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