Why Tensor Calculus?

bit.ly/PavelPatreon
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

[@wagsman9999]

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 KZbin! Thanks professor. I will check out you other videos too.

[@MathTheBeautiful]

+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]

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

[@thane9]

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]

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]

@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]

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]

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.

[@jackdkendall]

"When you start your discussion by choosing your coordinates, you are doomed, because you end up with expressions that have as much to do with the coordinate system as they have to do with the problem you're solving." Most important piece of the lecture

[@bonbonpony]

True. And I see this problem all the time in physics. Suppose I want to solve Shrödinger's equation for the wave function of the electron. For a "free particle", I have to use one coordinate system and end up with one set of solutions. For a "particle in a box", I have to use another set of coordinates and end up with another set of solutions. For "hydrogen atom", it gets extremely worse! Because now I have to translate everything to spherical coordinates (and if you've ever seen how the Laplacian looks like in spherical coordinates, and how many pages of paper you have to waste to derive it, you know what I mean!), then separate the variables tediously, then solve the (hard!) differential equations of second order with VARIABLE coefficients, and finally I end up with a set of very fancy functions as my basis set of solutions (spherical harmonics), only to find out that it doesn't tell me much about the nature of electrons, because it doesn't account for spin :P And the solutions I end up in each of these problems look extremely different from each other! But, on the other hand, the same spherical harmonics you find when trying to solve extremely different physics problems, like the spatial characteristics of the EM field around antennas, or seismic waves, or microwave resonance cavities of spherical shape. Which makes you wonder: is this something peculiar to the problem you're trying to solve? Or perhaps it has more to do with the coordinate system you're working with? Because you can solve the "hydrogen atom" in rectangular coordinates as well (except it is thrice as hard!) and you'll find out that you end up with a completely different set of solutions, aligned to rectangular axes! :P Which suggests that is't more of a quirk of the coordinate system than the geometry of the physics problem itself :P

[@OttoFazzl]

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

[@thekkl]

"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.

[@iqdx]

This is a deeply satisfying lecture. I have struggled on and off for decades to attain insights found here in disarming simplicity. These set the stage for and should motivate the epic efforts required to master tensors. Thank you!

[@yanniphone6729]

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

[@cfriedalek]

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]

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

[@henryalferink1941]

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 KZbin, particularly in understanding the intuition behind things. Thank you very much!

[@Sillybb9142]

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

[@stevedeltoid5413]

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

[@pablojaviervaz]

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

[@Ector521]

+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]

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

[@lightfreak999]

+Ector521 I see what you did there ;)

[@comprehensiveboycomprehens8786]

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

[@RalphDratman]

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.

[@260bossute]

This guy is a wonderful lecturer.

[@MathTheBeautiful]

Thank you!

[@aravindradhakrishnan1521]

Lectures are very good sir. I was pondering over how to learn GR but this playlist gave me some hope as I am learning on my own... Thank you sir!!

[@mkmaharana]

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.

[@mehrdadkeneshlou7898]

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]

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.

[@PiotrSupski]

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

[@NothingMaster]

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

[@MistressGlowWorm]

This is beautiful! Thank you for the post.

[@nathanas64]

What a great Professor! Really enjoyed this lecture!

[@MyJuicehole]

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]

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]

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 )

[@pol...]

Thank you very much for this! I have an exam coming on classical field theory (special relativity and electromagnetism) and we use the language of tensors. Thus these lectures are gonna be very helpful. :D

[@nemanjastankovic941]

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

[@ramnewton8936]

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

[@MathTheBeautiful]

Thanks Ram! Check out the new Vector Calculus series.

[@Faustian10]

Thanks for the excellent introduction to tensor calculus; insight and understanding (first) instead of calculation (second).

[@Unidentifying]

I enjoyed this a lot, many thanks !

[@andrewwells6323]

Thank you. I'm never satisfied with how much I understand math, I always want to learn more :)

[@MathTheBeautiful]

Great attitude, keep it up!

[@cesargarcia458]

Absolutely brilliant! It truly opened my eyes.

[@rammadhavan4056]

I at least bought two books on Tensor calculus in the last couple of months and I pretty much quit reading them beyond few pages. I really enjoyed this lecture and I'm amazed by the clarity with which the professor is able to explain the subject. I couldn't wait to go through the rest of the lectures.

[@MathTheBeautiful]

Hope you kept the receipt for the books.

[@Al.Mo.]

@56:00 "Having my heart filled with despair", this mathematician understands our pain

[@zombiedude347]

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.

[@guilhermesobrinho1329]

beautiful indeed... thank you for sharing.

[@intellectelite]

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

[@rickandelon9374]

where you can draw straight lines!

[@bonbonpony]

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]

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]

Malcolm Forbes really?

[@iyalovecky]

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 :(

[@iaggocapitanio7909]

Beautiful and easy to be understood.

[@prabhatp654]

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

[@MathTheBeautiful]

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

[@johnfresen1013]

Great lecture. Many thanks.

[@AJ-et3vf]

Awesome video! Thank you!

[@alexanderkurz3621]

What a beautiful lecture. It has everything. Intriguing math puzzles to play around with. Beautiful surprising solutions. Fascinating history. Deep ideas.

[@MathTheBeautiful]

Thank you! That was very nice to read.

[@jorgerive7335]

phenomenal teacher!

[@ndmath]

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

[@nosnibor800]

Well I was hoping for tensor Calculus - but instead got a fascinating insight which made me smile. I never considered the geometric v analytic before. The optimisation problems a good example where you can "see" what is happening. (we engineers like this). Also I like your down to earth honesty - student things "god I don't understand this" - professor thinks "I don't understand this too, hope they don't ask me a hard question about it" :-) haha we have all been there ! I am trying to understand GR and therefore teach myself some Tensor Calculus - I was never taught it at the Poly during my EE course. Thanks !

[@angusjcampbell79]

Brilliant. Down to earth teacher.

[@justforknowledge6367]

I am already watching the first tutorial and feeling good. The tutorials appear to be classroom lectures, thereby increasing the time for download and file size. I understand that the series were recorded from the real classroom scenario. I have followed a couple of Stanford Online courses via coursera. They are very fast, lasting for 20 minutes. The advantage with recorded courses is that the recordings can be stopped at anytime, effectively increasing their delivery time to suit individual style and pace of learning. Sir, my request to would be for you to tighten up the delivery time so that our bandwidth could be saved. Please note that we can't fast forward recorded lectures but could pause or repeat them :) according to our need for time - converting them virtually to self-paced learning material :D

[@j.l.dedion6421]

MathTB-Thank you for putting these lectures online. I've wanted to learn tensors for years. The indices always looked so tedious that they killed my interest. Thanks for reawakening my interest and showing me a better entry point! I recently got the Intro book, too. I may be the millionth person since Heron to do so, but I noticed a variation on the solution to his problem. In addition to image village C, add an image of B and call it D. Then segments AD and BC are of equal length, are shortest paths by virtue of being straight, and intersect at point P on the river again creating the shortest path BPA. You've reawakened my interest in geometry, too!

[@ethanwinchester4585]

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

[@MathTheBeautiful]

Thank you, much appreciated.

[@adarshkishore6666]

This is amazing! I don't think I've seen someone explain so advanced topics starting from such a basic and yet complete viewpoint. Thank you, sir. Just asking, btw, do you plan to put out exercises for these topics other than LA on Lemma?

[@scientificresearch1400]

I had never seen such beautiful lectures.

[@MathTheBeautiful]

Thank you, that means a lot!

[@RenderMyBalls]

These are the math videos I'm looking for. Thanks for posting.

[@alexandra-stefaniamoloiu2431]

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

[@firstevidentenigma]

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

[@alikarimi-langroodi5402]

Excellant. Thank you

[@sorinsuciu8675]

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

[@zorro20010]

Very thoughtful and interesting lecture. I have watched it twice There were many things i did not understand completely specially point no 13 and Toricellis problem but there were always things in them which i can readily appreciate Heron's problem i think i have completely understood both its geometrical insight and th algebraic xplanation and th Archimedes problem also Explanation of Steiners Problem and Minimum surface area according to euler were very insightful I have always visualized stress strain tensors with th help of coordinates th subccripts of elements of th stress or strain tensor r coordinates So i really have dificulty in understanding Professor wen he says something akin to Tensors make us independant of any coordinate system Too many subscripts and superscripts really make it less neat than the Vector Calculas notations in whose formalism i have learnt about Maxwells equations about Electromagnetism but th math behind th Tensor formalism makes it easier to understand Modern Science of Relativity Very enlightning thanx again Try to finish the whole lecture series to get to th bottom of it

[@steviewonder9209]

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.

[@panchgani74]

Dr, Grinfed Thank you for the wonderful lectures, I would very much appreciate it if you could direct me to diagrams for questions 7-12 as it is not in the solution manual. I have great difficulty understanding the questions with out pictures.

[@christophermoore5389]

I love this guy thank you so much for sharing

[@MathTheBeautiful]

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

[@fizixx]

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]

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

[@bigpapi3636]

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!

[@ac-dp3jk]

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.

[@itzani8051]

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 ❤️✨

[@thcoura]

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.

[@MrJesuswebes]

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]

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

[@yyc3491]

How I wish I could find your channel earlier!

[@kevinbyrne4538]

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]

Did he mention where is this grave located?

[@jacopopispola9925]

@@bonbonpony Syracuse, Sicily

[@robabanque3253]

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 !!

[@FelipeZucchetti]

Nice lecture...

[@BarriosGroupie]

You definitely have an air of someone that has gone to the sources and studied them, rather than most who instead copy some author's text book. So I'm not surprised to find your book on Amazon.com has thirteen 5* and one 4* out of 14 ratings which is outstanding. Einstein does a pretty good job of introducing Tensors in his General Relativity paper compared to most authors even today, but I still had to go to the original source for a better introduction: The Absolute Differential Calculus (Calculus of Tensors) (Dover Books on Mathematics) Tullio Levi-Civita Unfortunately though, it's very old and a lot has gone on since then, which requires a modern book written by someone who knows the in depth history etc. When I have the time, I'll definitely be taking a look at your book, having ordered it, as well as: Tensor Geometry: The Geometric Viewpoint and its Uses (Graduate Texts in Mathematics)23 Nov 2009 by Christopher T. J. Dodson and Timothy Poston Sadly, I still think tensors are still very badly taught today in most universities, going by the number of questions on physics sites asked today, and the seemingly confused, conflicting definitions.

[@MathTheBeautiful]

My original source was www.tensorcalculus.org/book-reviews/mcconnell/ which I enthusiastically recommend!

[@bonbonpony]

The link is dead. It shows some domain registrar website that display ads. Good that we have the Web Archive, I found the original titlte there: "Applications of Tensor Analysis" by A.J. McConnell. web.archive.org/web/20170417113350/www.tensorcalculus.org/book-reviews/mcconnell/

[@joeyquiet4020]

can't thank you enough! thank you thank you

[@kevinbyrne4538]

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

[@MegaMaistro123]

thank u so much guys

[@monahcreations633]

This was a wonderful lecture,I understood nothing at all after a certain point since I am in high school,but I learned to further appreciate the elegance of a geometric approach as they make problems a lot simpler,cuz I was solving a problem i have never seen before in a competition,it had to do with finding the center of rotation.I tried an analytical approach but the equations got so nasty,i then tried a geometric approach and easily got a solution with no nastiness whatsoeverThanks prof.

[@MathTheBeautiful]

Thank you, I'm so glad you enjoyed it!

[@happytouch7104]

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.

[@jonasvogler2662]

Beautiful

[@tawabullas5058]

Big thumbs up to the professor.

[@dharma6662013]

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".

[@Jipzorowns]

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

[@MathTheBeautiful]

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

[@Jipzorowns]

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

[@comprehensiveboycomprehens8786]

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

[@Trotskisty]

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]

Trotskisty dope!

[@christophermoore5389]

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

[@w.hoffman3308]

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

[@Charles-uo9mo]

56:38 "Divergence of a what the hell is this.." that's Vector Analysis for ya.

[@bonbonpony]

As funny as it seems, I can see more clearly through this vector calculus stuff than through endless strings of Greek letters in upper and lower indexes which all look the same. True, one has to know more weird symbols in vector calculus, but once one understands its meaning, they're much easier to distinguish in formulas, because they don't look all the same.

[@chadliampearcy]

@@bonbonpony Look up Geometric Algebra!

[@bonbonpony]

@@chadliampearcy Too late, I already did ;> Clifford/Grassmann algebras are quite interesting indeed. However I still see some deficiencies in it as well, some things that, in my opinion, are not quite right yet. My remark above, though, was only about the notation, which is a bit too repetitive and monotone in tensor calculus than in "ordinary" vector calculus notation, where the symbols are a bit more distinctive. The upper and lower indices in tensor calculus are a bit too similar to exponent notation (which, in my opinion, is much more important than keeping track of indexes).

[@SussyBacca]

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.

[@alijoueizadeh8477]

Thank you for sharing this video. About finding the point D att 22:54, I would suggest to turn the triangle 60 deg. in the opposite direction. You get a point B*. Draw the line AB*. D is the intersection of A'B and AB*. Am I right?

[@tharanathakula3588]

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?

[@michalisstathakopoulos1166]

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??

[@redfire1508]

Si fueras tan amable de activar en tus vídeos la opción para los subtítulos automáticos por favor.

[@saikat93ify]

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.

[@profapotema]

I think there must be a wrong sign in Lagrange's differential equation for minimal surfaceses, because of the lack of simmetry in spite of the simmetry of the formula for the area. Beautiful lecture!!

[@hazememam4248]

I just love this ❤❤❤

[@MathTheBeautiful]

So do I!

[@vasanthir6273]

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

[@bernhardriemann3821]

Hahaha , lol

[@chaoshengzhe]

Beautiful! One question, the nonlinear ODE due to Euler at 35:17 Is the RHS 0 or 1? I did not find this on the book. Thanks!

[@MathTheBeautiful]

Good catch! It's 1, since mean curvature is given by rr’’-r’^2-1. Thank you.

[@rhitabrata08]

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

[@jimdogma1537]

Check out the professors neat trick at 46:25. LOLOLOLOL

[@bonbonpony]

It would be more surprising if he reappeared from the other end :)

[@ameeraislam1680]

Ahahhaha

[@ichigo_nyanko]

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]

Hi Luuucy, Glad to hear that! Pavel

[@steenpedersen8244]

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

[@xtenkfarpl]

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

[@waynelast1685]

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)?