Tensor Calculus and the Calculus of Moving Surfaces by Pavel Grinfeld

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source: MathTheBeautiful    2014年2月3日
Textbook: http://bit.ly/ITCYTNew Solutions: http://bit.ly/ITACMS_Sol_Set_YT Errata:http://bit.ly/ITAErrata
McConnell's classic: http://bit.ly/MCTensors
Weyl's masterpiece: http://bit.ly/SpaceTimeMatter
Levi-Civita's classic: http://bit.ly/LCTensors
Linear Algebra Videos: http://bit.ly/LAonYT
Table of Contents of http://bit.ly/ITCYTNew

Tensor Calculus: What Tensors Are For! 1:01:39
1: The Rules of the Game 40:05
2: The Two Definitions of the Gradient 36:05
2a: Two Geometric Gradient Examples 11:56
3: The Covariant Basis 37:01
3a: Change of Coordinates 39:13
4: The Tensor Notation 1:28:39
4squeeze: Fundamental Objects in Euclidean Spaces 29:41
4a: A Few Tensor Notation Exercises 18:50
4b: Quadratic Form Minimization 11:43
4c: Decomposition by Dot Product 13:25
4c+: The Relationship Between the Covariant and the Contravariant Bases 6:07
4d: Index Juggling 25:49
5: The Tensor Property 1:42:05
5b: Invariants Are Tensors 9:56
6a: The Christoffel Symbol 21:06
6b: The Covariant Derivative 50:11
6c: The Covariant Derivative 2 43:52
6d: Velocity, Acceleration, Jolt and the New δ/δt-derivative 43:11
7a: Determinants and Cofactors 1:08:49
7b: Relative Tensors 18:02
7c: The Levi-Civita Tensors 9:41
7d: The Voss-Weyl Formula 20:07
8: Embedded Surfaces and the Curvature Tensor 34:42
8b: The Surface Derivative of the Normal 32:44
8c: The Curvature Tensor On The Sphere Of Radius R 30:57
8d: The Christoffel Symbol on the Sphere of Radius R 12:33
8e: The Riemann Christoffel Tensor & Gauss's Remarkable Theorem 50:04
9a: The Equations of Surface and the Shift Tensor 59:42
9b: The Components of the Normal Vector 16:53
10a: The Covariant Surface Derivative in Its Full Generality 50:19
10b: The Normal Derivative 31:13
10c: The Second Order Normal Derivative 17:04
11a: Gauss' Theorema Egregium, Part 1 36:27
11b: Gauss' Theorema Egregium, Part 2 23:08
12a: Linear Transformations in Tensor Notation 18:47
12b: Inner Products in Tensor Notation 6:41
12c: The Self-Adjoint Property in Tensor Notation 11:16
13a: Integration - The Arithmetic Integral 8:30
13b: Integration - The Divergence Theorem 20:13
14a: Non-hypersurfaces 12:53
14b: Examples of Curves in 3D 11:23
14c: Non-hypersurfaces - Relationship Among The Shift Tensors 10:46
14d: Non-hypersurfaces - Relationship Among Curvature Tensors 1 12:08
14e: Non-hypersurfaces - Relationship Among Curvature Tensors 2 16:00
14f: Principal Curvatures 17:52
15: Geodesic Curvature Preview 13:57
Derivative of a Basis Vector Illustrated (e_r in polar coordinates) 0:20