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Matrices, Vectors, and Vector Calculus | |

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Introduction | |

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Concept of a Scalar | |

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Coordinate Transformations | |

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Properties of Rotation Matrices | |

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Matrix Operations | |

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Further Definitions | |

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Geometrical Significance of Transformation Matrices | |

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Definitions of a Scalar and a Vector in Terms of Transformation Properties | |

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Elementary Scalar and Vector Operations | |

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Scalar Product of Two Vectors | |

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Unit Vectors | |

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Vector Product of Two Vectors | |

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Differentiation of a Vector with Respect to a Scalar | |

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Examples of Derivatives--Velocity and Acceleration | |

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Angular Velocity | |

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Gradient Operator | |

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Integration of Vectors | |

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Problems | |

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Newtonian Mechanics--Single Particle | |

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Introduction | |

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Newton's Laws | |

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Frames of Reference | |

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The Equation of Motion for a Particle | |

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Conservation Theorems | |

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Energy | |

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Limitations of Newtonian Mechanics | |

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Problems | |

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Oscillations | |

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Introduction | |

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Simple Harmonic Oscillator | |

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Harmonic Oscillations in Two Dimensions | |

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Phase Diagrams | |

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Damped Oscillations | |

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Sinusoidal Driving Forces | |

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Physical Systems | |

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Principle of Superposition--Fourier Series | |

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The Response of Linear Oscillators to Impulsive Forcing Functions (Optional) | |

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Problems | |

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Nonlinear Oscillations and Chaos | |

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Introduction | |

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Nonlinear Oscillations | |

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Phase Diagrams for Nonlinear Systems | |

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Plane Pendulum | |

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Jumps, Hysteresis, and Phase Lags | |

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Chaos in a Pendulum | |

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Mapping | |

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Chaos Identification | |

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Problems | |

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Gravitation | |

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Introduction | |

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Gravitational Potential | |

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Lines of Force and Equipotential Surfaces | |

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When Is the Potential Concept Useful? | |

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Ocean Tides | |

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Problems | |

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Some Methods in the Calculus of Variations | |

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Introduction | |

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Statement of the Problem | |

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Euler's Equation | |

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The "Second Form" of the Euler Equation | |

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Functions with Several Dependent Variables | |

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Euler Equations When Auxiliary Conditions Are Imposed | |

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The [delta] Notation | |

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Problems | |

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Hamilton's Principle--Lagrangian and Hamiltonian Dynamics | |

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Introduction | |

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Hamilton's Principle | |

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Generalized Coordinates | |

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Lagrange's Equations of Motion in Generalized Coordinates | |

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Lagrange's Equations with Undetermined Multipliers | |

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Equivalence of Lagrange's and Newton's Equations | |

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Essence of Lagrangian Dynamics | |

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A Theorem Concerning the Kinetic Energy | |

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Conservation Theorems Revisited | |

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Canonical Equations of Motion--Hamiltonian Dynamics | |

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Some Comments Regarding Dynamical Variables and Variational Calculations in Physics | |

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Phase Space and Liouville's Theorem (Optional) | |

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Virial Theorem (Optional) | |

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Problems | |

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Central-Force Motion | |

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Introduction | |

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Reduced Mass | |

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Conservation Theorems--First Integrals of the Motion | |

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Equations of Motion | |

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Orbits in a Central Field | |

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Centrifugal Energy and the Effective Potential | |

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Planetary Motion--Kepler's Problem | |

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Orbital Dynamics | |

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Apsidal Angles and Precession (Optional) | |

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Stability of Circular Orbits (Optional) | |

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Problems | |

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Dynamics of a System of Particles | |

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Introduction | |

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Center of Mass | |

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Linear Momentum of the System | |

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Angular Momentum of the System | |

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Energy of the System | |

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Elastic Collisions of Two Particles | |

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Kinematics of Elastic Collisions | |

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Inelastic Collisions | |

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Scattering Cross Sections | |

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Rutherford Scattering Formula | |

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Rocket Motion | |

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Problems | |

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Motion in a Nonintertial Reference Frame | |

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Introduction | |

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Rotating Coordinate Systems | |

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Centrifugal and Coriolis Forces | |

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Motion Relative to the Earth | |

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Problems | |

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Dynamics of Rigid Bodies | |

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Introduction | |

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Simple Planar Motion | |

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Inertia Tensor | |

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Angular Momentum | |

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Principal Axes of Inertia | |

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Moments of Inertia for Different Body Coordinate Systems | |

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Further Properties of the Inertia Tensor | |

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Eulerian Angles | |

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Euler's Equations for a Rigid Body | |

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Force-Free Motion of a Symmetric Top | |

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Motion of a Symmetric Top with One Point Fixed | |

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Stability of Rigid-Body Rotations | |

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Problems | |

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Coupled Oscillations | |

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Introduction | |

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Two Coupled Harmonic Oscillators | |

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Weak Coupling | |

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General Problem of Coupled Oscillations | |

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Orthogonality of the Eigenvectors (Optional) | |

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Normal Coordinates | |

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Molecular Vibrations | |

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Three Linearly Coupled Plane Pendula--an Example of Degeneracy | |

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The Loaded String | |

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Problems | |

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Continuous Systems; Waves | |

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Introduction | |

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Continuous String as a Limiting Case of the Loaded String | |

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Energy of a Vibrating String | |

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Wave Equation | |

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Forced and Damped Motion | |

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General Solutions of the Wave Equation | |

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Separation of the Wave Equation | |

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Phase Velocity, Dispersion, and Attenuation | |

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Group Velocity and Wave Packets | |

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Problems | |

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Special Theory of Relativity | |

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Introduction | |

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Galilean Invariance | |

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Lorentz Transformation | |

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Experimental Verification of the Special Theory | |

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Relativistic Doppler Effect | |

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Twin Paradox | |

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Relativistic Momentum | |

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Energy | |

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Spacetime and Four-Vectors | |

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Lagrangian Function in Special Relativity | |

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Relativistic Kinematics | |

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Problems | |

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Appendices | |

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Taylor's Theorem | |

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Problems | |

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Elliptic Integrals | |

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Elliptic Integrals of the First Kind | |

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Elliptic Integrals of the Second Kind | |

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Elliptic Integrals of the Third Kind | |

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Problems | |

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Ordinary Differential Equations of Second Order | |

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Linear Homogeneous Equations | |

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Linear Inhomogeneous Equations | |

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Problems | |

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Useful Formulas | |

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Binomial Expansion | |

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Trigonometric Relations | |

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Trigonometric Series | |

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Exponential and Logarithmic Series | |

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Complex Quantities | |

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Hyperbolic Functions | |

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Problems | |

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Useful Integrals | |

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Algebraic Functions | |

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Trigonometric Functions | |

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Gamma Functions | |

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Differential Relations in Different Coordinate Systems | |

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Rectangular Coordinates | |

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Cylindrical Coordinates | |

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Spherical Coordinates | |

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A "Proof" of the Relation [characters not reproducible] = [characters not reproducible] | |

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Numerical Solution for Example 2.7 | |

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Selected References | |

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Bibliography | |

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Answers to Even-Numbered Problems | |

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Index | |