Hardback (ISBN-13: 9780521877619)
Paperback (ISBN-13: 9780521701471)
Gauss’s law for electric fields, Gauss’s law for magnetic fields, Faraday’s law, and the Ampere-Maxwell law are four of the most influential equations in science. In this guide for students, each equation is the subject of an entire chapter, with detailed, plain-language explanations of the physical meaning of each symbol in the equation, for both the integral and differential forms. The final chapter shows how Maxwell’s equations may be combined to produce the wave equation, the basis for the electromagnetic theory of light. This book is a wonderful resource for undergraduate and graduate courses in electromagnetism and electromagnetics. A website hosted by the author at www.cambridge.org/9780521701471 contains interactive solutions to every problem in the text as well as audio podcasts to walk students through each chapter.
* Features an interactive website with complete solutions to every problem within the text, as well as audio podcasts explaining key concepts
* Plain-language explanations of the symbols used in the equations
* Modular approach allows reader to find relevant material easily
Contents
Preface; 1. Gauss’s law for electric fields; 2. Gauss’s law for magnetic fields; 3. Faraday’s law; 4. The Ampere-Maxwell law; 5. From Maxwell’s equations to the wave equation; Appendix; Further Reading; Index.
Hardback (ISBN-13: 9780521886291)
Paperback (ISBN-13: 9780521713900)
This self-contained textbook presents an exposition of the well-known classical two-dimensional geometries, such as Euclidean, spherical, hyperbolic, and the locally Euclidean torus, and introduces the basic concepts of Euler numbers for topological triangulations, and Riemannian metrics. The careful discussion of these classical examples provides students with an introduction to the more general theory of curved spaces developed later in the book, as represented by embedded surfaces in Euclidean 3-space, and their generalization to abstract surfaces equipped with Riemannian metrics. Themes running throughout include those of geodesic curves, polygonal approximations to triangulations, Gaussian curvature, and the link to topology provided by the Gauss-Bonnet theorem. Numerous diagrams help bring the key points to life and helpful examples and exercises are included to aid understanding. Throughout the emphasis is placed on explicit proofs, making this text ideal for any student with a basic background in analysis and algebra.
* A concrete approach to the theory, with emphasis on self-contained explicit proofs; uses the classical geometries to motivate the basic ideas of elementary differential geometry.
* Provides a link between basic undergraduate courses on Analysis and Algebra, and more advanced theoretical courses in Geometry
* Rigorous treatment of the classical geometries, via analytical ideas with exercises at the end of each chapter, reinforcing the material in the text
* A novel approach to defining curvature on abstract surfaces, and to proving the topological invariance of the Euler number
* Coverage of a wide range of topics, starting with very elementary material and concluding with rather more advanced mathematical ideas
* Certain geometrical themes, such as geodesics, curvature, and the Gauss-Bonnet theorem, running throughout the book, provide a unifying philosophy
Contents
Preface; 1. Euclidean geometry; 2. Spherical geometry; 3. Triangulations and Euler numbers; 4. Riemannian metrics; 5. Hyperbolic geometry; 6. Smooth embedded surfaces; 7. Geodesics; 8. Abstract surfaces and Gauss-Bonnet.
Series: Cambridge Studies in Advanced Mathematics (No. 108)
Hardback (ISBN-13: 9780521883368)
Line up a deck of 52 cards on a table. Randomly choose two cards and switch them. How many switches are needed in order to mix up the deck? Starting from a few concrete problems such as random walks on the discrete circle and the finite ultrametric space this book develops the necessary tools for the asymptotic analysis of these processes. This detailed study culminates with the case-by-case analysis of the cut-off phenomenon discovered by Persi Diaconis. This self-contained text is ideal for students and researchers working in the areas of representation theory, group theory, harmonic analysis and Markov chains. Its topics range from the basic theory needed for students new to this area, to advanced topics such as the theory of Green’s algebras, the complete analysis of the random matchings, and a presentation of the presentation theory of the symmetric group.
* Can be used as a textbook for advanced undergraduate and graduate students, and as a reference for researchers
* First book with a complete treatment of the theory of Gelfand pairs
* Contains 140 exercises, with solutions or generous hints, and over 60 fully-worked examples
Contents
Part I. Preliminaries, Examples and Motivations: 1. Finite Markov chains; 2. Two basic examples on Abelian groups; Part II. Representation Theory and Gelfand Pairs: 3. Basic representation theory of finite groups; 4. Finite Gelfand pairs; 5. Distance regular graphs and the Hamming scheme; 6. The Johnson Scheme and the Laplace-Bernoulli diffusion model; 7. The ultrametric space; Part III. Advanced theory: 8. Posets and the q-analogs; 9. Complements on representation theory; 10. Basic representation theory of the symmetric group; 11. The Gelfand Pair (S2n, S2 o Sn) and random matchings; Appendix 1. The discrete trigonometric transforms; Appendix 2. Solutions of the exercises; Bibliography; Index.
Series: Encyclopedia of Mathematics and its Applications (No. 117)
Hardback (ISBN-13: 9780521885720)
The behaviour of systems occurring in real life is often modelled by partial differential equations. This book investigates how a user or observer can influence the behaviour of such systems mathematically and computationally. A thorough mathematical analysis of controllability problems is combined with a detailed investigation of methods used to solve them numerically, these methods being validated by the results of numerical experiments. In Part I of the book the authors discuss the mathematics and numerics relating to the controllability of systems modelled by linear and non-linear diffusion equations; Part II is dedicated to the controllability of vibrating systems, typical ones being those modelled by linear wave equations; finally, Part III covers flow control for systems governed by the Navier-Stokes equations modelling incompressible viscous flow. The book is accessible to graduate students in applied and computational mathematics, engineering and physics; it will also be of use to more advanced practitioners.
* Computationally oriented with a thorough discussion of the solution methods employed in the various chapters (finite element methods, conjugate gradient algorithms and more)
* Blends mathematical analysis and numerical analysis and illustrates with a large variety of numerical experiments
* One of the few books on controllability issues for systems modelled by partial differential equations from mechanics and physics, a hot topic at the moment
Contents
Preface; Introduction; Part I. Diffusion Models: 1. Distributed and point-wise control for linear diffusion equations; 2. Boundary control; 3. Control of the Stokes system; 4. Control of nonlinear diffusion systems; 5. Dynamic programming for linear diffusion equations; Part II. Wave Models: 6. Wave equations; 7. Helmholtz equation; 8. Coupled systems; Part III. Flow Control: 9. Optimal control of Navier-Stokes equations: drag reduction; Epilogue; Further acknowledgements; References.
Series: New Mathematical Monographs (No. 6)
Hardback (ISBN-13: 9780521878579)
Detailing the main methods in the theory of involutive systems of complex vector fields this book examines the major results from the last twenty five years in the subject. One of the key tools of the subject - the Baouendi-Treves approximation theorem - is proved for many function spaces. This in turn is applied to questions in partial differential equations and several complex variables. Many basic problems such as regularity, unique continuation and boundary behaviour of the solutions are explored. The local solvability of systems of partial differential equations is studied in some detail. The book provides a solid background for others new to the field and also contains a treatment of many recent results which will be of interest to researchers in the subject.
* Details the main tools and methods in the theory of involutive systems of complex vector fields
* The Baouendi-Treves approximation theorem is proved for many function spaces
* Provides a solid background for beginners in the field and also contains a treatment of many recent results of interest to researchers in the subject
Contents
Preface; 1. Locally integrable structures; 2. The Baouendi-Treves approximation formula; 3. Sussmann’s orbits and unique continuation; 4. Local solvability of vector fields; 5. The FBI transform and some applications; 6. Some boundary properties of solutions; 7. The differential complex associated to a formally integrable structure; 8. Local solvability in locally integrable structures; Epilogue; Bibliography; A. Hardy space lemmas.