feel like I learned something about Tensors and what they are used for!
Glad you enjoyed the video!
great video :)
This was very very good. Thank you.
Thank you very much!
Nice video! Your explanations are neat and your didactic skills are of course very good, but what I liked the most is that you get on eye level with the viewer.
I presume you are a physicist and as such learned the classical tensor calculus. I wonder, are you aware that there are other „schools“ of tensor calculus? For example, the Berlin way of doing tensor calculus further distinguishes „observer transformations“, „coordinate transformations“ and „basis transformation“.
Anyway, nicely done!
I may not know the names of these specific methods of teaching, but I have certainly searched up and down the internet finding an explanation that doesn't suck :P
I have other videos planned, though when they will be released, is up to the whims of the god of motivation
Very clear presentation of a topic that really is not easy.
Thank you very much! It took a while to make it just right.
As a mathematician, I only know of the word tensor in the term tensor product, how do these two things relate?
When a mathematician says tensor, they mean a map, or a function which sums things up like how a vector is a sum of basis vectors.
When a physicist says tensor, they mean a vector with extra dimensions/ranks.
A tensor product combines tensors of both kinds. I can’t go into more detail without a university course, but someone out there on youtube should have talked about this before.
If not, I’m making a video on it soon.
In an ideal world, I would really have liked to see how one actually uses a tensor mathematically to solve some straightforward problem. You gave an inkling of that, but did all the work pre-video and just showed the results, leaving me still unsure of how they are actually used. In contrast, it's pretty common to see matrices explained with concrete examples of using a matrix to do solve some sort of problem.
On the other hand, since this was meant as an entry to SoME, you did a great job in that regard! And I can understand that fitting such an example into this video might have been too much for a broader audience. One thing I did really like was the lamp post example and also the alternative example of frequency, and how different measurements may need to react in opposite ways. Good examples!
I'll check out some of your other videos to see if you have anything else about tensors. Thanks, and cheers!
Great introductory video! Thank you
Wouldnt the lamp be 8 hm instead of 8 m when using a hm ruler?
The idea is that it "should" be 16 hm. But, if we only blindly use numbers, then switching units would throw off the measurement
Its height doesn't change. . Units don't mean anything if they're not consistent. If you want the ultimate consistency, use fundamental constants, like C. We actually already do that.
Properties don’t, but think of how whether its inches or meters, physics remains the same.
really good video,
but what is a tensor anyway?
Technically, a tensor in maths is defined by the transformation, so anything which follows the transformation rule I mentioned in the video (transforming with coordinate changes) is a tensor. Vectors are tensors, scalars are also tensors.
If you want what physicists call tensors, Eigenchris does an amazing series on the maths, which you can watch here.
th-cam.com/play/PLJHszsWbB6hrkmmq57lX8BV-o-YIOFsiG.html
It is far harder, but is very meaty.
@@mindmaster107 thanks
Hey, I thought you were going to explain the math behind tensors, like, the matrices and such. Missed it :(
I might make long form videos in the future, but they are much harder to do for less views and hence less ears getting advice.
Stick around, and you might get what you are looking for. I'm not out of ideas yet.
1:20 For that matter, then almost all physical quantities are badly defined because they have units of measurement and hence depend on the definition of the unit and of which unit you’re using.
Its better to treat measurements like ratios, as while the exact number isn’t well defined, the actual amount it represents is real
I don't think a moving charge in free space without any contact with any other fields produces any magnetic field. Even to accelerate that charge requires it interact with some field . Also , electrons moving in a conductor reside within a very complex interplay of fields , and any motion would , of course , cause something to happen . A magnetc field perhaps ?
Well, the argument works the other way around. If you move past a charge, it has a magnetic field. This has been measured.
The complex interplay of forces, and hence accelerations, is super interesting, and is the entire field of electrodynamics, but there is a reason that topic is for 4th year university students.
@@mindmaster107 If you move past a charge and can detect ANYTHING then you are interacting with that charge . That's how measurement works . If you move past it without interacting , my guess is that there is no magnetic field. Mathematics can also be a great obfuscator and in and of itself won't get to the reality .
Yes, anything can happen when you aren’t looking, but that isn’t a scientific statement as thats always true (unfalsifiable).
@@mindmaster107 When an electron is in a magnetic field there is no detectable magnetic field but as soon as it moves within that field it does have a magnetic response . No magnetic response is going to appear in an electron just because you walked past it unless you were carrying a magnetic field yourself that the electron could detect . The whole thing is deeply entrenched . I've seen some of these experimental proofs and they are sketchy at best . To get to the bottom of it would require more talent than I think I have but It irritates me no end when I hear some of the accepted crap that even 4th yr university students must swallow .
Quantum field theory, literally has virtual particles which do the mediation of the magnetic field. There is literally something there which does the magnetic field.
Yes, you are correct that fields were originally just a formalism, and did not exist without a force acting on it, but experiments since the 1910’s revealed that the universe deals in fields and not forces. At larger scales, only equations of fields explain what we can observe, in General Relativity and The Standard Model.
Isn't the August 15th the deadline?
This was last year's lol
I'm making another one this year, so look forwards to that
Thank you very much for this video, mathematician definitions of tensors are so terse and boring for a physics grad like me :)
Definitions. The bane of our existence.
I'd just like to interject for a moment. What you are referring to as a "tensor", is in fact, a tensor field, or as I've taken to call it, an element of the tensor product of the tangent space of a pseudo-riemannian manifold and it's dual.
Vectors are not tensors unto themselves, but rather another component of a fully functioning vector field made useful by the theory of modules over infinite commutative division rings comprising a full vector space as defined by linear algebra.
🤓🤓🤓🤓🤓🤓🤓
Welcome to physics, where everything is always a smooth and well-defined field.
@@mindmaster107 the mathematics of spherical cows
Where every term beyond the first order doesn’t exist
Virgin number
Chad Tensor
LMAOOOOOOOOO 🤣🤣🤣🤣🤣🤣
EDIT: Help!! I can't stop laughing
Chad tensor reacts to coordinates. Chad tensor maintains it’s physical significance.
Ditch the sound effects.