Tensors for Physics

Part I A Primer on Vectors and Tensors


1 Introduction
1.1 Preliminary Remarks on Vectors
1.1.1 Vector Space
1.1.2 Norm and Distance
1.1.3 Vectors for Classical Physics
1.1.4 Vectors for Special Relativity
1.2 Preliminary Remarks on Tensors
1.3 Remarks on History and Literature
1.4 Scope of the Book

2 Basics
2.1 Coordinate System and Position Vector
2.1.1 Cartesian Components
2.1.2 Length of the Position Vector, Unit Vector
2.1.3 Scalar Product
2.1.4 Spherical Polar Coordinates
2.2 Vector as Linear Combination of Basis Vectors
2.2.1 Orthogonal Basis
2.2.2 Non-orthogonal Basis
2.3 Linear Transformations of the Coordinate System
2.3.1 Translation
2.3.2 Affine Transformation
2.4 Rotation of the Coordinate System
2.4.1 Orthogonal Transformation
2.4.2 Proper Rotation
2.5 Definitions of Vectors and Tensors in Physics
2.5.1 Vectors
2.5.2 What is a Tensor?
2.5.3 Multiplication by Numbers and Addition of Tensors
2.5.4 Remarks on Notation
2.5.5 Why the Emphasis on Tensors?
2.6 Parity
2.6.1 Parity Operation
2.6.2 Parity of Vectors and Tensors
2.6.3 Consequences for Linear Relations
2.6.4 Application: Linear and Nonlinear Susceptibility Tensors
2.7 Differentiation of Vectors and Tensors with Respect to a Parameter
2.7.1 Time Derivatives
2.7.2 Trajectory and Velocity
2.7.3 Radial and Azimuthal Components of the Velocity
2.8 Time Reversal

3 Symmetry of Second Rank Tensors, Cross Product
3.1 Symmetry
3.1.1 Symmetric and Antisymmetric Parts
3.1.2 Isotropic, Antisymmetric and Symmetric Traceless Parts
3.1.3 Trace of a Tensor
3.1.4 Multiplication and Total Contraction of Tensors, Norm
3.1.5 Fourth Rank Projections Tensors
3.1.6 Preliminary Remarks on ``Antisymmetric Part and Vector''
3.1.7 Preliminary Remarks on the Symmetric Traceless Part
3.2 Dyadics
3.2.1 Definition of a Dyadic Tensor
3.2.2 Products of Symmetric Traceless Dyadics
3.3 Antisymmetric Part, Vector Product
3.3.1 Dual Relation
3.3.2 Vector Product
3.4 Applications of the Vector Product
3.4.1 Orbital Angular Momentum
3.4.2 Torque
3.4.3 Motion on a Circle
3.4.4 Lorentz Force
3.4.5 Screw Curve

4 Epsilon-Tensor
4.1 Definition, Properties
4.1.1 Link with Determinants
4.1.2 Product of Two Epsilon-Tensors
4.1.3 Antisymmetric Tensor Linked with a Vector
4.2 Multiple Vector Products
4.2.1 Scalar Product of Two Vector Products
4.2.2 Double Vector Products
4.3 Applications
4.3.1 Angular Momentum for the Motion on a Circle
4.3.2 Moment of Inertia Tensor
4.4 Dual Relation and Epsilon-Tensor in 2D
4.4.1 Definitions and Matrix Notation

5 Symmetric Second Rank Tensors
5.1 Isotropic and Symmetric Traceless Parts
5.2 Principal Values
5.2.1 Principal Axes Representation
5.2.2 Isotropic Tensors
5.2.3 Uniaxial Tensors
5.2.4 Biaxial Tensors
5.2.5 Symmetric Dyadic Tensors
5.3 Applications
5.3.1 Moment of Inertia Tensor of Molecules
5.3.2 Radius of Gyration Tensor
5.3.3 Molecular Polarizability Tensor
5.3.4 Dielectric Tensor, Birefringence
5.3.5 Electric and Magnetic Torques
5.4 Geometric Interpretation of Symmetric Tensors
5.4.1 Bilinear Form
5.4.2 Linear Mapping
5.4.3 Volume and Surface of an Ellipsoid
5.5 Scalar Invariants of a Symmetric Tensor
5.5.1 Definitions
5.5.2 Biaxiality of a Symmetric Traceless Tensor
5.6 Hamilton-Cayley Theorem and Consequences
5.6.1 Hamilton-Cayley Theorem
5.6.2 Quadruple Products of Tensors
5.7 Volume Conserving Affine Transformation
5.7.1 Mapping of a Sphere onto an Ellipsoid
5.7.2 Uniaxial Ellipsoid

6 Summary: Decomposition of Second Rank Tensors

7 Fields, Spatial Differential Operators
7.1 Scalar Fields,Gradient
7.1.1 Graphical Representation of Potentials
7.1.2 Differential Change of a Potential, Nabla Operator
7.1.3 Gradient Field, Force
7.1.4 Newton's Equation of Motion, One and More Particles
7.1.5 Special Force Fields
7.2 Vector Fields, Divergence and Curl or Rotation
7.2.1 Examples for Vector Fields
7.2.2 Differential Change of a Vector Fields
7.3 Special Types of Vector Fields
7.3.1 Vorticity Free Vector Fields, Scalar Potential
7.3.2 Poisson Equation, Laplace Operator
7.3.3 Divergence Free Vector Fields, Vector Potential
7.3.4 Vorticity Free and Divergence Free Vector Fields, Laplace Fields
7.3.5 Conventional Classification of Vector Fields
7.3.6 Second Spatial Derivatives of Spherically Symmetric Scalar Fields
7.4 Tensor Fields
7.4.1 Graphical Representations of Symmetric Second Rank Tensor Fields
7.4.2 Spatial Derivatives of Tensor Fields
7.4.3 Local Mass and Momentum Conservation, Pressure Tensor
7.5 Maxwell Equations in Differential Form
7.5.1 Four-Field Formulation
7.5.2 Special Cases
7.5.3 Electromagnetic Waves in Vacuum
7.5.4 Scalar and Vector Potentials
7.5.5 Magnetic Field Tensors
7.6 Rules for Nabla and Laplace Operators
7.6.1 Nabla
7.6.2 Application: Orbital Angular Momentum Operator
7.6.3 Radial and Angular Parts of the Laplace Operator
7.6.4 Application: Kinetic Energy Operator in Wave Mechanics

8 Integration of Fields
8.1 Line Integrals
8.1.1 Definition, Parameter Representation
8.1.2 Closed Line Integrals
8.1.3 Line Integrals for Scalar and Vector Fields
8.1.4 Potential of a Vector Field
8.1.5 Computation of the Potential for a Vector Field
8.2 Surface Integrals, Stokes
8.2.1 Parameter Representation of Surfaces
8.2.2 Examples for Parameter Representations of Surfaces
8.2.3 Surface Integrals as Integrals Over Two Parameters
8.2.4 Examples for Surface Integrals
8.2.5 Flux of a Vector Field
8.2.6 Generalized Stokes Law
8.2.7 Application: Magnetic Field Around an Electric Wire
8.2.8 Application: Faraday Induction
8.3 Volume Integrals, Gauss
8.3.1 Volume Integrals in R3
8.3.2 Application: Mass Density, Center of Mass
8.3.3 Application: Moment of Inertia Tensor
8.3.4 Generalized Gauss Theorem
8.3.5 Application: Gauss Theorem in Electrodynamics, Coulomb Force
8.3.6 Integration by Parts
8.4 Further Applications of Volume Integrals
8.4.1 Continuity Equation, Flow Through a Pipe
8.4.2 Momentum Balance, Force on a Solid Body
8.4.3 The Archimedes Principle
8.4.4 Torque on a Rotating Solid Body
8.5 Further Applications in Electrodynamics
8.5.1 Energy and Energy Density in Electrostatics
8.5.2 Force and Maxwell Stress in Electrostatics
8.5.3 Energy Balance for the Electromagnetic Field
8.5.4 Momentum Balance for the Electromagnetic Field, Maxwell Stress Tensor
8.5.5 Angular Momentum in Electrodynamics


Part II Advanced Topics


9 Irreducible Tensors
9.1 Definition and Examples
9.2 Products of Irreducible Tensors
9.3 Contractions, Legendre Polynomials
9.4 Cartesian and Spherical Tensors
9.4.1 Spherical Components of a Vector
9.4.2 Spherical Components of Tensors
9.5 Cubic Harmonics
9.5.1 Cubic Tensors
9.5.2 Cubic Harmonics with Full Cubic Symmetry

10 Multipole Potentials
10.1 Descending Multipoles
10.1.1 Definition of the Multipole Potential Functions
10.1.2 Dipole, Quadrupole and Octupole Potentials
10.1.3 Source Term for the Quadrupole Potential
10.1.4 General Properties of Multipole Potentials
10.2 Ascending Multipoles
10.3 Multipole Expansion and Multipole Moments in Electrostatics
10.3.1 Coulomb Force and Electrostatic Potential
10.3.2 Expansion of the Electrostatic Potential
10.3.3 Electric Field of Multipole Moments
10.3.4 Multipole Moments for Discrete Charge Distributions
10.3.5 Connection with Legendre Polynomials
10.4 Further Applications in Electrodynamics
10.4.1 Induced Dipole Moment of a Metal Sphere
10.4.2 Electric Polarization as Dipole Density
10.4.3 Energy of Multipole Moments in an External Field
10.4.4 Force and Torque on Multipole Moments in an External Field
10.4.5 Multipole--Multipole Interaction
10.5 Applications in Hydrodynamics
10.5.1 Stationary and Creeping Flow Equations
10.5.2 Stokes Force on a Sphere

11 Isotropic Tensors
11.1 General Remarks on Isotropic Tensors
11.2 .-Tensors
11.2.1 Definition and Examples
11.2.2 General Properties of .-Tensors
11.2.3 .-Tensors as Derivatives of Multipole Potentials
11.3 Generalized Cross Product, -Tensors
11.3.1 Cross Product via the -Tensor
11.3.2 Properties of -Tensors
11.3.3 Action of the Differential Operator calL on Irreducible Tensors
11.3.4 Consequences for the Orbital Angular Momentum Operator
11.4 Isotropic Coupling Tensors
11.4.1 Definition of .(ell,2,ell)-Tensors
11.4.2 Tensor Product of Second Rank Tensors
11.5 Coupling of a Vector with Irreducible Tensors
11.6 Coupling of Second Rank Tensors with Irreducible Tensors
11.7 Scalar Product of Three Irreducible Tensors
11.7.1 Scalar Invariants
11.7.2 Interaction Potential for Uniaxial Particles

12 Integral Formulae and Distribution Functions
12.1 Integrals Over Unit Sphere
12.1.1 Integrals of Products of Two Irreducible Tensors
12.1.2 Multiple Products of Irreducible Tensors
12.2 Orientational Distribution Function
12.2.1 Orientational Averages
12.2.2 Expansion with Respect to Irreducible Tensors
12.2.3 Anisotropic Dielectric Tensor
12.2.4 Field-Induced Orientation
12.2.5 Kerr Effect, Cotton-Mouton Effect, Non-linear Susceptibility
12.2.6 Orientational Entropy
12.2.7 Fokker-Planck Equation for the Orientational Distribution
12.3 Averages Over Velocity Distributions
12.3.1 Integrals Over the Maxwell Distribution
12.3.2 Expansion About an Absolute Maxwell Distribution
12.3.3 Kinetic Equations, Flow Term
12.3.4 Expansion About a Local Maxwell Distribution
12.4 Anisotropic Pair Correlation Function and Static Structure Factor
12.4.1 Two-Particle Density, Two-Particle Averages
12.4.2 Potential Contributions to the Energy and to the Pressure Tensor
12.4.3 Static Structure Factor
12.4.4 Expansion of g(r)
12.4.5 Shear-Flow Induced Distortion of the Pair Correlation
12.4.6 Plane Couette Flow Symmetry
12.4.7 Cubic Symmetry
12.4.8 Anisotropic Structure Factor
12.5 Selection Rules for Electromagnetic Radiation
12.5.1 Expansion of the Wave Function
12.5.2 Electric Dipole Transitions
12.5.3 Electric Quadrupole Transitions

13 Spin Operators
13.1 Spin Commutation Relations
13.1.1 Spin Operators and Spin Matrices
13.1.2 Spin 1/2 and Spin 1 Matrices
13.2 Magnetic Sub-states
13.2.1 Magnetic Quantum Numbers and Hamilton Cayley
13.2.2 Projection Operators into Magnetic Sub-states
13.3 Irreducible Spin Tensors
13.3.1 Defintions and Examples
13.3.2 Commutation Relation for Spin Tensors
13.3.3 Scalar Products
13.4 Spin Traces
13.4.1 Traces of Products of Spin Tensors
13.4.2 Triple Products of Spin Tensors
13.4.3 Multiple Products of Spin Tensors
13.5 Density Operator
13.5.1 Spin Averages
13.5.2 Expansion of the Spin Density Operator
13.5.3 Density Operator for Spin 1/2 and Spin 1
13.6 Rotational Angular Momentum of Linear Molecules, Tensor Operators
13.6.1 Basics and Notation
13.6.2 Projection into Rotational Eigenstates, Traces
13.6.3 Diagonal Operators
13.6.4 Diagonal Density Operator,Averages
13.6.5 Anisotropic Dielectric Tensor of a Gas of Rotating Molecules
13.6.6 Non-diagonal Tensor Operators

14 Rotation of Tensors
14.1 Rotation of Vectors
14.1.1 Infinitesimal and Finite Rotation
14.1.2 Hamilton Cayley and Projection Tensors
14.1.3 Rotation Tensor for Vectors
14.1.4 Connection with Spherical Components
14.2 Rotation of Second Rank Tensors
14.2.1 Infinitesimal Rotation
14.2.2 Fourth Rank Projection Tensors
14.2.3 Fourth Rank Rotation Tensor
14.3 Rotation of Tensors of Rank ell
14.4 Solution of Tensor Equations
14.4.1 Inversion of Linear Equations
14.4.2 Effect of a Magnetic Field on the Electrical Conductivity
14.5 Additional Formulas Involving Projectors

15 Liquid Crystals and Other Anisotropic Fluids
15.1 Remarks on Nomenclature and Notations
15.1.1 Nematic and Cholesteric Phases, Blue Phases
15.1.2 Smectic Phases
15.2 Isotropic Nematic Phase Transition
15.2.1 Order Parameter Tensor
15.2.2 Landau-de Gennes Theory
15.2.3 Maier-Saupe Mean Field Theory
15.3 Elastic Behavior of Nematics
15.3.1 Director Elasticity, Frank Coefficients
15.3.2 The Cholesteric Helix
15.3.3 Alignment Tensor Elasticity
15.4 Cubatics and Tetradics
15.4.1 Cubic Order Parameter
15.4.2 Landau Theory for the Isotropic-Cubatic Phase Transition
15.4.3 Order Parameter Tensor for Regular Tetrahedra
15.5 Energetic Coupling of Order Parameter Tensors
15.5.1 Two Second Rank Tensors
15.5.2 Second-Rank Tensor and Vector
15.5.3 Second- and Third-Rank Tensors

16 Constitutive Relations
16.1 General Principles
16.1.1 Curie Principle
16.1.2 Energy Principle
16.1.3 Irreversible Thermodynamics, Onsager Symmetry Principle
16.2 Elasticity
16.2.1 Elastic Deformation of a Solid, Stress Tensor
16.2.2 Voigt Coefficients
16.2.3 Isotropic Systems
16.2.4 Cubic System
16.2.5 Microscopic Expressions for Elasticity Coefficients
16.3 Viscosity and Non-equilibrium Alignment Phenomena
16.3.1 General Remarks, Simple Fluids
16.3.2 Influence of Magnetic and Electric Fields
16.3.3 Plane Couette and Plane Poiseuille Flow
16.3.4 Senftleben-Beenakker Effect of the Viscosity
16.3.5 Angular Momentum Conservation, Antisymmetric Pressure and Angular Velocity
16.3.6 Flow Birefringence
16.3.7 Heat-Flow Birefringence
16.3.8 Visco-Elasticity
16.3.9 Nonlinear Viscosity
16.3.10 Vorticity Free Flow
16.4 Viscosity and Alignment in Nematics
16.4.1 Well Aligned Nematic Liquid Crystals and Ferro Fluids
16.4.2 Perfectly Oriented Ellipsoidal Particles
16.4.3 Free Flow of Nematics, Flow Alignment and Tumbling
16.4.4 Fokker-Planck Equation Applied to Flow Alignment
16.4.5 Unified Theory for Isotropic and Nematic Phases
16.4.6 Limiting Cases: Isotropic Phase, Weak Flow in the Nematic Phase
16.4.7 Scaled Variables, Model Parameters
16.4.8 Spatially Inhomogeneous Alignment

17 Tensor Dynamics
17.1 Time-Correlation Functions and Spectral Functions
17.1.1 Definitions
17.1.2 Depolarized Rayleigh Scattering
17.1.3 Collisional and Diffusional Line Broadening
17.2 Nonlinear Relaxation, Component Notation
17.2.1 Second-Rank Basis Tensors
17.2.2 Third-Order Scalar Invariant and Biaxiality Parameter
17.2.3 Component Equations
17.2.4 Stability of Stationary Solutions
17.3 Alignment Tensor Subjected to a Shear Flow
17.3.1 Dynamic Equations for the Components
17.3.2 Types of Dynamic States
17.3.3 Flow Properties
17.4 Nonlinear Maxwell Model
17.4.1 Formulation of the Model
17.4.2 Special Cases

18 From 3D to 4D: Lorentz Transformation, Maxwell Equations
18.1 Lorentz Transformation
18.1.1 Invariance Condition
18.1.2 4-Vectors
18.1.3 Lorentz Transformation Matrix
18.1.4 A Special Lorentz Transformation
18.1.5 General Lorentz Transformations
18.2 Lorentz-Vectors and Lorentz-Tensors
18.2.1 Lorentz-Tensors
18.2.2 Proper Time, 4-Velocity and 4-Acceleration
18.2.3 Differential Operators, Plane Waves
18.2.4 Some Historical Remarks
18.3 The 4D-Epsilon Tensor
18.3.1 Levi-Civita Tensor
18.3.2 Products of Two Epsilon Tensors
18.3.3 Dual Tensor, Determinant
18.4 Maxwell Equations in 4D-Formulation
18.4.1 Electric Flux Density and Continuity Equation
18.4.2 Electric 4-Potential and Lorentz Scaling
18.4.3 Field Tensor Derived from the 4-Potential
18.4.4 The Homogeneous Maxwell Equations
18.4.5 The Inhomogeneous Maxwell Equations
18.4.6 Inhomogeneous Wave Equation
18.4.7 Transformation Behavior of the Electromagnetic Fields
18.4.8 Lagrange Density and Variational Principle
18.5 Force Density and Stress Tensor
18.5.1 4D Force Density
18.5.2 Maxwell Stress Tensor

Appendix Exercises: Answers and Solutions

References

Index