MathTensor: A System for Doing Tensor Analysis by Computer

Table of Contents

1. A Brief Look at MathTensor

• 1.1 Creating Tensors With Symmetries
• 1.2 Simplification
• 1.3 Derivatives
• 1.4 Differential Forms
• 1.5 Calculating Curvature
• 1.6 Calculating Invariants
• 1.7 Multiple Types Of Indices
• 1.8 Rules With Dummy Indices
• 1.9 Coordinate transformations

Part A. Tensor Analysis in MathTensor

• 2.1 Tensors of Rank 1: Vectors
  • 2.1.1 Covariant vectors
  • 2.1.2 Contravariant vectors
• 2.2 Tensors of Higher Rank
• 2.3 Exterior Differential Forms
• 2.4 The Lie Derivative
• 2.5 Affine Connection, Covariant Derivative, Torsion
• 2.6 The Metric Tensor
  • 2.6.1 Affine Connection and the Metric
  • 2.6.2 Geodesics
  • 2.6.3 Lowering and Raising indices
• 2.7 The Riemann Curvature Tensor
  • 2.7.1 The Ricci, Einstein, and Weyl Tensors

• Starting MathTensor
• 3.1 Tensors and Their Symmetries
  • 3.1.1 Indices
  • 3.1.2 Some Common Tensors
  • 3.1.3 Symmetries
  • 3.1.4 Defining Your Own Tensors
• 3.2 Contraction and the Metric Tensor
  • 3.2.1 Forming New Tensors by Contraction
  • 3.2.2 Raising and Lowering Indices with the Metric Tensor
  • 3.2.3 Preventing Automatic Absorption of the Metric
• 3.3 Simplification
  • 3.3.1 Methods of Simplifying Tensor Expressions
  • 3.3.2 Simplifying with Symmetries
  • 3.3.3 Simplifying Traces and Contractions of Products
  • 3.3.4 Canonical Ordering of Indices
• 3.4 Operations on Tensors
  • 3.4.1 Symmetrization and Antisymmetrization
  • 3.4.2 Differential Form Operations
  • 3.4.3 Levi-Civita Epsilon Tensor
  • 3.4.4 Derivative Operators
• 3.5 Creating Rules and Definitions
  • 3.5.1 Simple Definitions and Rules
  • 3.5.2 Pitfalls of Dummy Indices
  • 3.5.3 Making Rules and Definitions Containing Dummy Indices
  • 3.5.4 Building Your Own Knowledge Base Files
• 3.6 The Riemann and Related Rules
  • 3.6.1 Organization of Rules
  • 3.6.2 Automatic Riemann Rules
  • 3.6.3 Riemann Rules Used with the ApplyRules Functions
  • 3.6.4 Rules Used Individually
  • 3.6.5 Conversion Rules
  • 3.6.6 Other Rules Functions
• 3.7 Using Contexts With MathTensor

• 4.1 Components and Their Values
  • 4.1.1 Integers as Indices
  • 4.1.2 Using Integer and Symbolic indices
  • 4.1.3 Flat Metric in Cartesian Coordinates
  • 4.1.4 Calculating the Components of Curvature Tensors
  • 4.1.5 Invariants From Curvature Components
  • 4.1.6 Simplification and Series Expansion in Components
  • 4.1.7 Schwarzschild With Alternate Simplification
  • 4.1.8 Asymptotic Behavior In Schwarzschild
  • 4.1.9 Perturbations of Flat Spacetime
  • 4.1.10 Evaluating General Tensor Expressions for Particular Metrics
• 4.2 Coordinate Transformations of Tensors
  • 4.2.1 Ttransform: Syntax and Explanation
  • 4.2.2 Four Examples
  • 4.2.3 Example One: Familiar Calculation
  • 4.2.4 Example Two: Simple Shear
  • 4.2.5 Example Three: Special Relativity
  • 4.2.6 Example Four: General Relativity --- Kruskal coordinates
• 4.3 Using More Than One Type of Index
  • 4.3.1 Types of Indices
  • 4.3.2 Tests and Commands for Various Types of Indices
  • 4.3.3 Transformation Rules with Multiple Types of Indices
  • 4.3.4 Warnings About Using Kdelta
  • 4.3.5 Pauli Spin Matrices
  • 4.3.6 Summing Expressions With Several Types of Indices
• 4.4 Manipulating Equations in MathTensor
• 4.5 Unit Systems, Constants, Sign Conventions
• 4.6 Electromagnetism: Objects and Rules
• 4.7 Flags and Miscellaneous Commands or Objects

Part B. Differential Forms in MathTensor

• Introduction
• 5.1 Basic Differential Form Operations
  • 5.1.1 Defining Forms and Taking Their Components
  • 5.1.2 The Exterior Product
  • 5.1.3 The Exterior Derivative
• 5.2 More Advanced Differential Form Operations
  • 5.2.1 The Lie Derivative
  • 5.2.2 The Interior Product
  • 5.2.3 The Hodge Star Operator
  • 5.2.4 The Codifferential Operator
  • 5.2.5 The Hodge and Bochner Laplacian Operators

• 6.1 The Theorem of Stokes
• 6.2 Differential Forms and Maxwell's Equations
• 6.3 Tensor Valued Forms: Cartan Structure Equations
• 6.4 Structure Equations In Euclidean Space

Part C. Applications of MathTensor

• 7.1 Maxwell's Electromagnetic Field Equations
  • 7.1.1 Units and Maxwell's Equations in {\psl MathTensor
  • 7.1.2 Electromagnetic Wave Equation in Curved Spacetime
  • 7.1.3 Lorentz Force Density from the Maxwell Stress Tensor
• 7.2 Special Relativistic Collisions
  • 7.2.1 Elastic Collision of Two Particles of Equal Mass
  • 7.2.2 Solution in Center-Of-Momentum Frame
  • 7.2.3 Transformation to Laboratory Frame

• 8.1 Finite Deformation of a Body
  • 8.1.1 Deformation Metrics and Strain
  • 8.1.2 Basis Vectors
  • 8.1.3 Relation to Cauchy-Green Strain Tensors
  • 8.1.4 The Stress Tensor
  • 8.1.5 Displacement and Strain
  • 8.1.6 Stress and Strain in an Elastic Body
  • 8.1.7 Elastically Isotropic Body
• 8.2 Simple Shear Deformation of a Solid
  • 8.2.1 Finding the Metric Tensors
  • 8.2.2 Finding the Invariants I1, I2, and I3
  • 8.2.3 Finding the Stress Tensor
  • 8.2.4 Finding the Surface Forces
• 8.3 Cylinder Twisted About Its Axis
  • 8.3.1 Metrics
  • 8.3.2 Invariants I1, I2, and I3
  • 8.3.3 Stress Tensor
  • 8.3.4 Equilibrium Equations
  • 8.3.5 Surface Forces on the Deformed Cylinder

• 9.1 Reissner-Nordstrom Invariants
  • 9.1.1 Computing the Riemann Tensor and Related Objects
  • 9.1.2 Invariant Definitions
  • 9.1.3 Evaluation of Invariants
• 9.2 Variations and Field Equations
  • 9.2.1 Metric Variations and the Einstein Action
  • 9.2.2 The Gauss-Gonnet Invariant

Bibliography

Index

To Order MathTensor

Send comments or questions to steve@smc.vnet.net