Tensor Calculus Made Simple

Preface
Nomenclature
1: Preliminaries
1.1: Historical Overview of Development & Use of Tensor Calculus
1.2: General Conventions
1.3: General Mathematical Background
1.3.1: Coordinate Systems
1.3.2: Vector Algebra and Calculus
1.3.3: Matrix Algebra
1.4: Exercises
2: Tensors
2.1: General Background about Tensors
2.2: General Terms and Concepts
2.3: General Rules
2.4: Examples of Tensors of Different Ranks
2.5: Applications of Tensors
2.6: Types of Tensor
2.6.1: Covariant and Contravariant Tensors
2.6.2: True and Pseudo Tensors
2.6.3: Absolute and Relative Tensors
2.6.4: Isotropic and Anisotropic Tensors
2.6.5: Symmetric and Anti-symmetric Tensors
2.7: Exercises
3: Tensor Operations
3.1: Addition and Subtraction
3.2: Multiplication of Tensor by Scalar
3.3: Tensor Multiplication
3.4: Contraction
3.5: Inner Product
3.6: Permutation
3.7: Tensor Test: Quotient Rule
3.8: Exercises
4: delta and epsilon Tensors
4.1: Kronecker delta
4.2: Permutation epsilon
4.3: Useful Identities Involving delta or/and epsilon
4.3.1: Identities Involving delta
4.3.2: Identities Involving epsilon
4.3.3: Identities Involving delta and epsilon
4.4: Generalized Kronecker delta
4.5: Exercises
5: Applications of Tensor Notation and Techniques
5.1: Common Definitions in Tensor Notation
5.2: Scalar Invariants of Tensors
5.3: Common Differential Operations in Tensor Notation
5.3.1: Cartesian Coordinate System
5.3.2: Cylindrical Coordinate System
5.3.3: Spherical Coordinate System
5.3.4: General Orthogonal Coordinate System
5.4: Common Identities in Vector and Tensor Notation
5.5: Integral Theorems in Tensor Notation
5.6: Examples of Using Tensor Techniques to Prove Identities
5.7: Exercises
6: Metric Tensor
6.1: Exercises
7: Covariant Differentiation
7.1: Exercises
References
Footnotes