Why Tensor Calculus?

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

[wagsman]

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

[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.

[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.

[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

[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!

[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!

[yanniphone]

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

[Sillybb]

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

[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.

[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.

[rammadhavan]

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.

[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.

[intellectelite]

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

[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.

[stevedeltoid]

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

[henryalferink]

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

[aravindradhakrishnan]

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!!

[prabhatp]

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

[ramnewton]

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