Tensor Calculus 3a: The Covariant Basis

This course will eventually continue on Patreon at http://bit.ly/PavelPatreon
Textbook: http://bit.ly/ITCYTNew
Errata: http://bit.ly/ITAErrata
McConnell's classic: http://bit.ly/MCTensors

Table of Contents of http://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