Physics of Continuous Matter: Exotic and Everyday Phenomena in the Macroscopic World, 2nd Edition (Hardback) book cover

Physics of Continuous Matter

Exotic and Everyday Phenomena in the Macroscopic World, 2nd Edition

By B. Lautrup

CRC Press

696 pages | 135 B/W Illus.

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Physics of Continuous Matter: Exotic and Everyday Phenomena in the Macroscopic World, Second Edition provides an introduction to the basic ideas of continuum physics and their application to a wealth of macroscopic phenomena. The text focuses on the many approximate methods that offer insight into the rich physics hidden in fundamental continuum mechanics equations. Like its acclaimed predecessor, this second edition introduces mathematical tools on a "need-to-know" basis.

New to the Second Edition

This edition includes three new chapters on elasticity of slender rods, energy, and entropy. It also offers more margin drawings and photographs and improved images of simulations. Along with reorganizing much of the material, the author has revised many of the physics arguments and mathematical presentations to improve clarity and consistency. The collection of problems at the end of each chapter has been expanded as well. These problems further develop the physical and mathematical concepts presented.

With worked examples throughout, this book clearly illustrates both qualitative and quantitative physics reasoning. It emphasizes the importance in understanding the physical principles behind equations and the conditions underlying approximations. A companion website provides a host of ancillary materials, including software programs, color figures, and additional problems.

Reviews

"With its elegant presentation and comprehensive treatment of the subject, Physics of Continuous Matter does a fantastic job of illustrating how the physics of the classical world around us is profound, beautiful, and often counterintuitive."

—Sujit S. Datta, Pure and Applied Geophysics, 170 (2013)

"I completely agree with the reviewer of the first edition that this book provides an excellent, modern introduction to the field of continuum mechanics. The second edition has been streamlined, and the structure of the presentation has been improved. … on its best way to become a classic text in the field. The text is exceptionally clear and well structured, and the breadth of the fields from which the author chooses his illustrating examples is impressive. … I can warmly recommend this book to everyone with an interest in continuum mechanics, lecturers and students alike. Lecturers will find various historical anecdotes, innumerable examples and applications, and a modern account of almost all basic aspects of continuum mechanics that will provide an excellent foundation for a lecture course on this subject. Students can benefit from the author’s deep physical insight into many difficult problems as well as his mastery of mathematical analysis."

—Thomas Peters, Contemporary Physics, January 2013

Praise for the First Edition:

"… this book satisfies with great style. Although it starts from the very beginning of the subject, it also reaches advanced topics, but without discontinuities along the way. … A good introductory course could be based on this material. …The emphasis is on understanding the problems and obtaining analytical solutions, but there are two chapters on computational methods, for static elasticity and for fluid dynamics. … This is an excellent text, which ought to inspire students and teachers alike with the richness of behaviour that is contained within a few continuum equations – equations that are easy to derive but often far from easy to solve. The subject may have its roots in the nineteenth century, but this book shows that it is still alive, relevant and challenging in the twenty-first."

—Tony Harker, Department of Physics and Astronomy, University College London, Physical Sciences Educational Reviews, Vol. 7, Issue 1, May 2006

"…a superb text on continuum theory and applications with absolutely outstanding graphics. The graphs and figures in the side panels I expect will be extremely helpful…The book is at the same time introductory and advanced, and the range of topics is exceptionally wide."

—Professor Richard Lovelace, Cornell University, Ithaca, New York, USA

"…perhaps the only modern advanced undergraduate introduction to the subject…put together very elegantly and intelligently, illustrated by many examples from geophysics, astrophysics and other fields…a pleasure to teach. For a student who has already encountered solid and fluid mechanics, the text offers rigor and breadth; nothing is asserted, everything is derived."

—Predrag Cvitanovic, Glen P. Robinson Chair in Nonlinear Sciences, School of Physics, Georgia Institute of Technology

Table of Contents

Contents

Preface xi

1 Continuous matter 1

1.1 Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 The continuum approximation . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Newtonian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Reference frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Cartesian coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.6 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

I Fluids at rest 19

2 Pressure 21

2.1 What is pressure? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 The pressure field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Hydrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5 Bulk modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.6 Application: Earth’s homentropic atmosphere . . . . . . . . . . . . . . . . . 34

2.7 Application: The Sun’s convective envelope . . . . . . . . . . . . . . . . . . 38

3 Buoyancy and stability 41

3.1 Archimedes’ principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 The gentle art of ballooning . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Stability of floating bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4 Ship stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 Hydrostatic shapes 57

4.1 Fluid interfaces in hydrostatic equilibrium . . . . . . . . . . . . . . . . . . . 57

4.2 The centrifugal force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3 The figure of Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4 The Earth, the Moon, and the tides . . . . . . . . . . . . . . . . . . . . . . . 62

4.5 Application: The tides of Io . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5 Surface tension 69

5.1 Basic physics of surface tension . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2 Soap bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3 Pressure discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.4 The Rayleigh–Plateau instability . . . . . . . . . . . . . . . . . . . . . . . . 78

5.5 Contact angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

vi PHYSICS OF CONTINUOUS MATTER

5.6 Meniscus at a flat wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.7 Meniscus in a cylindrical tube . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.8 Application: Sessile drops and captive bubbles . . . . . . . . . . . . . . . . 88

5.9 Application: Pendant drops and tethered bubbles . . . . . . . . . . . . . . . 90

II Solids at rest 95

6 Stress 97

6.1 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2 Stress fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.3 The nine components of stress . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.4 Mechanical equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.5 Asymmetric stress tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7 Strain 109

7.1 Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.2 The displacement field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.3 Geometrical meaning of the strain tensor . . . . . . . . . . . . . . . . . . . 116

7.4 Work and energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.5 Large deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8 Hooke’s law 125

8.1 Young’s modulus and Poisson’s ratio . . . . . . . . . . . . . . . . . . . . . . 125

8.2 Hooke’s law in isotropic matter . . . . . . . . . . . . . . . . . . . . . . . . . 128

8.3 Static uniform deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

8.4 Elastic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

9 Basic elastostatics 139

9.1 Equations of elastostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

9.2 Standing up to gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

9.3 Bending a beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

9.4 Twisting a shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

9.5 Application: Radial deformation of a spherical body . . . . . . . . . . . . . 153

9.6 Application: Radial deformation of a cylindrical body . . . . . . . . . . . . . 156

10 Slender rods 163

10.1 Small deflections without torsion . . . . . . . . . . . . . . . . . . . . . . . . 163

10.2 Buckling instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

10.3 Large deflections without torsion . . . . . . . . . . . . . . . . . . . . . . . . 169

10.4 Mixed bending and twisting . . . . . . . . . . . . . . . . . . . . . . . . . . 171

10.5 Application: The helical spring . . . . . . . . . . . . . . . . . . . . . . . . . 173

11 Computational elastostatics 177

11.1 Theory of the numeric method . . . . . . . . . . . . . . . . . . . . . . . . . 177

11.2 Discretization of space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

11.3 Application: Gravitational settling in two dimensions . . . . . . . . . . . . . 182

III Fluids in motion 187

12 Continuum dynamics 189

12.1 The velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

12.2 Incompressible flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

CONTENTS vii

12.3 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

12.4 Equations of continuum dynamics . . . . . . . . . . . . . . . . . . . . . . . 198

12.5 Application: Big Bang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

12.6 Application: Newtonian cosmology . . . . . . . . . . . . . . . . . . . . . . 201

13 Nearly ideal flow 207

13.1 Euler equation for incompressible ideal flow . . . . . . . . . . . . . . . . . . 207

13.2 Application: Collapse of a spherical cavity . . . . . . . . . . . . . . . . . . 209

13.3 Steady incompressible ideal flow . . . . . . . . . . . . . . . . . . . . . . . . 211

13.4 Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

13.5 Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

13.6 Potential flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

13.7 Application: Cylinder in uniform crosswind . . . . . . . . . . . . . . . . . . 222

13.8 Application: Sphere in a uniform stream . . . . . . . . . . . . . . . . . . . . 225

13.9 d’Alembert’s paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

14 Compressible flow 229

14.1 Small-amplitude sound waves . . . . . . . . . . . . . . . . . . . . . . . . . 229

14.2 Steady compressible flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

14.3 Application: The Laval nozzle . . . . . . . . . . . . . . . . . . . . . . . . . 237

15 Viscosity 243

15.1 Shear viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

15.2 Velocity-driven planar flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

15.3 Dynamics of incompressible Newtonian fluids . . . . . . . . . . . . . . . . . 250

15.4 Classification of flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

15.5 Dynamics of compressible Newtonian fluids . . . . . . . . . . . . . . . . . . 255

15.6 Application: Viscous attenuation of sound . . . . . . . . . . . . . . . . . . . 257

16 Channels and pipes 261

16.1 Steady, incompressible, viscous flow . . . . . . . . . . . . . . . . . . . . . . 261

16.2 Pressure-driven channel flow . . . . . . . . . . . . . . . . . . . . . . . . . . 262

16.3 Gravity-driven planar flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

16.4 Laminar pipe flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

16.5 Phenomenology of turbulent pipe flow . . . . . . . . . . . . . . . . . . . . . 274

16.6 Laminar cylindric flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

16.7 Secondary flow and Taylor vortices . . . . . . . . . . . . . . . . . . . . . . 281

17 Creeping flow 287

17.1 Stokes flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

17.2 Creeping flow around a solid ball . . . . . . . . . . . . . . . . . . . . . . . . 289

17.3 Beyond Stokes’ law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

17.4 Lubrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

17.5 Application: Loaded journal bearing . . . . . . . . . . . . . . . . . . . . . . 304

18 Rotating fluids 309

18.1 Fictitious forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

18.2 Steady flow in a rotating system . . . . . . . . . . . . . . . . . . . . . . . . 312

18.3 The Ekman layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

18.4 Application: Steady bathtub vortex . . . . . . . . . . . . . . . . . . . . . . . 318

18.5 Debunking an urban legend . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

19 Computational fluid dynamics 323

19.1 Unsteady, incompressible flow . . . . . . . . . . . . . . . . . . . . . . . . . 323

viii PHYSICS OF CONTINUOUS MATTER

19.2 Temporal discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

19.3 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

19.4 Application: Laminar channel entry flow . . . . . . . . . . . . . . . . . . . 330

IV Balance and conservation 337

20 Mechanical balances 339

20.1 Quantities and sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

20.2 Mass balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

20.3 Momentum balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

20.4 Angular momentum balance . . . . . . . . . . . . . . . . . . . . . . . . . . 346

20.5 Kinetic energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

20.6 Mechanical energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

21 Action and reaction 355

21.1 Reaction force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

21.2 Reaction moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

21.3 Application: The Francis turbine . . . . . . . . . . . . . . . . . . . . . . . . 366

22 Energy 371

22.1 First Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 371

22.2 Incompressible fluid at rest . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

22.3 Incompressible fluid in motion . . . . . . . . . . . . . . . . . . . . . . . . . 380

22.4 General homogeneous isotropic fluids . . . . . . . . . . . . . . . . . . . . . 384

23 Entropy 393

23.1 Entropy in classical thermodynamics . . . . . . . . . . . . . . . . . . . . . . 393

23.2 Entropy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

23.3 Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

V Selected topics 401

24 Elastic vibrations 403

24.1 Elastodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

24.2 Harmonic vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

24.3 Refraction and reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

24.4 Surface waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

25 Gravity waves 419

25.1 Basic wave concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

25.2 Harmonic surface waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

25.3 Open surface gravity waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

25.4 Capillary waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

25.5 Internal waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

25.6 Global wave properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

25.7 Statistics of wind-generated ocean waves . . . . . . . . . . . . . . . . . . . 439

26 Jumps and shocks 443

26.1 Hydraulic jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

26.2 Circular jump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448

26.3 Stationary shocks in uniformly moving fluids . . . . . . . . . . . . . . . . . 450

26.4 Application: Atmospheric blast wave . . . . . . . . . . . . . . . . . . . . . 454

CONTENTS ix

27 Whirls and vortices 461

27.1 Free cylindrical vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

27.2 Basic vortex theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464

27.3 Line vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

27.4 Advective vortex spin-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

27.5 Steady vortex sustained by secondary flow . . . . . . . . . . . . . . . . . . . 472

27.6 Application: The bathtub vortex . . . . . . . . . . . . . . . . . . . . . . . . 475

28 Boundary layers 481

28.1 Basic physics of boundary layers . . . . . . . . . . . . . . . . . . . . . . . . 481

28.2 Boundary layer theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

28.3 The Blasius layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

28.4 Turbulence in the Blasius layer . . . . . . . . . . . . . . . . . . . . . . . . . 492

28.5 Planar stagnation flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496

28.6 Self-similar boundary layers . . . . . . . . . . . . . . . . . . . . . . . . . . 498

28.7 Laminar boundary layer separation . . . . . . . . . . . . . . . . . . . . . . . 501

28.8 Wall-anchored model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

28.9 Wall derivative plus momentum balance . . . . . . . . . . . . . . . . . . . . 505

28.10 Momentum plus energy balance . . . . . . . . . . . . . . . . . . . . . . . . 506

28.11 Integral approximation to separation . . . . . . . . . . . . . . . . . . . . . . 508

29 Subsonic flight 513

29.1 Aircraft controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

29.2 Aerodynamic forces and moments . . . . . . . . . . . . . . . . . . . . . . . 516

29.3 Steady flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

29.4 Estimating lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

29.5 Estimating drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

29.6 Lift, drag, and the trailing wake . . . . . . . . . . . . . . . . . . . . . . . . 532

29.7 Two-dimensional airfoil theory . . . . . . . . . . . . . . . . . . . . . . . . . 537

29.8 The distant laminar wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541

30 Convection 547

30.1 Heat-driven convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

30.2 Convective instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552

30.3 Linear stability analysis of convection . . . . . . . . . . . . . . . . . . . . . 555

30.4 Application: Rayleigh–B´enard convection . . . . . . . . . . . . . . . . . . . 557

31 Turbulence 565

31.1 Scaling in fully developed turbulence . . . . . . . . . . . . . . . . . . . . . 565

31.2 Mean flow and fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 571

31.3 Universal inner layer near a smooth wall . . . . . . . . . . . . . . . . . . . . 574

31.4 The outer layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

31.5 Application: Turbulent channel flow . . . . . . . . . . . . . . . . . . . . . . 581

31.6 Application: Turbulent pipe flow . . . . . . . . . . . . . . . . . . . . . . . . 582

31.7 Turbulence modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584

VI Appendices 587

A Newtonian mechanics 589

A.1 Dynamic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

A.2 Force and momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590

A.3 Moment of force and angular momentum . . . . . . . . . . . . . . . . . . . 591

x PHYSICS OF CONTINUOUS MATTER

A.4 Power and kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592

A.5 Internal and external forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 593

A.6 Hierarchies of particle interactions . . . . . . . . . . . . . . . . . . . . . . . 594

B Cartesian coordinates 595

B.1 Cartesian vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595

B.2 Vector algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596

B.3 Basis vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598

B.4 Index notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

B.5 Cartesian coordinate transformations . . . . . . . . . . . . . . . . . . . . . . 601

B.6 Scalars, vectors, and tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 604

C Field calculus 611

C.1 Spatial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611

C.2 Spatial integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613

C.3 Fundamental integral theorems . . . . . . . . . . . . . . . . . . . . . . . . . 614

C.4 Proofs of the fundamental integral theorems . . . . . . . . . . . . . . . . . . 615

C.5 Field transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616

D Curvilinear coordinates 619

D.1 Cylindrical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619

D.2 Spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623

E Ideal gases 627

E.1 Internal energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627

E.2 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629

Answers to problems 631

References 665

Index 673

Subject Categories

BISAC Subject Codes/Headings:
SCI055000
SCIENCE / Physics
SCI077000
SCIENCE / Solid State Physics
TEC021000
TECHNOLOGY & ENGINEERING / Material Science