An Introduction to Mathematical Modeling: A Course in Mechanics -- Contents -- Preface -- I Nonlinear Continuum Mechanics -- 1 Kinematics of Deformable Bodies -- 1.1 Motion -- 1.2 Strain and Deformation Tensors -- 1.3 Rates of Motion -- 1.4 Rates of Deformation -- 1.5 The Piola Transformation -- 1.6 The Polar Decomposition Theorem -- 1.7 Principal Directions and Invariants of Deformation and Strain -- 1.8 The Reynolds' Transport Theorem -- 2 Mass and Momentum -- 2.1 Local Forms of the Principle of Conservation of Mass -- 2.2 Momentum -- 3 Force and Stress in Deformable Bodies -- 4 The Principles of Balance of Linear and Angular Momentum -- 4.1 Cauchy's Theorem: The Cauchy Stress Tensor -- 4.2 The Equations of Motion (Linear Momentum) -- 4.3 The Equations of Motion Referred to the Reference Configuration: The Piola-Kirchhoff Stress Tensors -- 4.4 Power -- 5 The Principle of Conservation of Energy -- 5.1 Energy and the Conservation of Energy -- 5.2 Local Forms of the Principle of Conservation of Energy -- 6 Thermodynamics of Continua and the Second Law -- 7 Constitutive Equations -- 7.1 Rules and Principles for Constitutive Equations -- 7.2 Principle of Material Frame Indifference -- 7.2.1 Solids -- 7.2.2 Fluids -- 7.3 The Coleman-Noll Method: Consistency with the Second Law of Thermodynamics -- 8 Examples and Applications -- 8.1 The Navier-Stokes Equations for Incompressible Flow -- 8.2 Flow of Gases and Compressible Fluids: The Compressible Navier-Stokes Equations -- 8.3 Heat Conduction -- 8.4 Theory of Elasticity -- II Electromagnetic Field Theory and Quantum Mechanics -- 9 Electromagnetic Waves -- 9.1 Introduction -- 9.2 Electric Fields -- 9.3 Gauss's Law -- 9.4 Electric Potential Energy -- 9.4.1 Atom Models -- 9.5 Magnetic Fields -- 9.6 Some Properties of Waves -- 9.7 Maxwell's Equations -- 9.8 Electromagnetic Waves.

10 Introduction to Quantum Mechanics -- 10.1 Introductory Comments -- 10.2 Wave and Particle Mechanics -- 10.3 Heisenberg's Uncertainty Principle -- 10.4 Schrödinger's Equation -- 10.4.1 The Case of a Free Particle -- 10.4.2 Superposition in Rn -- 10.4.3 Hamiltonian Form -- 10.4.4 The Case of Potential Energy -- 10.4.5 Relativistic Quantum Mechanics -- 10.4.6 General Formulations of Schrödinger's Equation -- 10.4.7 The Time-Independent Schrödinger Equation -- 10.5 Elementary Properties of the Wave Equation -- 10.5.1 Review -- 10.5.2 Momentum -- 10.5.3 Wave Packets and Fourier Transforms -- 10.6 The Wave-Momentum Duality -- 10.7 Appendix: A Brief Review of Probability Densities -- 11 Dynamical Variables and Observables in Quantum Mechanics: The Mathematical Formalism -- 11.1 Introductory Remarks -- 11.2 The Hilbert Spaces L2(R) (or L2(Rd)) and H1(R) (or H1(Rd)) -- 11.3 Dynamical Variables and Hermitian Operators -- 11.4 Spectral Theory of Hermitian Operators: The Discrete Spectrum -- 11.5 Observables and Statistical Distributions -- 11.6 The Continuous Spectrum -- 11.7 The Generalized Uncertainty Principle for Dynamical Variables -- 11.7.1 Simultaneous Eigenfunctions -- 12 Applications: The Harmonic Oscillator and the Hydrogen Atom -- 12.1 Introductory Remarks -- 12.2 Ground States and Energy Quanta: The Harmonic Oscillator -- 12.3 The Hydrogen Atom -- 12.3.1 Schrödinger Equation in Spherical Coordinates -- 12.3.2 The Radial Equation -- 12.3.3 The Angular Equation -- 12.3.4 The Orbitals of the Hydrogen Atom -- 12.3.5 Spectroscopic States -- 13 Spin and Pauli's Principle -- 13.1 Angular Momentum and Spin -- 13.2 Extrinsic Angular Momentum -- 13.2.1 The Ladder Property: Raising and Lowering States -- 13.3 Spin -- 13.4 Identical Particles and Pauli's Principle -- 13.5 The Helium Atom -- 13.6 Variational Principle -- 14 Atomic and Molecular Structure.

14.1 Introduction -- 14.2 Electronic Structure of Atomic Elements -- 14.3 The Periodic Table -- 14.4 Atomic Bonds and Molecules -- 14.5 Examples of Molecular Structures -- 15 Ab Initio Methods: Approximate Methods and Density Functional Theory -- 15.1 Introduction -- 15.2 The Born-Oppenheimer Approximation -- 15.3 The Hartree and the Hartree-Fock Methods -- 15.3.1 The Hartree Method -- 15.3.2 The Hartree-Fock Method -- 15.3.3 The Roothaan Equations -- 15.4 Density Functional Theory -- 15.4.1 Electron Density -- 15.4.2 The Hohenberg-Kohn Theorem -- 15.4.3 The Kohn-Sham Theory -- III Statistical Mechanics -- 16 Basic Concepts: Ensembles, Distribution Functions, and Averages -- 16.1 Introductory Remarks -- 16.2 Hamiltonian Mechanics -- 16.2.1 The Hamiltonian and the Equations of Motion -- 16.3 Phase Functions and Time Averages -- 16.4 Ensembles, Ensemble Averages, and Ergodic Systems -- 16.5 Statistical Mechanics of Isolated Systems -- 16.6 The Microcanonical Ensemble -- 16.6.1 Composite Systems -- 16.7 The Canonical Ensemble -- 16.8 The Grand Canonical Ensemble -- 16.9 Appendix: A Brief Account of Molecular Dynamics -- 16.9.1 Newtonian's Equations of Motion -- 16.9.2 Potential Functions -- 16.9.3 Numerical Solution of the Dynamical System -- 17 Statistical Mechanics Basis of Classical Thermodynamics -- 17.1 Introductory Remarks -- 17.2 Energy and the First Law of Thermodynamics -- 17.3 Statistical Mechanics Interpretation of the Rate of Work in Quasi-Static Processes -- 17.4 Statistical Mechanics Interpretation of the First Law of Thermodynamics -- 17.4.1 Statistical Interpretation of Q -- 17.5 Entropy and the Partition Function -- 17.6 Conjugate Hamiltonians -- 17.7 The Gibbs Relations -- 17.8 Monte Carlo and Metropolis Methods -- 17.8.1 The Partition Function for a Canonical Ensemble -- 17.8.2 The Metropolis Method.

17.9 Kinetic Theory: Boltzmann's Equation of Nonequilibrium Statistical Mechanics -- 17.9.1 Boltzmann's Equation -- 17.9.2 Collision Invariants -- 17.9.3 The Continuum Mechanics of Compressible Fluids and Gases: The Macroscopic Balance Laws -- Exercises -- Bibliography -- Index.