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1. Introduction to Tensor Calculus
2. The Rules of the Game
3. The Two Definitions of the Gradient
4. Two Geometric Gradient Examples
5. The Covariant Basis
6. Change of Coordinates
7. The Tensor Notation
8. Fundamental Objects in Euclidean Spaces
9. A Few Tensor Notation Exercises
10. Quadratic Form Minimization
11. Decomposition by Dot Product
12. The Relationship Between the Covariant and the Contravariant Bases
13. Index Juggling
14. The Tensor Property
15. Invariants Are Tensors
16. The Christoffel Symbol
17. The Covariant Derivative
18. The Covariant Derivative II
19. Velocity, Acceleration, Jolt and the New δ/δt-derivative
20. Determinants and Cofactors
21. Relative Tensors
22. The Levi-Civita Tensors
23. The Voss-Weyl Formula
24. Embedded Surfaces and the Curvature Tensor
25. The Surface Derivative of the Normal
26. The Curvature Tensor On The Sphere Of Radius R
27. The Christoffel Symbol on the Sphere of Radius R
28. The Riemann Christoffel Tensor & Gauss's Remarkable Theorem
29. The Equations of Surface and the Shift Tensor
30. The Components of the Normal Vector
31. The Covariant Surface Derivative in Its Full Generality
32. The Normal Derivative
33. The Second Order Normal Derivative
34. Gauss' Theorema Egregium (Part 1)
35. Gauss' Theorema Egregium (Part 2)
36. Linear Transformations in Tensor Notation
37. Inner Products in Tensor Notation
38. The Self-Adjoint Property in Tensor Notation
39. Integration: The Arithmetic Integral
40. Integration: The Divergence Theorem
41. Non-hypersurfaces
42. Examples of Curves in 3D
43. Non-hypersurfaces: Relationship Among The Shift Tensors
44. Non-hypersurfaces: Relationship Among Curvature Tensors I
45. Non-hypersurfaces: Relationship Among Curvature Tensors II
46. Principal Curvatures
47. Geodesic Curvature Preview

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