Introduction to Tensor Calculus

Author: Taha Sochi

• Department of Physics & Astronomy, University College London
• Email: [email protected]
• Lecture Date: May 25, 2016

Preface

• General notes on tensor calculus from personal studies
• Intended as a short reference for an introductory course
• Encourages understanding of basic calculus and linear algebra
• Plans for a series of documents increasing in complexity

Contents Overview

1. Notation, Nomenclature, and Conventions
2. Preliminaries
   • Introduction to Tensors
   • General Rules of Tensor Calculus
   • Examples and Applications
   • Types of Tensors
3. ** and Tensors**
4. Applications of Tensor Notation and Techniques
5. Metric Tensor
6. Covariant Differentiation

Key Topics

1. Notation, Nomenclature, and Conventions

• Scalars characterized by magnitude and sign
• Vectors characterized by magnitude and direction
• Tensors are arrays of mathematical objects
• Rank of tensor signifies complexity:
  • Rank-0: Scalars
  • Rank-1: Vectors
  • Rank-2: Dyads
  • Rank-3: Triads
• Notation:
  • Scalars: light face Latin letters
  • Vectors: bold face Latin letters
  • Tensors: upper case bold face Latin letters
  • Indexed notation for components

2. Preliminaries

2.1 Introduction

• Tensors transform according to coordinate changes
• Tensors ensure the invariance of physical laws
• Tensor equality and operations defined

2.2 General Rules

• Free and dummy indices
• Summation convention for dummy indices
• Covariant (subscript) and contravariant (superscript) indices
• Consistency in index usage across expressions and equations

2.3 Examples of Tensors of Different Ranks

• Scalars: Energy, Mass, Temperature
• Vectors: Displacement, Force
• Rank-2 Tensors: Stress, Strain
• Rank-3 Tensors: Levi-Civita Tensor

2.4 Applications of Tensors

• Used in fluid mechanics, general relativity, structural engineering
• Compact formulation of equations

2.5 Types of Tensors

• Types: Covariant, Contravariant, Mixed
• True vs. Pseudo Tensors
• Absolute vs. Relative Tensors
• Isotropic vs. Anisotropic Tensors
• Symmetric vs. Anti-symmetric Tensors

2.6 Tensor Operations

• Addition/Subtraction
• Scalar multiplication
• Tensor multiplication (outer product)
• Contraction
• Inner product
• Permutation

3. and Tensors

3.1 Kronecker

• Rank-2 symmetric tensor
• Identity matrix in 3D space

3.2 Permutation

• Rank equals space dimensions
• Totally anti-symmetric

3.3 Useful Identities

• Various identities involving Kronecker delta and Levi-Civita

4. Applications of Tensor Notation and Techniques

4.1 Common Definitions

• Trace and determinant of matrices

4.2 Scalar Invariants of Tensors

• Magnitude of vector as an invariant
• Three main scalar invariants for rank-2 tensors

4.3 Differential Operations

• Gradient, Divergence, Curl in Cartesian and other coordinates

4.4 Common Identities

• Vector calculus identities

4.5 Integral Theorems

• Divergence and Stokes’ theorem in tensor notation

4.6 Proving Identities

• Examples of tensor techniques in proving identities

5. Metric Tensor

• Generalizes concept of distance
• Invariance in different coordinate systems

6. Covariant Differentiation

• Ensures invariance of derivatives
• Utilizes Christoffel symbols

References

• Text references include works by Arfken, Bird, Boas, etc.
• Acknowledgment of online tutorials and resources