So glad this series is finally relevant to my life, I've been meaning to watch it forever but the math was a little over my head - now this video is saving my ass on a relativity assignment and it makes so much sense!
Thank you Andrew, as someone who recently just graduated with an undergrad degree in physics and is continuing to study physics on my own time (aka no grad degree), your videos are an excellent resource on helping me understand general relativity and quantum field theoy.
Just started a math methods in physics class, and I couldn't understand the idea of a metric tensor for my life through my lectures. Your explanation made so much sense to me and is saving my life (and grade) as I type. Thank you so much for this content.
Hi Andrew. I am a Brazilian student of mechanical engineering, and in the future I want to go to the area of physics. So it's amazing to watch your videos and keep up with your routine. Hugs from Brazil and career success. Continue with the videos, please.
Was pretty interesting seeing a physics POV of the metric; I'd always seen it as the defining characteristic of the inner product of a space, but thinking about it as a unit-tacking linear machine is an interesting proposition.
Yeah definitely not a formal way of looking at it, but just looking at tensors as things that transform objects into other objects has always helped me. "multi-linear maps" would probably be more accurate
Last time I left a comment saying Thanks, I was in Math Methods for Physics I learning about the Levi-Covina Tensor. Now I’m saying Thank you again, I’m in Math Methods for Physics II and was assigned a project about I think EC coordinates and the first question is “find the metric” and the textbook wasn’t helping me at all. Your video made calculating the metric essentially crystal clear. Thanks Andrew!
I have been getting serious withdrawal from your videos😂 I'm not even a physics major but here I am making notes trying to understand it😄 You look great by the way!
Hey! I have a video suggestion for the maths videos, this of course after the tensor videos: A clean derivation of the Fourier Transform! I would totally watch it. Btw, this video was AMAZING, this is my first exposure to actual tensors, and although I’m sure there is a lot more to it, I was able to understand all of that you explained! Very well put together!
Beard delete.... But seriously enjoyed this video. It's funny because my College Algebra I professor used the end of the semester to start teaching us about vectors and different coordinate systems and I had no idea how any of it applied to anything, but seeing you use the same things he taught us is great. Love these videos Andrew.
I'm a few weeks behind in my Uni physics course as tensors have not made any sense to me at all. Repeating the same chapters in my textbook and then going through Fleisch's student's guide. Now I've seen this video it all clicked. I think I'm going to be ok. x
One of the questions on my GR homework that was due today was to convert the metric tensor from cylindrical to spherical coordinates, but using different GR notation lol
Nice! Just stumbled upon your channel. Expanding to minkowski space would be nice, perhaps look at some applications on the Dirac eq? Just some thoughts
hey dude i have one question we already learned about Jacobian Metrix when we r freshmen but. i really confuse why Jacobian matrix is one of metric tensor ... is it 2 Tensor?
well andrew dotson, you can read minds that are in future too. thats all i gotta say...... really man you do this goddamn job so well m not gay but i love you man
Hey, can you recommend a good text book for tensor calc for physicists? I'm doing a course in GR but I really cant get my head round lots of the maths but Im not mathsy enough for a pure math textbook
Learn the basics of differential geometry instead, if you are a math boi. This will get you started, begins with topology, but the main goal is GR. ruclips.net/p/PLFeEvEPtX_0S6vxxiiNPrJbLu9aK1UVC_
Isn't this just the Jacobi matrix/determinant one uses in coordinate transforms? (I am 2 mins in, and have no hope to understand this since I am but a lowly EE major and have never worked with tensors)
Hey Andrew (or whoever else reads this comment)! I'm an incoming physics major who will be minoring in math. As such, several of the classes for the math minor will be satisfied via my major. However, after the satisfied courses and the requirements, I am left with a choice of any two of the following courses. Which would be most beneficial to someone who wishes to pursue graduate studies? I'd appreciate any advice. Beyond "beneficial," if you are particularly passionate about any of the subjects offered, I'd love to hear! I love math so all of these honestly sound pretty cool. Thanks! Symbolic Computations in Mathematics (3) Intermediate Analysis I (3) Intermediate Analysis II (3) Complex Variables (3) Introduction to Combinatorics (3) Introduction to Graph Theory (3) Introduction to Cryptography and Coding Theory (3) Numerical Analysis I (3) Numerical Analysis II (3) Theory of Computation (3) Differential Equations (3) Optimization (3) Introduction to Partial Differential Equations (3) Elementary Abstract Algebra (3) Elementary Abstract Algebra II (3) History of Modern Mathematics (3) Modern Geometry (3) Differential Geometry (3) Introduction to Topology (3) Introduction to Probability (3)
Haha I'm a computer science student so the physics aspect doesn't concern me (although it was interesting), but i need to know tensor calculus for an AI project i was working on and this was a really good tutorial which could be understood by a computer science student who is only really good at statistics lol
Ok, I'm rewatching this series to begin solidifying my knowledge, but I have a question... During the metric tensor derivation in spherical coordinates, dotting the basis vector with itself was a simple squaring of the terms, and not an algebraic distribution that you would do for polynomials. Why?
@@AndrewDotsonvideos, honestly, being a middle-aged guy, this series is phenomenal. Coupled with tensor and Relativity video series put out by eigenchris and 3Blue1Brown, it's stunningly fluid to understand all of the mathematics and concepts. This is everything I've ever wanted to learn 25 years ago! Additionally, if you look carefully, one can see parallels to quantum mechanics, though they are subtle. You are a fantastic teacher, Andrew, and I look forward to expanding my knowledge through you. Thank you.
@@user-ji2kd8sx3y glad to hear you’re watching eigenchris as well, his videos will take you a lot farther! The parallels with quantum are everywhere. If you go down the QFT road a bit and learn about gauge theory, you’ll encounter something called a gauge covariant derivative. Very similar to the covariant derivative in GR. You ask the question “what do I need to modify my derivative to be so that the action is invariant under ____ transformations?” And instead of adding christoffel symbols, you add a gauge field (like the photon). Thanks again for checking out the series. Just let me know if anything is unclear or if you have some feedback on what could be better 👌🏻
@@arecus54 Definition: Let (V, +, dot) be a vector space. An (r, s) - tensor T over V is a multilinear map T: (V* X V* X dots V*) X (V x V dots V) --------> R where V* is a covector and V is a vector in which the number of V*'s refers to r and the number of V's refers to s. So, for example, a (1, 1) - tensor would be T^i_j, a map T: V* X V -----> R. The upper indices of a tensor refers to the covector components and the bottom indices refer to the vector components. That's it. Nothing sketchy or tricky.
the dS^2 that you write out at 13:06 at the top of the board should be r^2dtheta^2. 8
Thanks!
I promise that SOMEDAY i will understand this
So glad this series is finally relevant to my life, I've been meaning to watch it forever but the math was a little over my head - now this video is saving my ass on a relativity assignment and it makes so much sense!
"A tensor is something that transforms like a tensor" -- Einstein Gravity in a Nutshell: A. Zee
I have always hated spherical coordinates because they dΦ common sense
Jajajaja!!! Nice one.
Pooo
especially in electrodynamics.
Haha
Thank you Andrew, as someone who recently just graduated with an undergrad degree in physics and is continuing to study physics on my own time (aka no grad degree), your videos are an excellent resource on helping me understand general relativity and quantum field theoy.
I love how you actually read the comments! A fellow nine year old here👏👏
Very clear, even for someone with minimal physics knowledge. Thanks!
i named my cat entropy. always tending towards disorder
Ha ha nice
Just started a math methods in physics class, and I couldn't understand the idea of a metric tensor for my life through my lectures. Your explanation made so much sense to me and is saving my life (and grade) as I type. Thank you so much for this content.
FINALLY I understand it as I work on my differential geometry final project. Thank you Andrew.
Hi Andrew. I am a Brazilian student of mechanical engineering, and in the future I want to go to the area of physics. So it's amazing to watch your videos and keep up with your routine. Hugs from Brazil and career success. Continue with the videos, please.
Was pretty interesting seeing a physics POV of the metric; I'd always seen it as the defining characteristic of the inner product of a space, but thinking about it as a unit-tacking linear machine is an interesting proposition.
Yeah definitely not a formal way of looking at it, but just looking at tensors as things that transform objects into other objects has always helped me. "multi-linear maps" would probably be more accurate
@@AndrewDotsonvideos _multilinear maps are super intuitive tf are you on bout'_
Will someone award this man a Nobel Prize already! 👍
Chill man
Hey man! Thank you so much. Your explanation is so simple & effective that even I can understand it. You are a great tutor!!!
glad to see this series coming back! just in time for me to start studying the stuff (:
Great video!! Your explanations are awesome!! Thank you
Last time I left a comment saying Thanks, I was in Math Methods for Physics I learning about the Levi-Covina Tensor. Now I’m saying Thank you again, I’m in Math Methods for Physics II and was assigned a project about I think EC coordinates and the first question is “find the metric” and the textbook wasn’t helping me at all. Your video made calculating the metric essentially crystal clear. Thanks Andrew!
Love to hear it, really glad it was helpful!
I have been getting serious withdrawal from your videos😂 I'm not even a physics major but here I am making notes trying to understand it😄 You look great by the way!
Very nice approach, greatly presented!
Let’s go!
Edit: Can you add this to the Tensor Playlist? I don’t want to miss any of these videos since we just started Rigid Body Rotation in Taylor.
It’s been 5 days without my boi. Feels good to see him again
Beautifully and very easily explained .....
Very beneficial playlist . Thank you
It was an honest explanation if I am supposed to be honest at the instant. Keep it up!!!
A very useful one indeed. Cool 😎
Hey! I have a video suggestion for the maths videos, this of course after the tensor videos: A clean derivation of the Fourier Transform!
I would totally watch it.
Btw, this video was AMAZING, this is my first exposure to actual tensors, and although I’m sure there is a lot more to it, I was able to understand all of that you explained! Very well put together!
Finally Arfken is making some sense to me! Thanks a lot
Beard delete....
But seriously enjoyed this video. It's funny because my College Algebra I professor used the end of the semester to start teaching us about vectors and different coordinate systems and I had no idea how any of it applied to anything, but seeing you use the same things he taught us is great. Love these videos Andrew.
Yes, been waiting for this
Please keep this series going :)
Thank you so much for this video, this spherical coordinate metric was driving me mad, you saved my arse
I'm a few weeks behind in my Uni physics course as tensors have not made any sense to me at all. Repeating the same chapters in my textbook and then going through Fleisch's student's guide. Now I've seen this video it all clicked. I think I'm going to be ok. x
Thanks! That was really really helpful...
You would love differential geometry, the metric tensor is the gram matrix of the parametrization in the coordinates you desire
Great video! A big fan of yours! Question: What textbook do you recommend for tensor calculus?
hmm this guys knows how to explain, keep it up!!
Good job man thank you so much
One of the questions on my GR homework that was due today was to convert the metric tensor from cylindrical to spherical coordinates, but using different GR notation lol
Nice! Just stumbled upon your channel. Expanding to minkowski space would be nice, perhaps look at some applications on the Dirac eq?
Just some thoughts
Thanks again dude!!!!🤙🤙🤙
I’ve always ignored your calculations intensive videos tbh but I’ll stick with this and see how it relates to relativity. Also, notification gang
Next video I get into (special) relativity!
Thank you!!
could we see how these metric tensor transformations are used in cartographic projection (ie spehrical to cylindric to cartesian)..? thanks for vid
Can you please show how to calculate the gradient on dot product, using tensor notation?
Please do a followup on general relativity
won a sub buddy you legend!
Great...tensor or any other video...do any video...it's gonna be great!!
Can You Do a Series In Special And General relativity Using tensors and Dirac Notation
what books we should follow to understand tensor???
You didn't go over using the jacobian matrix to calculate the metric. I noticed an interesting pattern when I used that method.
Please another one as soon as possible
8:28 why do we use subscripts for the Metric Tensor
Must watch later xD Gotta read over it before prob :D
Hello andrew , any advices for the Physics GRE , books materials apart of conquering the physics GRE ?
I'm not really the person to ask lol
hey dude i have one question we already learned about Jacobian Metrix when we r freshmen but. i really confuse
why Jacobian matrix is one of metric tensor ...
is it 2 Tensor?
Are all rank 2 tensors symmetric matrices?
well andrew dotson, you can read minds that are in future too. thats all i gotta say......
really man you do this goddamn job so well m not gay but i love you man
Why does index notation use superscripts instead of subscripts? Is there a specific reason or is just convention?
Does the result of multiplying by the metric tensor always convert coordinates to a unit of distance squared?
ds^2 =g_{mu nu}dx^mu dx^nu always has units of distance squared.
Dear Mr. Dotson, I think you've been recommending a book on the topic. Is it called Tensor calculus for physics by Neuenschwander?
yep
Yep.
Hey, can you recommend a good text book for tensor calc for physicists? I'm doing a course in GR but I really cant get my head round lots of the maths but Im not mathsy enough for a pure math textbook
"Tensor Analysis by Labedev and Cloud" is a great introduction to tensors written for physicists.
Learn the basics of differential geometry instead, if you are a math boi. This will get you started, begins with topology, but the main goal is GR.
ruclips.net/p/PLFeEvEPtX_0S6vxxiiNPrJbLu9aK1UVC_
Yay,now you look more beautiful.
Isn't this just the Jacobi matrix/determinant one uses in coordinate transforms? (I am 2 mins in, and have no hope to understand this since I am but a lowly EE major and have never worked with tensors)
Mate Toth the square root of the determinant of the metric is the jacobian, yup!
Hey Andrew (or whoever else reads this comment)! I'm an incoming physics major who will be minoring in math. As such, several of the classes for the math minor will be satisfied via my major. However, after the satisfied courses and the requirements, I am left with a choice of any two of the following courses. Which would be most beneficial to someone who wishes to pursue graduate studies? I'd appreciate any advice. Beyond "beneficial," if you are particularly passionate about any of the subjects offered, I'd love to hear! I love math so all of these honestly sound pretty cool. Thanks!
Symbolic Computations in Mathematics (3)
Intermediate Analysis I (3)
Intermediate Analysis II (3)
Complex Variables (3)
Introduction to Combinatorics (3)
Introduction to Graph Theory (3)
Introduction to Cryptography and Coding Theory (3)
Numerical Analysis I (3)
Numerical Analysis II (3)
Theory of Computation (3)
Differential Equations (3)
Optimization (3)
Introduction to Partial Differential Equations (3)
Elementary Abstract Algebra (3)
Elementary Abstract Algebra II (3)
History of Modern Mathematics (3)
Modern Geometry (3)
Differential Geometry (3)
Introduction to Topology (3)
Introduction to Probability (3)
I love you.
Haha I'm a computer science student so the physics aspect doesn't concern me (although it was interesting), but i need to know tensor calculus for an AI project i was working on and this was a really good tutorial which could be understood by a computer science student who is only really good at statistics lol
"Tensor boy out!" God I love this
Thanks a lot! Glad you’re getting a lot out of it!
Woah you shaved
He needs to get His teeth done
he also stopped calling phi as fee
Yass, yay
I'm a ninth grader going on 10 th and i want to pursue a physics degree so what math should i start learning to understand higher physics courses?
school maths
@@CubetasticMushrooms lmao roasted
Ok, I'm rewatching this series to begin solidifying my knowledge, but I have a question... During the metric tensor derivation in spherical coordinates, dotting the basis vector with itself was a simple squaring of the terms, and not an algebraic distribution that you would do for polynomials. Why?
Nevermind! I had a brain fart... No explanation needed! Great series!
Thank you! Would like to hear your feed back as you go farther!@@user-ji2kd8sx3y
@@AndrewDotsonvideos, honestly, being a middle-aged guy, this series is phenomenal. Coupled with tensor and Relativity video series put out by eigenchris and 3Blue1Brown, it's stunningly fluid to understand all of the mathematics and concepts. This is everything I've ever wanted to learn 25 years ago! Additionally, if you look carefully, one can see parallels to quantum mechanics, though they are subtle. You are a fantastic teacher, Andrew, and I look forward to expanding my knowledge through you. Thank you.
@@user-ji2kd8sx3y glad to hear you’re watching eigenchris as well, his videos will take you a lot farther!
The parallels with quantum are everywhere. If you go down the QFT road a bit and learn about gauge theory, you’ll encounter something called a gauge covariant derivative. Very similar to the covariant derivative in GR. You ask the question “what do I need to modify my derivative to be so that the action is invariant under ____ transformations?” And instead of adding christoffel symbols, you add a gauge field (like the photon).
Thanks again for checking out the series. Just let me know if anything is unclear or if you have some feedback on what could be better 👌🏻
Anyone explain the point at 14:00
I missed you
and I you
That Shit!! Hits different at 2x speed
😂😂😂
4:31 hmm
it,s a gravitational waves hahaha
5:44 usual physicist moment
Neck!!
Neck!!
Neeeeeeck!!!!!
What happened at 4:31, I think gravitational waves penetrated you, causing the picture to become distorted haha
Something on plasma physics.
Press F to pay respects to Andrew's beard
The beard!!!!
Am fresh undergrad, not sure what tensors are tbh._.
Other than matrixes in higher dimensions
they follow a specific transformation rule under coordinate transformations
do you still not know what tensors are 2 years later?
@@williammendez5209 no?
@@arecus54 Definition: Let (V, +, dot) be a vector space. An (r, s) - tensor T over V is a multilinear map T: (V* X V* X dots V*) X (V x V dots V) --------> R where V* is a covector and V is a vector in which the number of V*'s refers to r and the number of V's refers to s. So, for example, a (1, 1) - tensor would be T^i_j, a map T: V* X V -----> R. The upper indices of a tensor refers to the covector components and the bottom indices refer to the vector components. That's it. Nothing sketchy or tricky.
why did you shavee??
holy shit you shaved. rip in peace magnificent beard
liboga kaayu imong index ug imong power
Right then. Back to third grade I go. :-D
You look a few years younger.
"Tensor Boi out......"
I feel smarter somehow.
Those negative signs will get you
I am halfway through physics in highschool sooo...
Has anyone failed a math test once and still went on to get an A in the class?
You starved me of content for too long
5:45 that's dimensionless again, SHIT
🤬I need a bigger whiteboard!
His lacking of facial hair made me think this was an old video...
Why the beard why
Being gay is no excuse not to learn tensor calculus