Why Tensor Calculus?

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bit.ly/PavelPat...
Textbook: bit.ly/ITCYTNew
Errata: bit.ly/ITAErrata
McConnell's classic: bit.ly/MCTensors
Table of Contents of bit.ly/ITCYTNew
Rules of the Game
Coordinate Systems and the Role of Tensor Calculus
Change of Coordinates
The Tensor Description of Euclidean Spaces
The Tensor Property
Elements of Linear Algebra in Tensor Notation
Covariant Differentiation
Determinants and the Levi-Civita Symbol
The Tensor Description of Embedded Surfaces
The Covariant Surface Derivative
Curvature
Embedded Curves
Integration and Gauss’s Theorem
The Foundations of the Calculus of Moving Surfaces
Extension to Arbitrary Tensors
Applications of the Calculus of Moving Surfaces
Index:
Absolute tensor
Affine coordinates
Arc length
Beltrami operator
Bianchi identities
Binormal of a curve
Cartesian coordinates
Christoffel symbol
Codazzi equation
Contraction theorem
Contravaraint metric tensor
Contravariant basis
Contravariant components
Contravariant metric tensor
Coordinate basis
Covariant basis
Covariant derivative
Metrinilic property
Covariant metric tensor
Covariant tensor
Curl
Curvature normal
Curvature tensor
Cuvature of a curve
Cylindrical axis
Cylindrical coordinates
Delta systems
Differentiation of vector fields
Directional derivative
Dirichlet boundary condition
Divergence
Divergence theorem
Dummy index
Einstein summation convention
Einstein tensor
Equation of a geodesic
Euclidean space
Extrinsic curvature tensor
First groundform
Fluid film equations
Frenet formulas
Gauss’s theorem
Gauss’s Theorema Egregium
Gauss-Bonnet theorem
Gauss-Codazzi equation
Gaussian curvature
Genus of a closed surface
Geodesic
Gradient
Index juggling
Inner product matrix
Intrinsic derivative
Invariant
Invariant time derivative
Jolt of a particle
Kronecker symbol
Levi-Civita symbol
Mean curvature
Metric tensor
Metrics
Minimal surface
Normal derivative
Normal velocity
Orientation of a coordinate system
Orientation preserving coordinate change
Relative invariant
Relative tensor
Repeated index
Ricci tensor
Riemann space
Riemann-Christoffel tensor
Scalar
Scalar curvature
Second groundform
Shift tensor
Stokes’ theorem
Surface divergence
Surface Laplacian
Surge of a particle
Tangential coordinate velocity
Tensor property
Theorema Egregium
Third groundform
Thomas formula
Time evolution of integrals
Torsion of a curve
Total curvature
Variant
Vector
Parallelism along a curve
Permutation symbol
Polar coordinates
Position vector
Principal curvatures
Principal normal
Quotient theorem
Radius vector
Rayleigh quotient
Rectilinear coordinates
Vector curvature normal
Vector curvature tensor
Velocity of an interface
Volume element
Voss-Weyl formula
Weingarten’s formula
Applications: Differenital Geometry, Relativity

Опубликовано:

28 сен 2024

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Комментарии : 389

@wagsman9999 8 лет назад

I am just about through these lectures. I must say I have really enjoyed them. Years back I took the usual dose of calculus in high school / college, but I was never exposed to tensor analysis. While trying to swallow general relativity on my own, I realized the math was over my head. These lectures are the PERFECT entry point if you have no background in tensor analysis. A practical and clear presentation, without getting bogged down in the underlying theory. This is why I love RU-vid! Thanks professor. I will check out you other videos too.

@MathTheBeautiful 8 лет назад

+Mark Wagner Hi Mark, thank you, I'm very glad you enjoyed the lectures! If I may share my opinion on one thing, I would say that the "underlying theory" is very much there in my videos. What I've eschewed was the obfuscating formalism.

@wagsman9999 8 лет назад

+MathTheBeautiful Yes, that's a better way to put it. Thanks again.

@thane9 7 лет назад

I genuinely believe bearing witness to the beauty of mathematics really requires a lot that we learn in a math undergrad. Sure there's the wonderful elegance of things like Heron's, but for me it wasn't until I really understood the complexity of the messy answers that the beauty of the elegant emerged. Yes, it's such a shame that the uninitiated are missing out on so much.

@wagsman9999 7 лет назад

michael jordan if you have had a class in multivariable calc you should do just fine. The product and chain rules are used a lot. Hope you enjoy these as much as I did!

@David-km2ie 5 лет назад

@Jakob Jones I dont see tensors as multidimensional linear algebra, it are just summations. That's it. You can use it as alternative for linear algebra though.

@JumpUpNPullaco 7 лет назад

I'm watching these videos multiple times, until I really understand. I wish I had this replay option back in the nineties. Your teaching method, enthusiasm, and devotion is lovely. Don't stop. Many thanks to you from Victoria, British Columbia.

@TheDavidlloydjones 10 лет назад

It was thoughtful of you, when the light blanked out the slides on your screen, to add a graphic which made the illustration clear for the video viewers. Thinking about whether your medium is in fact intermediating is one of the signs of a pro, and of a person who takes enough pride in their work to think that it's worth getting it across to the audience. It's also a sign of a teacher who has some respect for their students. Thank you. -dlj.

@cfriedalek 10 лет назад

Having a look at tensor calculus just for fun after decades of having never understood it. With this first lecture you've given it meaning to me for the first time. Absolutely awesome. I look forward to the remainder of the series.

@MathTheBeautiful 6 лет назад

Thank you. Please check out the new Vector Calculus series, too! It's a prequel to this series.

@OttoFazzl 9 лет назад

At 39:35 "in order for it to be seen in nature, it needs to be a minimum". Really nice phrase and thought provoking!

@MyJuicehole 10 лет назад

I like how you talked about not really picking a specific coordinate system. I remember looking at a proof for Gauss' divergence theorem, and the way it went was that you first prove the divergence theorem for an arbitrary coordinate system containing some rectangular box, and then you expand and generalize it by assuming that a figure that can be represented as a rectangle in one coordinate system will appear arbitrarily shaped in another coordinate system. And by that, you assume that you can write the variables for the one coordinate system in terms of the other system. I remember spending hours writing the full proof because I just really wanted to wrap my mind around it, and it seriously just made me feel so dumb. I am honestly amazed that someone could have that much intuition and creativity in approaching a problem that is so difficult to comprehend, even after looking at the solution. I hope one day I can become that good.

@MathTheBeautiful 10 лет назад

The way I see Gauss' theorem is originally an algebraic theorem in the arithmetic space, which is then carried over to the geometric world. (You can find the proof on page 242 of the book.) I'm not sure about the proof with rectangles since the boundary rectangles don't have the right normals. I would be curious to see that argument.

@bonbonpony 5 лет назад

Yeah, Gauss's theorem really is more about the geometry (or should I say topology?) than calculus. It boils down to the observation that whatever comes from the inside of the surface to the outside of it, must go through that surface, whatever shape the surface has. And how much stuff goes through that surface (disappearing from the inside), depends on how fast it goes through that surface. And this observation SHOULD be coordinate-independend, and SHOULD not depend on the shape of the surface. Coordinate systems and calculus only make this harder to see. (Not to mention if one adds some physics shenanigans to the brew :P )

@Sillybb9142 7 лет назад

Really made me understand the motivation behind tensors that I was so confused about before. Thank you sir!

@yanniphone6729 10 лет назад

Been waiting for some decent tensor calc lectures for years now. Thanks fior posting!

@RalphDratman 4 года назад

This is really good in many respects. I haven't heard such a high level perspective on a subject, comparing approaches, especially on the first day.

@stevedeltoid5413 8 лет назад

This subject is completely irrelevant to my daily existence, yet I enjoyed watching. Great teacher.

@pablojaviervaz 8 лет назад

+B Meyer That's the spirit! Can you talk with my students please =)

@Ector521 8 лет назад

+Pablo Vaz *talk to To "talk with" should be used when talking with a person or multiple persons. "Talking to" should be used when you are talking to either no responsive people or things.

@pablojaviervaz 8 лет назад

+Ector521 thanks buddy as you can see, my english needs to be improved! Thank you!

@lightfreak999 8 лет назад

+Ector521 I see what you did there ;)

@comprehensiveboycomprehens8786 8 лет назад

Well, math helps me forget daily existence and I like that!

@intellectelite 7 лет назад

I absolutely love the you defined Euclidean space. A space where straight lines exist. So simple yet so brilliant. Thanks Professor!

@rickandelon9374 6 лет назад

where you can draw straight lines!

@bonbonpony 5 лет назад

and they don't turn out to be some circles or other curved lines in the end :) Another good way to characterize Euclidean space: it's the space in which parallel straight lines never cross.

@mdforbes500 9 лет назад

Beautiful lecture, though it took me a moment to retrieve the meaning in the English language from the mathematical language. I see why Einstein was so entranced by the beauty of tensors. Binge watching this series of lectures for fun.

@Hebrew247 6 лет назад

Malcolm Forbes really?

@ramnewton 6 лет назад

Professor, you are awesome. Really loved the way you talked about co - ordinate systems and the beauty in geometric solutions

@MathTheBeautiful 6 лет назад

Thanks Ram! Check out the new Vector Calculus series.

@henryalferink1941 4 года назад

Hi Sir, I just want to thank you for all your videos. I've been going through your linear algebra playlists, and wanted to say that your content has been the most useful I've found on RU-vid, particularly in understanding the intuition behind things. Thank you very much!

@mehrdadkeneshlou7898 2 года назад

This introduction lecture is one of the best lectures, or better to say probably the best one I've ever had in my life. As a geometry lover the proof of the Heron's problem was so amusing. This lectures allowed me understand why I've read continuum mechanics.

@jmafoko 4 года назад

Great teacher ever. This man makes this look so easy, which is how I think all subjects are in essence. In a hands of a great teacher, what seems formidable is made palatable.

@thekkl 10 лет назад

"and I don't know if you remember from multivariable calculus - I HOPE YOU DON'T" That's when I tried to like the video a second time.

@mkmaharana 9 лет назад

I enjoyed the video lectures. Thank you, I did learn a lot. It will help me understand the concepts in non-linear finite element analysis where a lot of tensor notation is used.

@NothingMaster 5 лет назад

This is nothing short of a priceless teaching of insightful mathematics; and all of it brilliantly put into historical perspective, as well. Bravo! 👏🏻

@prabhatp654 4 года назад

greatest introduction and the humility which was refrained by the professor for mathematicians was absolutly warming

@MathTheBeautiful 4 года назад

Thank you for the kind words. And you're right, I'm one of the most (if not *the* most) humble person I know

@sorinsuciu8675 7 лет назад

Awesome lecture. Small correction: @09:12 - Archimedes demonstration survived 2 millennia, not 2 centuries

@iyalovecky 3 года назад

Oh man, I love you so much!!! I just don't have time to study this beautiful area, I need to do full-stack development because it brings money to me right now :(

@fizixx 9 лет назад

I should have, but never did take this coursework in school. I've always wanted to learn it, but most of the resources I've found have been unsatisfactory --- for me at least. So I am eager to go through each of your videos here. You seem to have a good way of explaining things so I'm hopeful that, for the first time, I will get good insight to the material. Enough that I will understand enough that I can do more reading on my own. I have several advanced science degrees so I'm not unfamiliar with complicated material. Thanks for posting the videos.....I will also check out your book as I get a little further into your videos and I'm sure I'm following things well.

@w.hoffman3308 7 лет назад

+fizixx I feel as you. Good post and a thumb from me.

@kevinbyrne4538 9 лет назад

9:16 -- The Roman statesman Cicero (106-43 BC) actually visited Archimedes' grave, where he saw Archimedes' tombstone, which he described in his "Tusculan Disputations", book V, sections 64-66. It did indeed have a sphere and a cylinder.

@bonbonpony 5 лет назад

Did he mention where is this grave located?

@jacopopispola9925 3 года назад

@@bonbonpony Syracuse, Sicily

@PiotrSupski 8 лет назад

This is just what I needed. Thank you a lot, it's just ridiculously awesome content!

@steviewonder9209 10 лет назад

At about 22:15, to find the location of the Torricelli point on line A'B, just repeat the process using another vertex as the pivot. Using A as the pivot, for example, would generate B'C. The intersection of the two lines is the Torricelli point.

@Trotskisty 9 лет назад

Archimedes actually got his tombstone: and it used to even be a local Syracuse 'tourist attraction', in the ancient World. But sometime during the Middle Ages -- it apparently went missing... Maybe it'll show up somewhere, someday.

@intellectelite 7 лет назад

Trotskisty dope!

@christophermoore5389 4 года назад

I thought he was killed when Syracuse was invaded by the romans

@zombiedude347 7 лет назад

I wish I had the opportunity to learn this earlier, cause now I'm dealing with a ton of vector calculus and using tables to convert between coordinate systems.

@bigpapi3636 7 лет назад

Discovering Tensor Calculus just made my job incredibly easier. Taking coordinate drawings of complex exhaust systems and creating 3D insulation systems based on those drawings. Like discovering fire!

@nathanas64 4 года назад

What a great Professor! Really enjoyed this lecture!

@ac-dp3jk 6 лет назад

A great lecture thank you ! FOr those who like me got confused by the use of the world 'algebra' at 1:27 or 'algebraic geometry' later on: the lecturer means 'calculus' or 'analysis'. Algebraic geometry is a completely unrelated topic.

@thcoura 8 лет назад

Amazing classes! Here is my humble suggestion for the viewers. If you fell anxious asking yourself, where are the tensors!? Go without any compromise through other videos, see them. Finally when you start to know what you don't know, come back in peace to the first video and enjoy the course.

@happytouch7104 9 лет назад

Could you subtitle these excellent video lectures or provide transcripts? I think this will be most helpful to non-native English-speaking folks. If you can do this, I will always appreciate it.

@Jipzorowns 8 лет назад

I wish these videos were available a couple of years ago, when I was studying General Relativity. Anyway, great insight, thanks!

@MathTheBeautiful 8 лет назад

+jip laan They *were* available a couple of years ago. :)

@Jipzorowns 8 лет назад

+MathTheBeautiful heh, I meant 4 years ago or so :-). Thanks for uploading!

@comprehensiveboycomprehens8786 8 лет назад

+MathTheBeautiful Maybe they were not available two years ago in his frame of reference :)

@ndmath 9 лет назад

37:31 this was the perfect response I should have said to my math teachers.

@itzani8051 2 года назад

Maths is really the most beautiful thing...and I really appreciate that I've learned something new here,a new way of thinking...thank you sir ❤️love from India ❤️✨

@kevinbyrne4538 9 лет назад

The instructor is Pavel Grinfeld, associate professor of mathematics at Drexel University in Philadelphia, Pennsylvania.

@dharma6662013 8 лет назад

26:28 The third point needs to be in the same segment to have the same angle. The two fixed points generate a chord which splits the circle into two segments. The angles subtended in the two different segments are complementary, e.g. the angle would be 10 degrees in one segments and 170 degrees in the other. You seem to confuse "segment" with "chord".

@MrJesuswebes 8 лет назад

Great course!!! I recommend to follow this course along professor Susskind course about General Relativity: the maths needed and explained by Susskind can be strongly renforced in this course in order to really understand mathematical formalism and physical theory. Internet is really great, indeed, thanks to people like Susskind or Grinfeld. Thanks to both!

@arunbharadwaj145 8 лет назад

The very reason I came here tbh xD I started out with Suskind. Then I Realized I needed Tensor Calculus to understand.

@nemanjastankovic941 9 лет назад

I have really enjoyed your introduction. Can't wait to watch other lectures. Best regards from Serbia :)

@michalisstathakopoulos1166 6 лет назад

My dear friend you are a true great. It's a pitty that H.Rund (my favorite writer ) is not alive to see you. Try to prepare videos for grtensor and other stuff of computer algebra to help applying tensors even more in sciences. I recommend all the people who visit your channel to watch the proof of the Voss-Weyl formula and the Gauss integral theorem. I hope you always have the courage to create more and more lecture videos! Have you requested the Khan Academy to produce anything on tensor calculus??

@ichigo_nyanko 3 года назад

I have to admit I was a bit giddy at the diagram 12:50. When I has instantly jumped from having never having seen the diagram to fully understanding it, all of this advanced maths I have been doing is nice and I love it - but nothing beats that feeling of instant understanding (except maybe the opposite - proving something after banging your head against the wall for weeks). Made me feel like I was a kid again.

@MathTheBeautiful 3 года назад

Hi Luuucy, Glad to hear that! Pavel

@SussyBacca 6 лет назад

For those that want the gist of this video, skip to @50:30. You will see he finally drives the steak into the vegan vampire (no idea why I'm saying that)... tensors are right between (logical) expressions and (visual) geometry. You can always explain things best with tensor calculus. He does not, ever, go into exactly what tensors are, or how to think in terms of them. This video is great, but is a history lesson and progression through mathematical evolution and doesn't really get into tensors until the end. This is not the quick intro to tensors you may be seeking.

@yyc3491 4 года назад

How I wish I could find your channel earlier!

@ethanwinchester4585 2 года назад

You are a gem and help me learn so much about tensor calculus

@MathTheBeautiful 2 года назад

Thank you, much appreciated.

@MrJesuswebes 8 лет назад

Note: The only pitfall is that Susskind and Grinfeld use a very different notation... but that´s good in order to show yourself that you really understand both courses: Susskind´s oriented to not really formalized physics and Grinfeld´s more oriented to maths and physiscs in general. Just that.

@w.hoffman3308 7 лет назад

It's wonderful to listen to a topic presented with background as Math....does. I am reminded of my insight, achieved some years past my hs days' learning of algebra (and my dread of "work" problems), when I suddenly realized that another issue i had had problems with in studying circuit analysis and Kirchoff's Law were critically related, and that in an instant I finally understood the relationship of energy and work (the same units for heavens'sake!) apportioned between two actors. I can't specify the date, but do recall my associates, also chemistry graduate students who went on to get their Ph.D.s - as did I, fwiw - being surprised by my paper napkin and its markings. . Now that I've seen that Kirchoff did much more than just provide a basis for circuit analysis, I will reexamine other items of his.. In a similar vein, my limited understanding of Maxwell's equations has always irritated me. His insights into the real world, imo, were the most important before Einstein's, and even now promise more than AE's. but for relativity. Since his equations were formulated with vector and tensor features, I have been looking for a better handle on the ideas and hope Math.... helps me find a new entry point. Never quit learning. I do not plan to

@firstevidentenigma 8 лет назад

I have a lowly bachelors of applied math and I approve these lectures. lol

@christophermoore5389 4 года назад

I love this guy thank you so much for sharing

@MathTheBeautiful 4 года назад

I do, too, and I'm glad you liked it

@lucjannastalek9978 10 лет назад

Slide number 10: in the audience it says: "Shape of given volume and smaller surface area", whereas on the card shown on the side (for better clarity) it says: "Shape of given surface area and largest volume". I would advise, for clarity, to at least add a comment on that in the description of the video - so to make the reception easier and the content more consistent: why the change, and what effect does it have.

@alexandra-stefaniamoloiu2431 7 лет назад

This kind of videos make me love mathematics! Thank you!

@31173x 8 лет назад

Hello, I am an undergraduate in mathematics. I have taken my basic linear algebra, and multivariable calculus courses, along with some real analysis. Will I have the prerequisite mathematical tools to learn tensor analysis?

@MathTheBeautiful 8 лет назад

Sure!

@saikat93ify 7 лет назад

Hey .... I really enjoy your lectures and enthusiasm for Maths. I would probably have been better in Physics if someone could have shown me how they were related like how you did in your lectures ... I just want to point out an easier way to construct the Toricelli point. It is a point that looks at all other vertices at an angle of 120.

@robabanque3253 8 лет назад

hi would you consider doing a video course on General Relativity? I've tried a couple of times to understand it, using various books or watching other video courses, but I keep getting stuck at some point. Having watched and understood your course on Tensor Calculus, I think I might finally understand GR if you taught it. Fingers crossed !!

@stephanebeauchamp-kiss3181 6 лет назад

Awesome lecture! But...honestly, you're the best teacher in the world at x1.25 speed

@MathTheBeautiful 6 лет назад

Thanks! I do sound better up a major third.

@Bootmahoy88 Год назад

It's interesting to note that many people who use math are already using tensors; they just don't call them that. A scalar is a tensor. A basic two dimensional vector is a tensor, as is a three dimensional one. Once you grasp the basic concept of tensors, your mathematical acumen will be enhanced considerably in a practical fashion.

@MathTheBeautiful 6 месяцев назад

Thank you for your comment!

@vasanthir6273 3 года назад

CNN : mathematicians proud to unveil their new, harder math, the question is " when will they stop ¿¿

@bernhardriemann3821 3 года назад

Hahaha , lol

@Unidentifying 10 лет назад

I enjoyed this a lot, many thanks !

@ardiris2715 7 лет назад

You could reboot this course as Drone Telemetry and have 1 million views in just a few months.

@BDYH-ey8kd 7 лет назад

could rename this video as: 'PROOF- jesus resurrection after 3 days!'

@pacchutubu Месяц назад

Is there a video where above problems are solved using Tensors?

@heeraksharma1224 Год назад

Referring to the Heron's problem, should we expect the distance function to be continuous without explicitly calculating it? I'll leave a timestamp here: 14:54

@MathTheBeautiful 6 месяцев назад

Yes I think so. Even differentiable.

@tharanathakula3588 4 года назад

Lagrange's equation to find the minimal surface area may be made use of with some changes in the equation to sole the Poincare's conjecture?

@thevegg3275 2 года назад

Could you please explain mathematically how the g sub mn matrix if formed using ds^2=dr^2 + r^2 d (theta)^2 ? What was presented was g sub mn = [1 0, 0 r^2]. Note [1 0, 0 r^2] is a two by two matrix

@iamnorwegian 7 лет назад

I have a slightly off topic question about the proof that maximum area for given perimeter is a circle. I feel like the right angle part might not be needed, if we instead make repeated use of the symmetry argument, cutting the perimeter in half by angles 360/2^n, and noting that both halves must be equal. This should lead to the realization that this object has infinite symmetries, or said in a different way, that it consists of arbitrarily many arbitrarily small, equal, line segments. With a bit of polishing, I feel like this would be a proof on its own. Perhaps this usage of limits and behaviours towards infinity makes it a not entirely geometric proof, but doesn't this line of reasoning work as well?

@CecilWard 7 лет назад

Ah, I've done it backwards - in general for an arbitrary point, it isn't so, it this is the Torricelli point, somthe lengths have to be minimised so that is true if the long three-part line is straight, so that is why it is true for our special point only. That right? Finally got their, but I think I should have been ‘working backwards’ rather than trying to go forwards any more - if that makes sense.

@rhitabrata08 7 лет назад

Would you kindly give me the source from where I can get the subtitles of this lecture video?

@OttoFazzl 9 лет назад

At 35:00 I think the minimum surface would be for the surface described by relation r(x) = c, where c is the radius of the metal rings. I don't understand how the more complex solution will have smaller area than just simple cylindrical shape.

@PhilippeCarphin 8 лет назад

+Otto Fazzl It's the shortest line between those points. But since it's a surface of revolution, we have to think about how much small line segments contribute to the surface of revolution. The think is that line segments that are close to the center (with small r) contribute less area then line segments that are closer to c.

@cesargarcia458 9 лет назад

Absolutely brilliant! It truly opened my eyes.

@tharanathakula5508 6 лет назад

Our text book on geometry was by Euclid. "If Euclid has not kindled your youthful enthusiasm you were not born to be a scientific thinker" by -Albert Einstein.

@snnwstt 7 лет назад

The spherical coord. equation at 41:30 is clearly not equivalent to the cartesian coord at 38:10 since 38:10 gives z = f(x, y) (which cannot describe a sphere, as example, since a function returns a single value and z need two values, for a sphere.... we need at least something like z^2 = f(x, y), or more generally, 0 = P(z, x, z) ) while 41:30 gives r(theta, phi) = f(theta), ... no phi, so r( ) is a function of theta only, a single parameter, while z=f(x, y) depends of two parameters. Clearly, the two formulations are not equivalent to start with, so any comparison between the two is so so ... is it not?

@bonbonpony 5 лет назад

OK, you sold me the advantage of tensors pretty well. But the problem is, whatever book I pick on this subject and open it, I get flooded with unintelligible gibberish of symbols and advanced terminology, usually mixed with multivariable calculus and General Relativity (if having just the math weren't enough of a trouble :q ). But I haven't found any book yet that would explain tensors from the grass roots, beginning of a good conceptual definition of what the hell a tensor actually is, or wouldn't use some metaphors for a tensor that don't say much about what that tensor actually is anyway, geometrically. A good anti-example is Thulio Levi-Civita's book (the supposed father of tensor calculus). If the father can't explain it well enough, then I don't know if anyone else can :q And the current tensor notation doesn't help the case either, because it's very monotone and repepepetitititive. One can say it's good, because it's less to learn, but when you look at long expressions full of indexes, like the one you showed, it gets really hard to spot and distinguish every piece of the formula by sight. Sometimes I think that the only rule for those expressions is for the letters to not overlap each other :q That's how a mess it is already. I wonder if there could be some other notation that could help see the geometry behind it better. Because the way it looks today, reminds me of that algebraic notation of ancient Greeks - they also used Greek letters with upper and lower indexes to denote squares, cubes and other powers, as well as roots, products and sums, and all of that was mixed with numbers, which also used Greek letters, so it all mixed up into a blob of Greek letters. It took an entire millennium and a half until people come up with our modern algebraic notation which is much more clear. So maybe there's a better way to denote tensors as well?

@AJ-et3vf 2 года назад

Awesome video! Thank you!

@theoremus 2 года назад

When Maxwell wrote his equations, there were many of them. Vector calculus narrowed them down to four.

@gejyspa 3 года назад

I came to the is video to find out "what is a tensor?" Unfortunately, other than the title, that question is addressed nowhere! I still don't know what a tensor is. (OTOH, thanks for the beautiful proof on Heron's problem)

@MathTheBeautiful 3 года назад

You're right! I'm rewriting the accompanying textbook and in it, tensors are defined in Chapter 14.

@xtenkfarpl 9 лет назад

Whoa... what happens at around 33:20? Seems like there's a big discontinuity: as if some connecting material has been skipped somehow?

@rfyl 7 лет назад

To MathTheBeautiful -- Your series looks like it's going to be excellent! I am very much looking forward to going through the entire series. When I bookmark pages, I like to bookmark the name of the author. This page, and indeed the page for your channel, do not give your name. Am I correct in assuming that you are Pavel Grinfeld, since that is the author of the text you link to? Anyway, keep up the great work!

@rfyl 7 лет назад

Sorry, when I said "bookmark" I meant to say "tag".

@joepadlo4337 10 лет назад

Please share the text book name and author that you are using for the Tensor Calculus course.

@suluclacrosnet61 10 лет назад

We use Dr. Grinfeld's Introduction to Tensor Analysis And the Calculus of Moving Surfaces: goo.gl/IaCQC2

@powerinneed 10 лет назад

hey cud u contact me , i cud use some help in the matter . dragan.spear@gmail.com

@suluclacrosnet61 10 лет назад

Ayush Trivedi Hi Ayush, you can visit the forums on tensorcalculus.org where Dr. Grinfeld or one of us will be happy to answer your questions.

@alan2here 5 лет назад

Minimise N while moving successively through sequence M. N = distance M = {given point, any point on given line, given point} What would light do? Mirror, solved :) N = time M = {given point, any path between given parallel lines, given point} What would light do? Glass sheet, solved :) N = area M = {all of given circle X, all of given circle (parallel in position and identical in radius) to X} What would light do? Erm! 🤔 Maybe therefore it would? Reminds me of a string in physics. Maybe between 0D use a light ray, 1D a soap bubble film, 2D (a tiny water droplet volume?).

@samlaf92 4 года назад

I'm confused as to what exactly the different generalizations of Vector calculus are, and where exactly Tensor Calculus resides among them. In your book, you mention "Tensor calculus is not the only language for multivariable calculus and its applications. A popular alternative to tensors is the so-called modern language of differential geometry" Just to be clear, is Tensor calculus equivalent to Ricci Calculus? (Wiki en.wikipedia.org/wiki/Tensor_calculus says so) And is the "modern language of differential geometry" differential forms and the exterior calculus? In an interview, Shiing Shen Chern is asked "You are seen as one of the main exponents of global differential geometry. Like Cartan you have worked with *differential forms* and connections and so on. But the German school, of which Wilhelm P. A. Klingenberg is one of the exponents, does global geometry in a different way. They don’t like to use differential forms, they argue with *geodesics and comparison theorems*, and so on. How do you see this difference?" and simply answers: "There is *no essential difference*." So far the story makes sense. Except one thing. Wiki (en.wikipedia.org/wiki/Vector_calculus#Other_dimensions) says that the two generalizations of Vector calculus are 1) Geometric Algebra and 2) Differential Forms. This makes sense to me, since GA works with vectors, and DF with covectors. These are naturally dual, as Chern puts: " I usually like to say that vector fields is like a man, and differential forms is like a woman. Society must have two sexes. If you only have one, it’s not enough." But where does Tensor calculus reside then? There are two "dualities" above, and they don't seem to be equivalent... unless Geometric algebra and Tensor calculus are really one and the same thing? Tensors are made of both vectors and covectors though... so perhaps Tensor calculus is a generalization of GA and DFs that encompasses both? Someone please help me and throw some light onto this confusion!

@solten5184 2 года назад

is this the correct differential equation at 35:17? because my solution to it is a*e^(bx) and not coshx. what i did was taking the r'^2 to the oder side, divided through by r' integrated both sides (which is easy because on both sides the numerator is derivative of the denominator) and then you exponetiate the result and get a first order ode.

@waynelast1685 3 года назад

Is "working with all coordinate systems at once" similar to finding the optimum generalized coordinates in Lagrangian methods? Using Tensor calculus, can one find the most logical "coordinate system" (which may be coordinates not in a physical sense that we would expect)?

@palfers1 7 лет назад

You wrote down the Lagrange minimal surface expression wrong. By inspection it is not symmetrical but should be.

@MathTheBeautiful 7 лет назад

Yes, you're right, thank you.

@waynelast1685 3 года назад

I am doing my own physic research. I am comfortable with math and have taken advanced calculus, matrices, complex variables, and math methods for physicists. I tried searching for a useful outline and explanation of the different branches of mathematics on Google and RU-vid. So far I have been disappointed. Either there is very limited information or it is too long and detailed. I need a solid overview with example applications, but not take too long to do it. Thus, explanations for someone with a math background already. I do not want to learn it all just distinguish the branches. Can anyone help me? I am sure this information would be very well received by the vast majority of online searchers also.

@MathTheBeautiful 3 года назад

Yes, start with Linear Algebra (lem.ma/LA) and go from there.

@waynelast1685 3 года назад

@@MathTheBeautiful thank you. I should have clarified but I am hearing a lot about Lie Groups, group theory, tensors, set theory, topology, manifolds. I am trying to get perspective to really learn it well. Starting with linear algebra sounds like a plan.

@steenpedersen8244 7 лет назад

is there a table explaining which video correspond which chapter or section in the book?

@guilhermesobrinho1329 7 лет назад

beautiful indeed... thank you for sharing.

@kevinbyrne4538 9 лет назад

44:32 -- Jean Baptiste Marie Charles Meusnier de la Place (June 19, 1754 - June 13 , 1793)

@saadhassan9469 Год назад

Do I need to know about tensors to understand these lectures?

@MathTheBeautiful 6 месяцев назад

No, these lectures are intended to explain tensors.

@Nutrition-Facts-Tips 8 лет назад

What are the pre requisite to understand these lectures

@MathTheBeautiful 8 лет назад

+Afaq If you know linear algebra and multivariable calculus, tensor calculus is a framework that weaves geometry into it. So I would say, the prerequisites are multivariable calculus, linear algebra and a desire for clarity in applied mathematics.

@Nutrition-Facts-Tips 8 лет назад

+MathTheBeautiful Thankyou,,that means I am all set then.. Regards

@intellectelite 7 лет назад

MathTheBeautiful I've taken those classes! you just got a new subscriber. do you know anything about differential forms? I have some questions about them.