A modern approach to mathematical modeling, featuring unique applications from the field of mechanics
An Introduction to Mathematical Modeling: A Course in Mechanics is designed to survey the mathematical models that form the foundations of modern science and incorporates examples that illustrate how the most successful models arise from basic principles in modern and classical mathematical physics. Written by a world authority on mathematical theory and computational mechanics, the book presents an account of continuum mechanics, electromagnetic field theory, quantum mechanics, and statistical mechanics for readers with varied backgrounds in engineering, computer science, mathematics, and physics.
The author streamlines a comprehensive understanding of the topic in three clearly organized sections:
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Nonlinear Continuum Mechanics introduces kinematics as well as force and stress in deformable bodies; mass and momentum; balance of linear and angular momentum; conservation of energy; and constitutive equations
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Electromagnetic Field Theory and Quantum Mechanics contains a brief account of electromagnetic wave theory and Maxwell's equations as well as an introductory account of quantum mechanics with related topics including ab initio methods and Spin and Pauli's principles
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Statistical Mechanics presents an introduction to statistical mechanics of systems in thermodynamic equilibrium as well as continuum mechanics, quantum mechanics, and molecular dynamics
Each part of the book concludes with exercise sets that allow readers to test their understanding of the presented material. Key theorems and fundamental equations are highlighted throughout, and an extensive bibliography outlines resources for further study.
Extensively class-tested to ensure an accessible presentation, An Introduction to Mathematical Modeling is an excellent book for courses on introductory mathematical modeling and statistical mechanics at the upper-undergraduate and graduate levels. The book also serves as a valuable reference for professionals working in the areas of modeling and simulation, physics, and computational engineering.
Table of Contents:
Preface xiii
I Nonlinear Continuum Mechanics 1
1 Kinematics of Deformable Bodies 3
1.1 Motion 4
1.2 Strain and Deformation Tensors 7
1.3 Rates of Motion 10
1.4 Rates of Deformation 13
1.5 The Piola Transformation 15
1.6 The Polar Decomposition Theorem 19
1.7 Principal Directions and Invariants of Deformation and Strain 20
1.8 The Reynolds' Transport Theorem 23
2 Mass and Momentum 25
2.1 Local Forms of the Principle of Conservation of Mass 26
2.2 Momentum 28
3 Force and Stress in Deformable Bodies 29
4 The Principles of Balance of Linear and Angular Momentum 35
4.1 Cauchy's Theorem: The Cauchy Stress Tensor 36
4.2 The Equations of Motion (Linear Momentum) 38
4.3 The Equations of Motion Referred to the Reference Configuration: The Piola-Kirchhoff Stress Tensors 40
4.4 Power 42
5 The Principle of Conservation of Energy 45
5.1 Energy and the Conservation of Energy 45
5.2 Local Forms of the Principle of Conservation of Energy 47
6 Thermodynamics of Continua and the Second Law 49
7 Constitutive Equations 53
7.1 Rules and Principles for Constitutive Equations 54
7.2 Principle of Material Frame Indifference 57
7.2.1 Solids 57
7.2.2 Fluids 59
7.3 The Coleman-Noll Method: Consistency with the Second Law of Thermodynamics 60
8 Examples and Applications 63
8.1 The Navier-Stokes Equations for Incompressible Flow 63
8.2 Flow of Gases and Compressible Fluids: The Compressible Navier-Stokes Equations 66
8.3 Heat Conduction 67
8.4 Theory of Elasticity 69
II Electromagnetic Field Theory and Quantum Mechanics 73
9 Electromagnetic Waves 75
9.1 Introduction 75
9.2 Electric Fields 75
9.3 Gauss's Law 79
9.4 Electric Potential Energy 80
9.4.1 Atom Models 80
9.5 Magnetic Fields 81
9.6 Some Properties of Waves 84
9.7 Maxwell's Equations 87
9.8 Electromagnetic Waves 91
10 Introduction to Quantum Mechanics 93
10.1 Introductory Comments 93
10.2 Wave and Particle Mechanics 94
10.3 Heisenberg's Uncertainty Principle 97
10.4 Schrödinger's Equation 99
10.4.1 The Case of a Free Particle 99
10.4.2 Superposition in Rn 101
10.4.3 Hamiltonian Form 102
10.4.4 The Case of Potential Energy 102
10.4.5 Relativistic Quantum Mechanics 102
10.4.6 General Formulations of Schrödinger's Equation 103
10.4.7 The Time-Independent Schrödinger Equation 104
10.5 Elementary Properties of the Wave Equation 104
10.5.1 Review 104
10.5.2 Momentum 106
10.5.3 Wave Packets and Fourier Transforms 109
10.6 The Wave-Momentum Duality 110
10.7 Appendix: A Brief Review of Probability Densities 111
11 Dynamical Variables and Observables in Quantum Mechanics: The Mathematical Formalism 115
11.1 Introductory Remarks 115
11.2 The Hilbert Spaces L2(R) (or L2(Rd)) and H1(R) (or H1(Rd)) 116
11.3 Dynamical Variables and Hermitian Operators 118
11.4 Spectral Theory of Hermitian Operators: The Discrete Spectrum 121
11.5 Observables and Statistical Distributions 125
11.6 The Continuous Spectrum 127
11.7 The Generalized Uncertainty Principle for Dynamical Variables 128
11.7.1 Simultaneous Eigenfunctions 130
12 Applications: The Harmonic Oscillator and the Hydrogen Atom 131
12.1 Introductory Remarks 131
12.2 Ground States and Energy Quanta: The Harmonic Oscillator 131
12.3 The Hydrogen Atom 133
12.3.1 Schrödinger Equation in Spherical Coordinates 135
12.3.2 The Radial Equation 136
12.3.3 The Angular Equation 138
12.3.4 The Orbitals of the Hydrogen Atom 140
12.3.5 Spectroscopic States 140
13 Spin and Pauli's Principle 145
13.1 Angular Momentum and Spin 145
13.2 Extrinsic Angular Momentum 147
13.2.1 The Ladder Property: Raising and Lowering States 149
13.3 Spin 151
13.4 Identical Particles and Pauli's Principle 155
13.5 The Helium Atom 158
13.6 Variational Principle 161
14 Atomic and Molecular Structure 165
14.1 Introduction 165
14.2 Electronic Structure of Atomic Elements 165
14.3 The Periodic Table 169
14.4 Atomic Bonds and Molecules 173
14.5 Examples of Molecular Structures 180
15 Ab Initio Methods: Approximate Methods and Density Functional Theory 189
15.1 Introduction 189
15.2 The Born-Oppenheimer Approximation 190
15.3 The Hartree and the Hartree-Fock Methods 194
15.3.1 The Hartree Method 196
15.3.2 The Hartree-Fock Method 196
15.3.3 The Roothaan Equations 199
15.4 Density Functional Theory 200
15.4.1 Electron Density 200
15.4.2 The Hohenberg-Kohn Theorem 205
15.4.3 The Kohn-Sham Theory 208
III Statistical Mechanics 213
16 Basic Concepts: Ensembles, Distribution Functions, and Averages 215
16.1 Introductory Remarks 215
16.2 Hamiltonian Mechanics 216
16.2.1 The Hamiltonian and the Equations of Motion 218
16.3 Phase Functions and Time Averages 219
16.4 Ensembles, Ensemble Averages, and Ergodic Systems 220
16.5 Statistical Mechanics of Isolated Systems 224
16.6 The Microcanonical Ensemble 228
16.6.1 Composite Systems 230
16.7 The Canonical Ensemble 234
16.8 The Grand Canonical Ensemble 239
16.9 Appendix: A Brief Account of Molecular Dynamics 240
16.9.1 Newtonian's Equations of Motion 241
16.9.2 Potential Functions 242
16.9.3 Numerical Solution of the Dynamical System 245
17 Statistical Mechanics Basis of Classical Thermodynamics 249
17.1 Introductory Remarks 249
17.2 Energy and the First Law of Thermodynamics 250
17.3 Statistical Mechanics Interpretation of the Rate of Work in Quasi-Static Processes 251
17.4 Statistical Mechanics Interpretation of the First Law of Thermodynamics 254
17.4.1 Statistical Interpretation of Q 256
17.5 Entropy and the Partition Function 257
17.6 Conjugate Hamiltonians 259
17.7 The Gibbs Relations 261
17.8 Monte Carlo and Metropolis Methods 262
17.8.1 The Partition Function for a Canonical Ensemble 263
17.8.2 The Metropolis Method 264
17.9 Kinetic Theory: Boltzmann's Equation of Nonequilibrium Statistical Mechanics 265
17.9.1 Boltzmann's Equation 265
17.9.2 Collision Invariants 268
17.9.3 The Continuum Mechanics of Compressible Fluids and Gases: The Macroscopic Balance Laws 269
Exercises 273
Bibliography 317
Index 325
About the Author :
John Tinsley Oden, PhD, is Associate Vice President for Research and Director of the Institute for Computational Engineering and Sciences (ICES) at The University of Texas at Austin. He was the founding Director of the Institute, which was created in January of 2003 as an expansion of the Texas Institute for Computational and Applied Mathematics. A member of the U.S. National Academy of Engineering, the National Academies of Engineering of Mexico and of Brazil, and The American Academy of Arts and Sciences, he serves on numerous national and international organizational, scientific, and advisory committees including the NSF Blue Ribbon Panel on Simulation-Based Engineering Science and the Task Force on Cyber Science and Grand Challenge Communities and Virtual Organizations. Dr. Oden has worked extensively on the mathematical theory and implementation of numerical methods applied to problems in solid and fluid mechanics and, particularly, nonlinear continuum mechanics and, in recent years, multi-scale modeling, stochastic systems, and uncertainty quantification.
Review :
“The book also serves as a valuable reference for professionals working in the areas of modeling and simulation, physics, and computational engineering.” (Zentralblatt MATH, 2012)
