When you said “mathematicians” I was like “ok? That’s weird.” Then I saw the German Eularoid and almost spit out the water I was drinking. I love your subtle jokes.
I usually mostly watched joke sketches on this channel, so for the first half of the video I was thinking this is some _very_ elaborate set up for a joke.
Regarding 2:34, it's true that linear transformations need not map into the same vector space. A linear operator is a linear transformation that maps into the same vector space. HOWEVER We can really view any linear transformation as being a linear operator without any real loss of generality in an algebraic sense. Here's the important facts that make this is true: (i) A linear transformation can map into a different vector space. BUT, both vector spaces have to be over the same field F, just by definition. (ii) Two vector spaces of the same dimension over the same field F are isomorphic. (iii) A vector space of dimension n always has subspaces of dimension n-1, n-2, n-3,...,0. (iv) The rank-nullity theorem guarantees that the image T(V) has dimension less than or equal to dim(V), for any linear transformation T. NOW, in the general setting, I have vector spaces V, W over field F and I have a linear transformation T: V -> W. By (iv), I know I am mapping into a subspace of W (perhaps W itself) with dimension less than dim(V). By (iii), V has a subspace K of dimension dim(T(V)), since dim(T(V))
True. However, in general there is not a canonical isomorphism between such spaces. The metric tensor allows us to convert between covariant and contravariant indices. You can also use the Kronecker epsilon to assist with (iirc) mapping between rank k and rank n-k tensors in n-dimensional space; this is how we get the cross-product in 3-space, for example.
Maxwell stress tensor shows up somewhere in the second half of Griffiths EM, also towards the end of the text when it touches on relativity. Nothing crazy
The first tensor you will possibly see is a moment of inertia tensor in classical mechanics, either sophomore or junior year. Just get through your intro courses and everything will be fine
@@Ryan_Perrin Moment of inertia tensor is usually treated as just a matrix though. A discussion of covariant/contravariant tensors is mandatory if you take an optional general relativity class. Otherwise you might go through undergrad without taking them.
current undergrad here. Learned of tensors in theoretical physics 1, basically theoretical mechanics when learning about force of inertia, galilei transformation and how to express a curve in other systems via tensors using levi civita and kroneckers delta
Our boy Andrew finally understanding tensors as multi-linear maps from a vector space/dual space. I'm so proud. Fun fact, V = (V*)* [The vector space is isomorphic to the dual of its dual space] only for finite dimensions!
Nope. V=(V*)* is a property of so called 'reflective spaces'. A Hilbert space (not necessarily finite) is just an example of a reflective space. Got it ?
Something that I think would be good to make more explicit is that action by the metric tensor is specifically what raises or lowers indices. We can apply it to as many indices as we need.
Been a while since I covered this stuff. You have explained it better than I think I ever heard (I had professors from Sixty Symbols /Numberphile as well)
One forms... someone is starting to read some differential geometry I hope... you’re in for the tastiest meal you’ve ever eaten at the table of physics
Andrew if you contract a doubly covariant tensor with a contravariant vector the tensor contraction in this case means multiplying the 1x n vector (i.e. a row vector) with the matrix corresponding to the biliniar form. Hence, you get a row vector back. So it still makes sense to talk about the eigenvalue problem granted that the components of i is the same as the components of j. So that you get a an nxn matrix. Otherwise, if the matrix is nxm with n not equal to m then it doesn’t make sense to talk about it’s eigenvalues.
Hey Andrew im a big fan of yours. Can you make a video about the double slit experiment and also give your thoughts of why does the particles behave differently when a detector is used.
First person to find a "natural" instance (i.e. coming directly out of a field equation or metric) of a mix tensor that has ever been used for anything, gets a freakin medal. #notmyhilbertspace Ps: Still love your videos :D
I Dont know why this was recommended to me, all i heard was .......schrödinger equations....... Wave function....... Thats all i understood. Im Still super happy i watched it!
Being a past math/physics major, I was embarrassed that I didn't know what tensors were, because as much as I got through before my scholarship went bankrupt, never heard of them. Ended up asking my engineer brother, and it was really just that when I took my grad classes in undergrad, either it required basic math because of complexity, and then the one teacher just taught us her PhD thesis instead of the class which would have used tensors. Her thesis was terrible and boring as fuck haha.
@@AndrewDotsonvideos my chaos theory class was a physics theory, and my fluid dynamics and chaos theory that I took at Carnegie Mellon was physics too. The former was the terrible teacher but should have used tensors, the latter, our math was very simplified because you have to take out a dimension and a lot of different variables unless you go down the fluid dynamics route for your degree I think.
@@AndrewDotsonvideos though I did skip a lot of my math classes... I could teach myself all of that and I was busy working on a solo project in analytic number theory. I managed to piss off two departments doing that haha
I took linear algebra as a bio major the whole time I was like idk wtf all these thing are conceptually like an eigenvector, value, grand Schmidt etc mean but I can get you them for you 😂😂😂
All I hear when Andrew speaks is, blah blah Vectorspace... Blah Blah Blah Matrices, Blah Blah Blah T µν Blah Blah Tensors, Blah Blah I still have problems with those 63 Integrals, maybe should I ask Papa Flammy about them... Blah Blah what does he mean Euler Mascheroni constant? Blah Blah Transformations, Blah Blah, I wonder if Wendy's is open, Blah Blah I wish I had a Black Board and some Hagoromo Chalk, Blah Blah Blah thanks for watching. Oh shit I think I can hear Andrew's Inner Monologues...
It's been a couple years since I graduated.... but doesn't 4:31 break the normal rules on contracting indices...? I would have expected a scalar output, not a trace.
The trace of a tensor gives a scalar, there aren't any free indices since mu is summed over. As an example, the QCD lagrangian (which is a scalar) can be written in terms of the trace of the gluon field strength tensor.
It's 1.1 meters high and 2 meters wide. Or at least due to my assumptions I guess those are the measurements. Sure it's America so the closest number in feets and inches... perhaps, so 42 inches high and 78 inches wide.
When you said “mathematicians” I was like “ok? That’s weird.” Then I saw the German Eularoid and almost spit out the water I was drinking. I love your subtle jokes.
I usually mostly watched joke sketches on this channel, so for the first half of the video I was thinking this is some _very_ elaborate set up for a joke.
Regarding 2:34, it's true that linear transformations need not map into the same vector space. A linear operator is a linear transformation that maps into the same vector space.
HOWEVER
We can really view any linear transformation as being a linear operator without any real loss of generality in an algebraic sense. Here's the important facts that make this is true:
(i) A linear transformation can map into a different vector space. BUT, both vector spaces have to be over the same field F, just by definition.
(ii) Two vector spaces of the same dimension over the same field F are isomorphic.
(iii) A vector space of dimension n always has subspaces of dimension n-1, n-2, n-3,...,0.
(iv) The rank-nullity theorem guarantees that the image T(V) has dimension less than or equal to dim(V), for any linear transformation T.
NOW, in the general setting, I have vector spaces V, W over field F and I have a linear transformation T: V -> W.
By (iv), I know I am mapping into a subspace of W (perhaps W itself) with dimension less than dim(V).
By (iii), V has a subspace K of dimension dim(T(V)), since dim(T(V))
Underrated comment. Where my other math bois at?
Crap I just saw this
@@AndrewDotsonvideos Its okay, we still love you
True. However, in general there is not a canonical isomorphism between such spaces. The metric tensor allows us to convert between covariant and contravariant indices. You can also use the Kronecker epsilon to assist with (iirc) mapping between rank k and rank n-k tensors in n-dimensional space; this is how we get the cross-product in 3-space, for example.
I don't understand a thing but i feel smart watching your videos.
My boi with the another Great tensor series video
Keep up the good work! Love the content
I was like, hell yeah, I'll watch this once I wrap up this 15 page LaTeX document. Yeah that was 12 hours ago.
Can't wait to start physics at university so i can understand all of these vids lol
Maxwell stress tensor shows up somewhere in the second half of Griffiths EM, also towards the end of the text when it touches on relativity. Nothing crazy
The first tensor you will possibly see is a moment of inertia tensor in classical mechanics, either sophomore or junior year. Just get through your intro courses and everything will be fine
I'm starting in a few weeks!
@@Ryan_Perrin Moment of inertia tensor is usually treated as just a matrix though. A discussion of covariant/contravariant tensors is mandatory if you take an optional general relativity class. Otherwise you might go through undergrad without taking them.
current undergrad here. Learned of tensors in theoretical physics 1, basically theoretical mechanics when learning about force of inertia, galilei transformation and how to express a curve in other systems via tensors using levi civita and kroneckers delta
I love these clarification videos! They're extremely helpful in clearing up fundamental misconceptions.
I’m so glad I took a more abstract linear algebra course because I feel like that will make eventually learning tensors much easier
Our boy Andrew finally understanding tensors as multi-linear maps from a vector space/dual space. I'm so proud.
Fun fact, V = (V*)* [The vector space is isomorphic to the dual of its dual space] only for finite dimensions!
Nope. V=(V*)* is a property of so called 'reflective spaces'. A Hilbert space (not necessarily finite) is just an example of a reflective space. Got it ?
And to think one day he may be a professor. Oh those poor students...
my god... this madlad is rewriting the entirety of nonrelativistic quantum mechanics in tensor notation *gasp*
Something that I think would be good to make more explicit is that action by the metric tensor is specifically what raises or lowers indices. We can apply it to as many indices as we need.
Been a while since I covered this stuff. You have explained it better than I think I ever heard (I had professors from Sixty Symbols /Numberphile as well)
Next year I'm doing linear algebra at school yay
One forms... someone is starting to read some differential geometry I hope... you’re in for the tastiest meal you’ve ever eaten at the table of physics
This is so cool. I'm going to have to try implementing this in my current electronics project. Reminds me of the Hadamard transform.
Andrew if you contract a doubly covariant tensor with a contravariant vector the tensor contraction in this case means multiplying the 1x n vector (i.e. a row vector) with the matrix corresponding to the biliniar form. Hence, you get a row vector back. So it still makes sense to talk about the eigenvalue problem granted that the components of i is the same as the components of j. So that you get a an nxn matrix. Otherwise, if the matrix is nxm with n not equal to m then it doesn’t make sense to talk about it’s eigenvalues.
Hey Andrew im a big fan of yours. Can you make a video about the double slit experiment and also give your thoughts of why does the particles behave differently when a detector is used.
First person to find a "natural" instance (i.e. coming directly out of a field equation or metric) of a mix tensor that has ever been used for anything, gets a freakin medal. #notmyhilbertspace
Ps: Still love your videos :D
I Dont know why this was recommended to me, all i heard was .......schrödinger equations....... Wave function....... Thats all i understood. Im Still super happy i watched it!
Could you make a series of vídeos about differential geometry in similar fashion to the tensor series? That'd be really cool
"Good morning fellow mathematicians"
The word of Andrew dotson a.k.a the theory boi
Ayy, papa Dotson back at it!
Starts video, hits like, begins unzipping pants. Pulls notepad then begins taking notes
Being a past math/physics major, I was embarrassed that I didn't know what tensors were, because as much as I got through before my scholarship went bankrupt, never heard of them. Ended up asking my engineer brother, and it was really just that when I took my grad classes in undergrad, either it required basic math because of complexity, and then the one teacher just taught us her PhD thesis instead of the class which would have used tensors.
Her thesis was terrible and boring as fuck haha.
In my experience, what we call tensors, math people just call multi linear mappings
@@AndrewDotsonvideos my chaos theory class was a physics theory, and my fluid dynamics and chaos theory that I took at Carnegie Mellon was physics too. The former was the terrible teacher but should have used tensors, the latter, our math was very simplified because you have to take out a dimension and a lot of different variables unless you go down the fluid dynamics route for your degree I think.
@@AndrewDotsonvideos though I did skip a lot of my math classes... I could teach myself all of that and I was busy working on a solo project in analytic number theory. I managed to piss off two departments doing that haha
never ask an engineer its just embarrassing
i am in high school and .... my head just rebooted
2:41 - CaN'T bReAtHE 😂😂😂
Thanks Andrew, very cool
I took linear algebra as a bio major the whole time I was like idk wtf all these thing are conceptually like an eigenvector, value, grand Schmidt etc mean but I can get you them for you 😂😂😂
I don't understand a word of this but it is very interesting
All I hear when Andrew speaks is, blah blah Vectorspace... Blah Blah Blah Matrices, Blah Blah Blah T µν Blah Blah Tensors, Blah Blah I still have problems with those 63 Integrals, maybe should I ask Papa Flammy about them... Blah Blah what does he mean Euler Mascheroni constant? Blah Blah Transformations, Blah Blah, I wonder if Wendy's is open, Blah Blah I wish I had a Black Board and some Hagoromo Chalk, Blah Blah Blah thanks for watching.
Oh shit I think I can hear Andrew's Inner Monologues...
It's been a couple years since I graduated.... but doesn't 4:31 break the normal rules on contracting indices...? I would have expected a scalar output, not a trace.
The trace of a tensor gives a scalar, there aren't any free indices since mu is summed over. As an example, the QCD lagrangian (which is a scalar) can be written in terms of the trace of the gluon field strength tensor.
@@AndrewDotsonvideos looks like ive got some review to do haha. Thanks!
Tensors-are-matracies thing - Andrew Dotson 2019
Oh fucc a quickie
Bro. Wereyou the best in your class or one of the best in the country when you were in high school?
Do you have any idea of this board measurements ? i want to buy one like this one, it looks perfect to train my physics
It's 1.1 meters high and 2 meters wide. Or at least due to my assumptions I guess those are the measurements. Sure it's America so the closest number in feets and inches... perhaps, so 42 inches high and 78 inches wide.
@@livedandletdie I dont know how you know that, but thats a lot
Is it true that the only math I need to get into physics is Algebra, trig and calc?
Actually,Only linear algebra and calc is sufficient .But then, Everything comes under those two .
Does this mean exterior products are coming soon?
i dont understand any of these vids but when i start physics at uni ima rewatch this stuff and understand it lol
Australians be like: E mu...
Wait, you’re at 69k subscribers
Wait.. isn't g_\mu
u T^\mu
u by itself the trace (that is, a scalar)? Why mention it as T^\mu_\mu? (small edit: my head)
Emre Acartürk sorry I don’t think I understand the question. Yes g_munu T^mu nu is the trace, which is equivalent to T^mu_mu
@@AndrewDotsonvideos I meant... why include indices at all?
It _is_ a scalar, after all, what could we index?
So now I've got explenation for staying up late xD
Flamie has been quicced :(
I'm view #1000, yay!!
those mu's are so... make-fun-of-able (?
I did my best
just had my exam on general relativity this monday , what a releif...
Ahh, I am actually jealous. I wish I could learn general relativity at school. I am still at middle school (15 years old), it is so boring.. 😧