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The Geometry of Physics

Book Description

This book provides a working knowledge of those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms that are essential for a deeper understanding of both classical and modern physics and engineering. Included are discussions of analytical and fluid dynamics, electromagnetism (in flat and curved space), thermodynamics, the Dirac operator and spinors, and gauge fields, including Yang–Mills, the Aharonov–Bohm effect, Berry phase and instanton winding numbers, quarks and quark model for mesons. Before discussing abstract notions of differential geometry, geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space. The book is ideal for graduate and advanced undergraduate students of physics, engineering or mathematics as a course text or for self study. This third edition includes an overview of Cartan's exterior differential forms, which previews many of the geometric concepts developed in the text.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Dedication
  5. Contents
  6. Preface to the Third Edition
  7. Preface to the Second Edition
  8. Preface to the Revised Printing
  9. Preface to the First Edition
  10. Overview. An Informal Overview of Cartan’s Exterior Differential Forms, Illustrated with an Application to Cauchy’s Stress Tensor
    1. Introduction
      1. O.a. Introduction
    2. Vectors, 1-Forms, and Tensors
      1. O.b. Two Kinds of Vectors
      2. O.c. Superscripts, Subscripts, Summation Convention
      3. O.d. Riemannian Metrics
      4. O.e. Tensors
    3. Integrals and Exterior Forms
      1. O.f. Line Integrals
      2. O.g. Exterior 2-Forms
      3. O.h. Exterior p-Forms and Algebra in Rn
      4. O.i. The Exterior Differential d
      5. O.j. The Push-Forward of a Vector and the Pull-Back of a Form
      6. O.k. Surface Integrals and “Stokes’ Theorem”
      7. O.l. Electromagnetism, or, Is it a Vector or a Form?
      8. O.m. Interior Products
      9. O.n. Volume Forms and Cartan’s Vector Valued Exterior Forms
      10. O.o. Magnetic Field for Current in a Straight Wire
    4. Elasticity and Stresses
      1. O.p. Cauchy Stress, Floating Bodies, Twisted Cylinders, and Strain Energy
      2. O.q. Sketch of Cauchy’s “First Theorem”
      3. O.r. Sketch of Cauchy’s “Second Theorem,” Moments as Generators of Rotations
      4. O.s. A Remarkable Formula for Differentiating Line, Surface, and…, Integrals
  11. I: Manifolds, Tensors, and Exterior Forms
    1. 1. Manifolds and Vector Fields
      1. 1.1. Submanifolds of Euclidean Space
        1. 1.1a. Submanifolds of RN
        2. 1.1b. The Geometry of Jacobian Matrices: The “Differential”
        3. 1.1c. The Main Theorem on Submanifolds of RN
        4. 1.1d. A Nontrivial Example: The Configuration Space of a Rigid Body
      2. 1.2. Manifolds
        1. 1.2a. Some Notions from Point Set Topology
        2. 1.2b. The Idea of a Manifold
        3. 1.2c. A Rigorous Definition of a Manifold
        4. 1.2d. Complex Manifolds: The Riemann Sphere
      3. 1.3. Tangent Vectors and Mappings
        1. 1.3a. Tangent or “Contravariant” Vectors
        2. 1.3b. Vectors as Differential Operators
        3. 1.3c. The Tangent Space to Mn at a Point
        4. 1.3d. Mappings and Submanifolds of Manifolds
        5. 1.3e. Change of Coordinates
      4. 1.4. Vector Fields and Flows
        1. 1.4a. Vector Fields and Flows on Rn
        2. 1.4b. Vector Fields on Manifolds
        3. 1.4c. Straightening Flows
    2. 2. Tensors and Exterior Forms
      1. 2.1. Covectors and Riemannian Metrics
        1. 2.1a. Linear Functionals and the Dual Space
        2. 2.1b. The Differential of a Function
        3. 2.1c. Scalar Products in Linear Algebra
        4. 2.1d. Riemannian Manifolds and the Gradient Vector
        5. 2.1e. Curves of Steepest Ascent
      2. 2.2. The Tangent Bundle
        1. 2.2a. The Tangent Bundle
        2. 2.2b. The Unit Tangent Bundle
      3. 2.3. The Cotangent Bundle and Phase Space
        1. 2.3a. The Cotangent Bundle
        2. 2.3b. The Pull-Back of a Covector
        3. 2.3c. The Phase Space in Mechanics
        4. 2.3d. The Poincaré 1-Form
      4. 2.4. Tensors
        1. 2.4a. Covariant Tensors
        2. 2.4b. Contravariant Tensors
        3. 2.4c. Mixed Tensors
        4. 2.4d. Transformation Properties of Tensors
        5. 2.4e. Tensor Fields on Manifolds
      5. 2.5. The Grassmann or Exterior Algebra
        1. 2.5a. The Tensor Product of Covariant Tensors
        2. 2.5b. The Grassmann or Exterior Algebra
        3. 2.5c. The Geometric Meaning of Forms in Rn
        4. 2.5d. Special Cases of the Exterior Product
        5. 2.5e. Computations and Vector Analysis
      6. 2.6. Exterior Differentiation
        1. 2.6a. The Exterior Differential
        2. 2.6b. Examples in R3
        3. 2.6c. A Coordinate Expression for d
      7. 2.7. Pull-Backs
        1. 2.7a. The Pull-Back of a Covariant Tensor
        2. 2.7b. The Pull-Back in Elasticity
      8. 2.8. Orientation and Pseudoforms
        1. 2.8a. Orientation of a Vector Space
        2. 2.8b. Orientation of a Manifold
        3. 2.8c. Orientability and 2-Sided Hypersurfaces
        4. 2.8d. Projective Spaces
        5. 2.8e. Pseudoforms and the Volume Form
        6. 2.8f. The Volume Form in a Riemannian Manifold
      9. 2.9. Interior Products and Vector Analysis
        1. 2.9a. Interior Products and Contractions
        2. 2.9b. Interior Product in R3
        3. 2.9c. Vector Analysis in R3
      10. 2.10. Dictionary
    3. 3. Integration of Differential Forms
      1. 3.1. Integration over a Parameterized Subset
        1. 3.1a. Integration of a p-Form in Rp
        2. 3.1b. Integration over Parameterized Subsets
        3. 3.1c. Line Integrals
        4. 3.1d. Surface Integrals
        5. 3.1e. Independence of Parameterization
        6. 3.1f. Integrals and Pull-Backs
        7. 3.1g. Concluding Remarks
      2. 3.2. Integration over Manifolds with Boundary
        1. 3.2a. Manifolds with Boundary
        2. 3.2b. Partitions of Unity
        3. 3.2c. Integration over a Compact Oriented Submanifold
        4. 3.2d. Partitions and Riemannian Metrics
      3. 3.3. Stokes’s Theorem
        1. 3.3a. Orienting the Boundary
        2. 3.3b. Stokes’s Theorem
      4. 3.4. Integration of Pseudoforms
        1. 3.4a. Integrating Pseudo-n-Forms on an n-Manifold
        2. 3.4b. Submanifolds with Transverse Orientation
        3. 3.4c. Integration over a Submanifold with Transverse Orientation
        4. 3.4d. Stokes’s Theorem for Pseudoforms
      5. 3.5. Maxwell’s Equations
        1. 3.5a. Charge and Current in Classical Electromagnetism
        2. 3.5b. The Electric and Magnetic Fields
        3. 3.5c. Maxwell’s Equations
        4. 3.5d. Forms and Pseudoforms
    4. 4. The Lie Derivative
      1. 4.1. The Lie Derivative of a Vector Field
        1. 4.1a. The Lie Bracket
        2. 4.1b. Jacobi’s Variational Equation
        3. 4.1c. The Flow Generated by [X, Y]
      2. 4.2. The Lie Derivative of a Form
        1. 4.2a. Lie Derivatives of Forms
        2. 4.2b. Formulas Involving the Lie Derivative
        3. 4.2c. Vector Analysis Again
      3. 4.3. Differentiation of Integrals
        1. 4.3a. The Autonomous (Time-Independent) Case
        2. 4.3b. Time-Dependent Fields
        3. 4.3c. Differentiating Integrals
      4. 4.4. A Problem Set on Hamiltonian Mechanics
        1. 4.4a. Time-Independent Hamiltonians
        2. 4.4b. Time-Dependent Hamiltonians and Hamilton’s Principle
        3. 4.4c. Poisson brackets
    5. 5. The Poincaré Lemma and Potentials
      1. 5.1. A More General Stokes’s Theorem
      2. 5.2. Closed Forms and Exact Forms
      3. 5.3. Complex Analysis
      4. 5.4. The Converse to the Poincaré Lemma
      5. 5.5. Finding Potentials
    6. 6. Holonomic and Nonholonomic Constraints
      1. 6.1. The Frobenius Integrability Condition
        1. 6.1a. Planes in R3
        2. 6.1b. Distributions and Vector Fields
        3. 6.1c. Distributions and 1-Forms
        4. 6.1d. The Frobenius Theorem
      2. 6.2. Integrability and Constraints
        1. 6.2a. Foliations and Maximal Leaves
        2. 6.2b. Systems of Mayer–Lie
        3. 6.2c. Holonomic and Nonholonomic Constraints
      3. 6.3. Heuristic Thermodynamics via Caratheodory
        1. 6.3a. Introduction
        2. 6.3b. The First Law of Thermodynamics
        3. 6.3c. Some Elementary Changes of State
        4. 6.3d. The Second Law of Thermodynamics
        5. 6.3e. Entropy
        6. 6.3f. Increasing Entropy
        7. 6.3g. Chow’s Theorem on Accessibility
  12. II: Geometry and Topology
    1. 7. R3 and Minkowski Space
      1. 7.1. Curvature and Special Relativity
        1. 7.1a. Curvature of a Space Curve in R3
        2. 7.1b. Minkowski Space and Special Relativity
        3. 7.1c. Hamiltonian Formulation
      2. 7.2. Electromagnetism in Minkowski Space
        1. 7.2a. Minkowski’s Electromagnetic Field Tensor
        2. 7.2b. Maxwell’s Equations
    2. 8. The Geometry of Surfaces in R3
      1. 8.1. The First and Second Fundamental Forms
        1. 8.1a. The First Fundamental Form, or Metric Tensor
        2. 8.1b. The Second Fundamental Form
      2. 8.2. Gaussian and Mean Curvatures
        1. 8.2a. Symmetry and Self-Adjointness
        2. 8.2b. Principal Normal Curvatures
        3. 8.2c. Gauss and Mean Curvatures: The Gauss Normal Map
      3. 8.3. The Brouwer Degree of a Map: A Problem Set
        1. 8.3a. The Brouwer Degree
        2. 8.3b. Complex Analytic (Holomorphic) Maps
        3. 8.3c. The Gauss Normal Map Revisited: The Gauss–Bonnet Theorem
        4. 8.3d. The Kronecker Index of a Vector Field
        5. 8.3e. The Gauss Looping Integral
      4. 8.4. Area, Mean Curvature, and Soap Bubbles
        1. 8.4a. The First Variation of Area
        2. 8.4b. Soap Bubbles and Minimal Surfaces
      5. 8.5. Gauss’s Theorema Egregium
        1. 8.5a. The Equations of Gauss and Codazzi
        2. 8.5b. The Theorema Egregium
      6. 8.6. Geodesics
        1. 8.6a. The First Variation of Arc Length
        2. 8.6b. The Intrinsic Derivative and the Geodesic Equation
      7. 8.7. The Parallel Displacement of Levi-Civita
    3. 9. Covariant Differentiation and Curvature
      1. 9.1. Covariant Differentiation
        1. 9.1a. Covariant Derivative
        2. 9.1b. Curvature of an Affine Connection
        3. 9.1c. Torsion and Symmetry
      2. 9.2. The Riemannian Connection
      3. 9.3. Cartan’s Exterior Covariant Differential
        1. 9.3a. Vector-Valued Forms
        2. 9.3b. The Covariant Differential of a Vector Field
        3. 9.3c. Cartan’s Structural Equations
        4. 9.3d. The Exterior Covariant Differential of a Vector-Valued Form
        5. 9.3e. The Curvature 2-Forms
      4. 9.4. Change of Basis and Gauge Transformations
        1. 9.4a. Symmetric Connections Only
        2. 9.4b. Change of Frame
      5. 9.5. The Curvature Forms in a Riemannian Manifold
        1. 9.5a. The Riemannian Connection
        2. 9.5b. Riemannian Surfaces M2
        3. 9.5c. An Example
      6. 9.6. Parallel Displacement and Curvature on a Surface
      7. 9.7. Riemann’s Theorem and the Horizontal Distribution
        1. 9.7a. Flat metrics
        2. 9.7b. The Horizontal Distribution of an Affine Connection
        3. 9.7c. Riemann’s Theorem
    4. 10. Geodesics
      1. 10.1. Geodesics and Jacobi Fields
        1. 10.1a. Vector Fields Along a Surface in Mn
        2. 10.1b. Geodesics
        3. 10.1c. Jacobi Fields
        4. 10.1d. Energy
      2. 10.2. Variational Principles in Mechanics
        1. 10.2a. Hamilton’s Principle in the Tangent Bundle
        2. 10.2b. Hamilton’s Principle in Phase Space
        3. 10.2c. Jacobi’s Principle of “Least” Action
        4. 10.2d. Closed Geodesics and Periodic Motions
      3. 10.3. Geodesics, Spiders, and the Universe
        1. 10.3a. Gaussian Coordinates
        2. 10.3b. Normal Coordinates on a Surface
        3. 10.3c. Spiders and the Universe
    5. 11. Relativity, Tensors, and Curvature
      1. 11.1. Heuristics of Einstein’s Theory
        1. 11.1a. The Metric Potentials
        2. 11.1b. Einstein’s Field Equations
        3. 11.1c. Remarks on Static Metrics
      2. 11.2. Tensor Analysis
        1. 11.2a. Covariant Differentiation of Tensors
        2. 11.2b. Riemannian Connections and the Bianchi Identities
        3. 11.2c. Second Covariant Derivatives: The Ricci Identities
      3. 11.3. Hilbert’s Action Principle
        1. 11.3a. Geodesics in a Pseudo-Riemannian Manifold
        2. 11.3b. Normal Coordinates, the Divergence and Laplacian
        3. 11.3c. Hilbert’s Variational Approach to General Relativity
      4. 11.4. The Second Fundamental Form in the Riemannian Case
        1. 11.4a. The Induced Connection and the Second Fundamental Form
        2. 11.4b. The Equations of Gauss and Codazzi
        3. 11.4c. The Interpretation of the Sectional Curvature
        4. 11.4d. Fixed Points of Isometries
      5. 11.5. The Geometry of Einstein’s Equations
        1. 11.5a. The Einstein Tensor in a (Pseudo-)Riemannian Space–Time
        2. 11.5b. The Relativistic Meaning of Gauss’s Equation
        3. 11.5c. The Second Fundamental Form of a Spatial Slice
        4. 11.5d. The Codazzi Equations
        5. 11.5e. Some Remarks on the Schwarzschild Solution
    6. 12. Curvature and Topology: Synge’s Theorem
      1. 12.1. Synge’s Formula for Second Variation
        1. 12.1a. The Second Variation of Arc Length
        2. 12.1b. Jacobi Fields
      2. 12.2. Curvature and Simple Connectivity
        1. 12.2a. Synge’s Theorem
        2. 12.2b. Orientability Revisited
    7. 13. Betti Numbers and De Rham’s Theorem
      1. 13.1. Singular Chains and Their Boundaries
        1. 13.1a. Singular Chains
        2. 13.1b. Some 2-Dimensional Examples
      2. 13.2. The Singular Homology Groups
        1. 13.2a. Coefficient Fields
        2. 13.2b. Finite Simplicial Complexes
        3. 13.2c. Cycles, Boundaries, Homology and Betti Numbers
      3. 13.3. Homology Groups of Familiar Manifolds
        1. 13.3a. Some Computational Tools
        2. 13.3b. Familiar Examples
      4. 13.4. De Rham’s Theorem
        1. 13.4a. The Statement of de Rham’s Theorem
        2. 13.4b. Two Examples
    8. 14. Harmonic Forms
      1. 14.1. The Hodge Operators
        1. 14.1a. The * Operator
        2. 14.1b. The Codifferential Operator δ = d *
        3. 14.1c. Maxwell’s Equations in Curved Space–Time M4
        4. 14.1d. The Hilbert Lagrangian
      2. 14.2. Harmonic Forms
        1. 14.2a. The Laplace Operator on Forms
        2. 14.2b. The Laplacian of a 1-Form
        3. 14.2c. Harmonic Forms on Closed Manifolds
        4. 14.2d. Harmonic Forms and de Rham’s Theorem
        5. 14.2e. Bochner’s Theorem
      3. 14.3. Boundary Values, Relative Homology, and Morse Theory
        1. 14.3a. Tangential and Normal Differential Forms
        2. 14.3b. Hodge’s Theorem for Tangential Forms
        3. 14.3c. Relative Homology Groups
        4. 14.3d. Hodge’s Theorem for Normal Forms
        5. 14.3e. Morse’s Theory of Critical Points
  13. III: Lie Groups, Bundles, and Chern Forms
    1. 15. Lie Groups
      1. 15.1. Lie Groups, Invariant Vector Fields and Forms
        1. 15.1a. Lie Groups
        2. 15.1b. Invariant Vector Fields and Forms
      2. 15.2. One Parameter Subgroups
      3. 15.3. The Lie Algebra of a Lie Group
        1. 15.3a. The Lie Algebra
        2. 15.3b. The Exponential Map
        3. 15.3c. Examples of Lie Algebras
        4. 15.3d. Do the 1-Parameter Subgroups Cover G?
      4. 15.4. Subgroups and Subalgebras
        1. 15.4a. Left Invariant Fields Generate Right Translations
        2. 15.4b. Commutators of Matrices
        3. 15.4c. Right Invariant Fields
        4. 15.4d. Subgroups and Subalgebras
    2. 16. Vector Bundles in Geometry and Physics
      1. 16.1. Vector Bundles
        1. 16.1a. Motivation by Two Examples
        2. 16.1b. Vector Bundles
        3. 16.1c. Local Trivializations
        4. 16.1d. The Normal Bundle to a Submanifold
      2. 16.2. Poincaré’s Theorem and the Euler Characteristic
        1. 16.2a. Poincaré’s Theorem
        2. 16.2b. The Stiefel Vector Field and Euler’s Theorem
      3. 16.3. Connections in a Vector Bundle
        1. 16.3a. Connection in a Vector Bundle
        2. 16.3b. Complex Vector Spaces
        3. 16.3c. The Structure Group of a Bundle
        4. 16.3d. Complex Line Bundles
      4. 16.4. The Electromagnetic Connection
        1. 16.4a. Lagrange’s Equations Without Electromagnetism
        2. 16.4b. The Modified Lagrangian and Hamiltonian
        3. 16.4c. Schrodinger’s Equation in an Electromagnetic Field
        4. 16.4d. Global Potentials
        5. 16.4e. The Dirac Monopole
        6. 16.4f. The Aharonov–Bohm Effect
    3. 17. Fiber Bundles, Gauss–Bonnet, and Topological Quantization
      1. 17.1. Fiber Bundles and Principal Bundles
        1. 17.1a. Fiber Bundles
        2. 17.1b. Principal Bundles and Frame Bundles
        3. 17.1c. Action of the Structure Group on a Principal Bundle
      2. 17.2. Coset Spaces
        1. 17.2a. Cosets
        2. 17.2b. Grassmann Manifolds
      3. 17.3. Chern’s Proof of the Gauss–Bonnet–Poincaré Theorem
        1. 17.3a. A Connection in the Frame Bundle of a Surface
        2. 17.3b. The Gauss–Bonnet–Poincaré Theorem
        3. 17.3c. Gauss–Bonnet as an Index Theorem
      4. 17.4. Line Bundles, Topological Quantization, and Berry Phase
        1. 17.4a. A Generalization of Gauss–Bonnet
        2. 17.4b. Berry Phase
        3. 17.4c. Monopoles and the Hopf Bundle
    4. 18. Connections and Associated Bundles
      1. 18.1. Forms with Values in a Lie Algebra
        1. 18.1a. The Maurer–Cartan Form
        2. 18.1b. g-Valued p-Forms on a Manifold
        3. 18.1c. Connections in a Principal Bundle
      2. 18.2. Associated Bundles and Connections
        1. 18.2a. Associated Bundles
        2. 18.2b. Connections in Associated Bundles
        3. 18.2c. The Associated Ad Bundle
      3. 18.3. r-Form Sections of a Vector Bundle: Curvature
        1. 18.3a. r -Form sections of E
        2. 18.3b. Curvature and the Ad Bundle
    5. 19. The Dirac Equation
      1. 19.1. The Groups SO(3) and SU(2)
        1. 19.1a. The Rotation Group SO(3) of R3
        2. 19.1b. SU(2): The Lie algebra su(2)
        3. 19.1c. SU (2) is Topologically the 3-Sphere
        4. 19.1d. Ad : SU(2) → SO(3) in More Detail
      2. 19.2. Hamilton, Clifford, and Dirac
        1. 19.2a. Spinors and Rotations of R3
        2. 19.2b. Hamilton on Composing Two Rotations
        3. 19.2c. Clifford Algebras
        4. 19.2d. The Dirac Program: The Square Root of the d’Alembertian
      3. 19.3. The Dirac Algebra
        1. 19.3a. The Lorentz Group
        2. 19.3b. The Dirac Algebra
      4. 19.4. The Dirac Operator ∂ in Minkowski Space
        1. 19.4a. Dirac Spinors
        2. 19.4b. The Dirac Operator
      5. 19.5. The Dirac Operator in Curved Space–Time
        1. 19.5a. The Spinor Bundle
        2. 19.5b. The Spin Connection in SM
    6. 20. Yang–Mills Fields
      1. 20.1. Noether’s Theorem for Internal Symmetries
        1. 20.1a. The Tensorial Nature of Lagrange’s Equations
        2. 20.1b. Boundary Conditions
        3. 20.1c. Noether’s Theorem for Internal Symmetries
        4. 20.1d. Noether’s Principle
      2. 20.2. Weyl’s Gauge Invariance Revisited
        1. 20.2a. The Dirac Lagrangian
        2. 20.2b. Weyl’s Gauge Invariance Revisited
        3. 20.2c. The Electromagnetic Lagrangian
        4. 20.2d. Quantization of the A Field: Photons
      3. 20.3. The Yang–Mills Nucleon
        1. 20.3a. The Heisenberg Nucleon
        2. 20.3b. The Yang–Mills Nucleon
        3. 20.3c. A Remark on Terminology
      4. 20.4. Compact Groups and Yang–Mills Action
        1. 20.4a. The Unitary Group Is Compact
        2. 20.4b. Averaging over a Compact Group
        3. 20.4c. Compact Matrix Groups Are Subgroups of Unitary Groups
        4. 20.4d. Ad Invariant Scalar Products in the Lie Algebra of a Compact Group
        5. 20.4e. The Yang–Mills Action
      5. 20.5. The Yang–Mills Equation
        1. 20.5a. The Exterior Covariant Divergence ∇*
        2. 20.5b. The Yang–Mills Analogy with Electromagnetism
        3. 20.5c. Further Remarks on the Yang–Mills Equations
      6. 20.6. Yang–Mills Instantons
        1. 20.6a. Instantons
        2. 20.6b. Chern’s Proof Revisited
        3. 20.6c. Instantons and the Vacuum
    7. 21. Betti Numbers and Covering Spaces
      1. 21.1. Bi-invariant Forms on Compact Groups
        1. 21.1a. Bi-invariant p-Forms
        2. 21.1b. The Cartan p-Forms
        3. 21.1c. Bi-invariant Riemannian Metrics
        4. 21.1d. Harmonic Forms in the Bi-invariant Metric
        5. 21.1e. Weyl and Cartan on the Betti Numbers of G
      2. 21.2. The Fundamental Group and Covering Spaces
        1. 21.2a. Poincaré’s Fundamental Group π1 (M)
        2. 21.2b. The Concept of a Covering Space
        3. 21.2c. The Universal Covering
        4. 21.2d. The Orientable Covering
        5. 21.2e. Lifting Paths
        6. 21.2f. Subgroups of π1 (M)
        7. 21.2g. The Universal Covering Group
      3. 21.3. The Theorem of S. B. Myers: A Problem Set
      4. 21.4. The Geometry of a Lie Group
        1. 21.4a. The Connection of a Bi-invariant Metric
        2. 21.4b. The Flat Connections
    8. 22. Chern Forms and Homotopy Groups
      1. 22.1. Chern Forms and Winding Numbers
        1. 22.1a. The Yang–Mills “Winding Number”
        2. 22.1b. Winding Number in Terms of Field Strength
        3. 22.1c. The Chern Forms for a U(n) Bundle
      2. 22.2. Homotopies and Extensions
        1. 22.2a. Homotopy
        2. 22.2b. Covering Homotopy
        3. 22.2c. Some Topology of SU(n)
      3. 22.3. The Higher Homotopy Groups πk (M)
        1. 22.3a. πk(M)
        2. 22.3b. Homotopy Groups of Spheres
        3. 22.3c. Exact Sequences of Groups
        4. 22.3d. The Homotopy Sequence of a Bundle
        5. 22.3e. The Relation Between Homotopy and Homology Groups
      4. 22.4. Some Computations of Homotopy Groups
        1. 22.4a. Lifting Spheres from M into the Bundle P
        2. 22.4b. SU(n) Again
        3. 22.4c. The Hopf Map and Fibering
      5. 22.5. Chern Forms as Obstructions
        1. 22.5a. The Chern Forms cr for an SU(n) Bundle Revisited
        2. 22.5b. c2 as an “Obstruction Cocycle”
        3. 22.5c. The Meaning of the Integer j (Δ4)
        4. 22.5d. Chern’s Integral
        5. 22.5e. Concluding Remarks
  14. Appendix A. Forms in Continuum Mechanics
    1. A.a. The Equations of Motion of a Stressed Body
    2. A.b. Stresses are Vector Valued (n − 1) Pseudo-Forms
    3. A.c. The Piola–Kirchhoff Stress Tensors S and P
    4. A.d. Strain Energy Rate
    5. A.e. Some Typical Computations Using Forms
    6. A.f. Concluding Remarks
  15. Appendix B. Harmonic Chains and Kirchhoff’s Circuit Laws
    1. B.a. Chain Complexes
    2. B.b. Cochains and Cohomology
    3. B.c. Transpose and Adjoint
    4. B.d. Laplacians and Harmonic Cochains
    5. B.e. Kirchhoff’s Circuit Laws
  16. Appendix C. Symmetries, Quarks, and Meson Masses
    1. C.a. Flavored Quarks
    2. C.b. Interactions of Quarks and Antiquarks
    3. C.c. The Lie Algebra of SU(3)
    4. C.d. Pions, Kaons, and Etas
    5. C.e. A Reduced Symmetry Group
    6. C.f. Meson Masses
  17. Appendix D. Representations and Hyperelastic Bodies
    1. D.a. Hyperelastic Bodies
    2. D.b. Isotropic Bodies
    3. D.c. Application of Schur’s Lemma
    4. D.d. Frobenius–Schur Relations
    5. D.e. The Symmetric Traceless 3 × 3 Matrices Are Irreducible
  18. Appendix E. Orbits and Morse–Bott Theory in Compact Lie Groups
    1. E.a. The Topology of Conjugacy Orbits
    2. E.b. Application of Bott’s Extension of Morse Theory
  19. References
  20. Index