Titel: Fields, Flows and Waves
Autor/en: David F. Parker
An Introduction to Continuum Models.
5. Mai 2003 - kartoniert - 284 Seiten
This book serves as an introduction to the use of mathematics in describing collective phenomena in physics and biology. Derived from a course of innovative lectures, the book shows students early in their studies how many of the topics they have encountered - partial differential equations, differential equations, Fourier series, and linear algebra - are useful in constructing, analysing and interpreting phenomena present in the real world. Throughout, ideas are developed using worked examples and exercises with solution. The text does not assume a strong background in physics.
1. The Continuum Description.- 1.1 Densities and Fluxes.- 1.2 Conservation and Balance Laws in One Dimension.- 1.3 Heat Flow.- 1.4 Steady Radial Flow in Two Dimensions.- 1.5 Steady Radial Flow in Three Dimensions.- 2. Unsteady Heat Flow.- 2.1 Thermal Energy.- 2.1.1 Heat Balance in One-dimensional Problems.- 2.1.2 Some Special Solutions of Equation (2.3).- 2.2 Effects of Heat Supply.- 2.3 Unsteady, Spherically Symmetric Heat Flow.- 3. Fields and Potentials.- 3.1 Gradient of a Scalar.- 3.1.1 Some Applications.- 3.2 Gravitational Potential.- 3.2.1 Special Properties of the Function ?=r?1.- 3.3 Continuous Distributions of Mass.- 3.4 Electrostatics.- 3.4.1 Gauss's Law of Flux.- 3.4.2 Charge-free Regions.- 3.4.3 Surface Charge Density.- 4. Laplace's Equation and Poisson's Equation.- 4.1 The Ubiquitous Laplacian.- 4.2 Separable Solutions.- 4.3 Poisson's Equation.- 4.4 Dipole Solutions.- 4.4.1 Uses of Dipole Solutions to ?2?=0.- 4.4.2 Spherical Inclusions.- 5. Motion of an Elastic String.- 5.1 Tension and Extension; Kinematics and Dynamics.- 5.1.1 Dynamics.- 5.2 Planar Motions.- 5.2.1 Small Transverse Motions.- 5.2.2 Longitudinal Motions.- 5.3 Properties of the Wave Equation.- 5.3.1 Standing Waves.- 5.3.2 Superposition of Standing Waves.- 5.4 D'Alembert's Solution, Travelling Waves and Wave Reflections.- 5.4.1 Wave Reflections.- 5.5 Other One-dimensional Waves.- 5.5.1 Acoustic Vibrations in a' lUbe.- 5.5.2 Telegraphy and High-voltage Transmission.- 6. Fluid Flow.- 6.1 Kinematics and Streamlines.- 6.1.1 Some Important Examples of Steady Flow.- 6.2 Volume Flux and Mass Flux.- 6.2.1 Incompressible Fluids.- 6.2.2 Mass Conservation.- 6.3 Two-dimensional Flows of Incompressible Fluids.- 6.3.1 The Continuity Equation.- 6.3.2 Irrotational Flows and the Velocity Potential.- 6.3.3 The Stream Function.- 6.4 Pressure in a Fluid.- 6.4.1 Resultant Force.- 6.4.2 Hydrostatics and Archimedes' Principle.- 6.4.3 Momentum Density and Momentum Flux.- 6.5 Bernoulli's Equation.- 6.5.1 The Material (Advected) Derivative.- 6.5.2 Bernoulli's Equation and Dynamic Pressure.- 6.5.3 The Principle of Aerodynamic Lift.- 6.6 Three-dimensional, Incompressible Flows.- 6.6.1 The Continuity Equation.- 6.6.2 Irrotational Flows, the Velocity Potential and Laplace's Equation.- 7. Elastic Deformations.- 7.1 The Kinematics of Deformation.- 7.1.1 Deformation Gradient.- 7.1.2 Stretch and Rotation.- 7.2 Polar Decomposition.- 7.3 Stress.- 7.3.1 Traction Vectors.- 7.3.2 Components of Stress.- 7.3.3 Traction on a General Surface.- 7.4 Isotropic Linear Elasticity.- 7.4.1 The Constitutive Law.- 7.4.2 Stretching, Shear and Torsion.- 8. Vibrations and Waves.- 8.1 Wave Reflection and Refraction.- 8.1.1 Use of the Complex Exponential.- 8.1.2 Plane Waves.- 8.1.3 Reflection at a Rigid Wall.- 8.1.4 Refraction at an Interface.- 8.1.5 Total Internal Reflection.- 8.2 Guided Waves.- 8.2.1 Acoustic Waves in a Layer.- 8.2.2 Waveguides and Dispersion.- 8.3 Love Waves in Elasticity.- 8.4 Elastic Plane Waves.- 8.4.1 Elastic Shear Waves.- 8.4.2 Dilatational Waves.- 9. Electromagnetic VVaves and Light.- 9.1 Physical Background.- 9.1.1 The Origin of Maxwell's Equations.- 9.1.2 Plane Electromagnetic Waves.- 9.1.3 Reflection and Refraction of Electromagnetic Waves.- 9.2 Waveguides.- 9.2.1 Rectangular Waveguides.- 9.2.2 Circular Cylindrical Waveguides.- 9.2.3 An Introduction to Fibre Optics.- 10. Chemical and Biological Models.- 10.1 Diffusion of Chemical Species.- 10.1.1 Fick's Law of Diffusion.- 10.1.2 Self-similar Solutions.- 10.1.3 Travelling Wavefronts.- 10.2 Population Biology.- 10.2.1 Growth and Dispersal.- 10.2.2 Fisher's Equation and Self-limitation.- 10.2.3 Population-dependent Dispersivity.- 10.2.4 Competing Species.- 10.2.5 Diffusive Instability.- 10.3 Biological Waves.- 10.3.1 The Logistic Wavefront.- 10.3.2 Travelling Pulses and Spiral Waves.- Solutions.
From the reviews:"This textbook ... is designed for undergraduates who have followed a first 'mathematical methods' course and who are ready to study in more depth the mathematics underlying phenomena ... . Each chapter includes a number of worked examples and a dozen or so exercises, with solutions collected at the end of the book. The book is aimed at students of applied mathematics, physics and engineering. The emphasis on techniques and the frequent references to applications make it particularly suitable for this audience." (S.C. Russen, The Mathematical Gazette, Vol. 89 (516), 2005)"Fields, Flows, and Waves, is an introduction to continuum models based on the author's lectures ... . Ample illustrations and worked examples come with the exposition, and there are several exercises with varying degrees of difficulty; detailed solutions are included at the end of the text. ... I warmly recommend this book. It reads well and is in an attractive, concise format. ... It makes one yearn for a course in the curriculum where this material could be regularly taught." (SIAM Review, Vol. 46 (3), 2004)"This book is an introduction to the mathematical methods in classical fields theory. It is designed for the second-year undergraduate in physics, mathematics and engineering. ... The presentation is excellent, numerous examples of increasing difficulties are considered with details. ... The book ends with solutions to the exercises a short bibliography and an index. In conclusion, I warmly recommend this book to any students in physics because it's well written ... interesting and very useful." (Stéphane Métens, Physicalia, Vol. 26 (1), 2004)"This book ... is a first introduction to the mathematical description of fields, flows and waves. It shows students, early in their studies, how many of the topics they have encountered are useful ... . Designed for second-year undergraduate students in mathematics, mathematical physics, and engineering, it presumes only a limited familiarity with several variable calculus and vector fields. ... The ideas are developed through worked examples, and a range of exercises (with solutions) is provided to test understanding." (Läenseignement Mathematique, Vol. 49 (3-4), 2003)"This is another excellent readable book in the Springer Undergraduate Mathematics Series (SUMS). It is a refreshingly modern approach to Continuum Mechanics ... . Indeed Professor Parker has written this book so that it might be used directly as an elementary course ... . This is a carefully written, well structured book which contains a wealth of examples complete with solutions. ... a carefully structured book from which a modern undergraduate applied mathematics course may be taught directly." (Sean McKee, Journal of Fluid Mechanics, Vol.504, 2004)"Introductory books ... often struggle with the balance between the motivating physical problems and the formal mathematical structures. As the title suggests, Parker ... manages to keep the more technical mathematical structure in clear view. Particularly impressive is how carefully the author leads readers ... . the book has a completeness that makes it attractive as a self-contained resource as well as a textbook. ... complete solutions (not just answers) to all of the exercises makes the book particularly effective for independent study of this material. Summing Up: Highly recommended." (J. Feroe, CHOICE, December, 2003)"The book is well-written and illustrated by interesting figures which make the text easy to read and attractive. Of course undergraduate students in physics and maybe in mathematics will surely benefit of a lecture and practice of this book. Each of the ten chapters indeed contains some lists of significant exercises. The more or less detailed solutions of these exercises are gathered at the end of the book." (Alain Brillard, Zentralblatt MATH, 2003)"Continuum models ignoring the substructures of fluids are useful and widely applied for the descr