Details

An Introduction to Mathematical Modeling


An Introduction to Mathematical Modeling

A Course in Mechanics
Wiley Series in Computational Mechanics, Band 40 1. Aufl.

von: J. Tinsley Oden

116,99 €

Verlag: Wiley
Format: EPUB
Veröffentl.: 23.02.2012
ISBN/EAN: 9781118105740
Sprache: englisch
Anzahl Seiten: 348

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Beschreibungen

<p>A modern approach to mathematical modeling, featuring unique applications from the field of mechanics</p> <p>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.</p> <p>The author streamlines a comprehensive understanding of the topic in three clearly organized sections:</p> <ul> <li> <p>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</p> </li> <li> <p>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</p> </li> <li> <p>Statistical Mechanics presents an introduction to statistical mechanics of systems in thermodynamic equilibrium as well as continuum mechanics, quantum mechanics, and molecular dynamics</p> </li> </ul> <p>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.</p> <p>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.</p>
<p>Preface xiii</p> <p><b>I Nonlinear Continuum Mechanics 1</b></p> <p><b>1 Kinematics of Deformable Bodies 3</b></p> <p>1.1 Motion 4</p> <p>1.2 Strain and Deformation Tensors 7</p> <p>1.3 Rates of Motion 10</p> <p>1.4 Rates of Deformation 13</p> <p>1.5 The Piola Transformation 15</p> <p>1.6 The Polar Decomposition Theorem 19</p> <p>1.7 Principal Directions and Invariants of Deformation and Strain 20</p> <p>1.8 The Reynolds' Transport Theorem 23</p> <p><b>2 Mass and Momentum 25</b></p> <p>2.1 Local Forms of the Principle of Conservation of Mass 26</p> <p>2.2 Momentum 28</p> <p><b>3 Force and Stress in Deformable Bodies 29</b></p> <p><b>4 The Principles of Balance of Linear and Angular Momentum 35</b></p> <p>4.1 Cauchy's Theorem: The Cauchy Stress Tensor 36</p> <p>4.2 The Equations of Motion (Linear Momentum) 38</p> <p>4.3 The Equations of Motion Referred to the Reference Configuration: The Piola-Kirchhoff Stress Tensors 40</p> <p>4.4 Power 42</p> <p><b>5 The Principle of Conservation of Energy 45</b></p> <p>5.1 Energy and the Conservation of Energy 45</p> <p>5.2 Local Forms of the Principle of Conservation of Energy 47</p> <p><b>6 Thermodynamics of Continua and the Second Law 49</b></p> <p><b>7 Constitutive Equations 53</b></p> <p>7.1 Rules and Principles for Constitutive Equations 54</p> <p>7.2 Principle of Material Frame Indifference 57</p> <p>7.2.1 Solids 57</p> <p>7.2.2 Fluids 59</p> <p>7.3 The Coleman-Noll Method: Consistency with the Second Law of Thermodynamics 60</p> <p><b>8 Examples and Applications 63</b></p> <p>8.1 The Navier-Stokes Equations for Incompressible Flow 63</p> <p>8.2 Flow of Gases and Compressible Fluids: The Compressible Navier-Stokes Equations 66</p> <p>8.3 Heat Conduction 67</p> <p>8.4 Theory of Elasticity 69</p> <p><b>II Electromagnetic Field Theory and Quantum Mechanics 73</b></p> <p><b>9 Electromagnetic Waves 75</b></p> <p>9.1 Introduction 75</p> <p>9.2 Electric Fields 75</p> <p>9.3 Gauss's Law 79</p> <p>9.4 Electric Potential Energy 80</p> <p>9.4.1 Atom Models 80</p> <p>9.5 Magnetic Fields 81</p> <p>9.6 Some Properties of Waves 84</p> <p>9.7 Maxwell's Equations 87</p> <p>9.8 Electromagnetic Waves 91</p> <p><b>10 Introduction to Quantum Mechanics 93</b></p> <p>10.1 Introductory Comments 93</p> <p>10.2 Wave and Particle Mechanics 94</p> <p>10.3 Heisenberg's Uncertainty Principle 97</p> <p>10.4 Schrödinger's Equation 99</p> <p>10.4.1 The Case of a Free Particle 99</p> <p>10.4.2 Superposition in Rn 101</p> <p>10.4.3 Hamiltonian Form 102</p> <p>10.4.4 The Case of Potential Energy 102</p> <p>10.4.5 Relativistic Quantum Mechanics 102</p> <p>10.4.6 General Formulations of Schrödinger's Equation 103</p> <p>10.4.7 The Time-Independent Schrödinger Equation 104</p> <p>10.5 Elementary Properties of the Wave Equation 104</p> <p>10.5.1 Review 104</p> <p>10.5.2 Momentum 106</p> <p>10.5.3 Wave Packets and Fourier Transforms 109</p> <p>10.6 The Wave-Momentum Duality 110</p> <p>10.7 Appendix: A Brief Review of Probability Densities 111</p> <p><b>11 Dynamical Variables and Observables in Quantum Mechanics: The Mathematical Formalism 115</b></p> <p>11.1 Introductory Remarks 115</p> <p>11.2 The Hilbert Spaces L<sup>2</sup>(R) (or L<sup>2</sup>(R<sup>d</sup>)) and H<sup>1</sup>(R) (or H<sup>1</sup>(R<sup>d</sup>)) 116</p> <p>11.3 Dynamical Variables and Hermitian Operators 118</p> <p>11.4 Spectral Theory of Hermitian Operators: The Discrete Spectrum 121</p> <p>11.5 Observables and Statistical Distributions 125</p> <p>11.6 The Continuous Spectrum 127</p> <p>11.7 The Generalized Uncertainty Principle for Dynamical Variables 128</p> <p>11.7.1 Simultaneous Eigenfunctions 130</p> <p><b>12 Applications: The Harmonic Oscillator and the Hydrogen Atom 131</b></p> <p>12.1 Introductory Remarks 131</p> <p>12.2 Ground States and Energy Quanta: The Harmonic Oscillator 131</p> <p>12.3 The Hydrogen Atom 133</p> <p>12.3.1 Schrödinger Equation in Spherical Coordinates 135</p> <p>12.3.2 The Radial Equation 136</p> <p>12.3.3 The Angular Equation 138</p> <p>12.3.4 The Orbitals of the Hydrogen Atom 140</p> <p>12.3.5 Spectroscopic States 140</p> <p><b>13 Spin and Pauli's Principle 145</b></p> <p>13.1 Angular Momentum and Spin 145</p> <p>13.2 Extrinsic Angular Momentum 147</p> <p>13.2.1 The Ladder Property: Raising and Lowering States 149</p> <p>13.3 Spin 151</p> <p>13.4 Identical Particles and Pauli's Principle 155</p> <p>13.5 The Helium Atom 158</p> <p>13.6 Variational Principle 161</p> <p><b>14 Atomic and Molecular Structure 165</b></p> <p>14.1 Introduction 165</p> <p>14.2 Electronic Structure of Atomic Elements 165</p> <p>14.3 The Periodic Table 169</p> <p>14.4 Atomic Bonds and Molecules 173</p> <p>14.5 Examples of Molecular Structures 180</p> <p><b>15 Ab Initio Methods: Approximate Methods and Density Functional Theory 189</b></p> <p>15.1 Introduction 189</p> <p>15.2 The Born-Oppenheimer Approximation 190</p> <p>15.3 The Hartree and the Hartree-Fock Methods 194</p> <p>15.3.1 The Hartree Method 196</p> <p>15.3.2 The Hartree-Fock Method 196</p> <p>15.3.3 The Roothaan Equations 199</p> <p>15.4 Density Functional Theory 200</p> <p>15.4.1 Electron Density 200</p> <p>15.4.2 The Hohenberg-Kohn Theorem 205</p> <p>15.4.3 The Kohn-Sham Theory 208</p> <p><b>III Statistical Mechanics 213</b></p> <p><b>16 Basic Concepts: Ensembles, Distribution Functions, and Averages 215</b></p> <p>16.1 Introductory Remarks 215</p> <p>16.2 Hamiltonian Mechanics 216</p> <p>16.2.1 The Hamiltonian and the Equations of Motion 218</p> <p>16.3 Phase Functions and Time Averages 219</p> <p>16.4 Ensembles, Ensemble Averages, and Ergodic Systems 220</p> <p>16.5 Statistical Mechanics of Isolated Systems 224</p> <p>16.6 The Microcanonical Ensemble 228</p> <p>16.6.1 Composite Systems 230</p> <p>16.7 The Canonical Ensemble 234</p> <p>16.8 The Grand Canonical Ensemble 239</p> <p>16.9 Appendix: A Brief Account of Molecular Dynamics 240</p> <p>16.9.1 Newtonian's Equations of Motion 241</p> <p>16.9.2 Potential Functions 242</p> <p>16.9.3 Numerical Solution of the Dynamical System 245</p> <p><b>17 Statistical Mechanics Basis of Classical Thermodynamics 249</b></p> <p>17.1 Introductory Remarks 249</p> <p>17.2 Energy and the First Law of Thermodynamics 250</p> <p>17.3 Statistical Mechanics Interpretation of the Rate of Work in Quasi-Static Processes 251</p> <p>17.4 Statistical Mechanics Interpretation of the First Law of Thermodynamics 254</p> <p>17.4.1 Statistical Interpretation of Q 256</p> <p>17.5 Entropy and the Partition Function 257</p> <p>17.6 Conjugate Hamiltonians 259</p> <p>17.7 The Gibbs Relations 261</p> <p>17.8 Monte Carlo and Metropolis Methods 262</p> <p>17.8.1 The Partition Function for a Canonical Ensemble 263</p> <p>17.8.2 The Metropolis Method 264</p> <p>17.9 Kinetic Theory: Boltzmann's Equation of Nonequilibrium Statistical Mechanics 265</p> <p>17.9.1 Boltzmann's Equation 265</p> <p>17.9.2 Collision Invariants 268</p> <p>17.9.3 The Continuum Mechanics of Compressible Fluids and Gases: The Macroscopic Balance Laws 269</p> <p>Exercises 273</p> <p>Bibliography 317</p> <p>Index 325</p>
<p>“The book also serves as a valuable reference for professionals working in the areas of modeling and simulation, physics, and computational engineering.” (<i>Zentralblatt MATH</i>, 2012)</p>
<b>John Tinsley Oden</b>, 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.
<p><b>A modern approach to mathematical modeling, featuring unique applications from the field of mechanics</b></p> <p><i>An Introduction to Mathematical Modeling: A Course in Mechanics</i> 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.</p> <p>The author streamlines a comprehensive understanding of the topic in three clearly organized sections:</p> <ul> <li> <p>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</p> </li> <li> <p>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</p> </li> <li> <p>Statistical Mechanics presents an introduction to statistical mechanics of systems in thermodynamic equilibrium as well as continuum mechanics, quantum mechanics, and molecular dynamics</p> </li> </ul> <p>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.</p> <p>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.</p>

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