This course will continue on Patreon at http://bit.ly/PavelPatreon

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

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