Brownian Motion and Potential Theory, Modern and Classical (1版)

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【簡介】 In this book, potential theory is presented in an inclusive and accessible manner, with the emphasis reaching from classical to modern, from analytic to probabilistic, and from Newtonian to abstract or axiomatic potential theory (including Dirichlet spaces). The reader is guided through stochastic analysis featuring Brownian motion in its early chapters to potential theory in its latter sections. This path covers the following themes: martingales, diffusion processes, semigroups and potential operators, analysis of super harmonic functions, Dirichlet problems, balayage, boundaries, and Green functions. The wide range of applications encompasses random walk models, especially reversible Markov processes, and statistical inference in machine learning models. However, the present volume considers the analysis from the point of view of function space theory, using Dirchlet energy as an inner product. This present volume is an expanded and revised version of an original set of lectures in the Aarhus University Mathematics Institute Lecture Note Series. 【目錄】 Contents: Introduction Martingales and Markov Processes Brownian Motion and Ito-Calculus Semi-Groups of Operators, Potentials, and Diffusion Equations Harmonic Functions, Dynkin, and Transforms Superharmonic Functions and Riesz Measures Green Functions, Boundary Value Problems, and Kernels Potential Theory, Capacity, Boundaries, Dirichlet Spaces, and Applications Appendix: Kernels and More General Classes of Gaussian Processes

Classical Mechanics and Electrodynamics (2版)

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【簡介】 The book gives a general introduction to classical theoretical physics, in the fields of mechanics, relativity and electromagnetism. It is analytical in approach and detailed in the derivations of physical consequences from the fundamental principles in each of the fields. This second edition has a new part, namely Classical Field Theory. Highlighting a close connection between this part and earlier parts of the book, where particles, rather than fields are the center of attention. As a general introduction to classical theoretical physics, the book is different from most textbooks at this level, which focus either on classical mechanics or classical electrodynamics but not both. The book will in particular be useful as a textbook for physics courses with such a broader approach to classical physics. For a wider group of students, the book may be of interest for self study. The new inclusion on classical field theory, will give students greater understanding on previous parts of the book, such as examining the Lagrangian formulation of Maxwell's equations with Noether's theorem. This is central in the use of Lagrangian on fields, as also discussed. The text is illustrated with many figures, most of these in color. There are many useful examples and exercises which complement the derivations in the text. 【目錄】 Analytical Mechanics: Generalized Coordinates Lagrange's Equations Hamiltonian Dynamics Relativity: The Four-Dimensional Space-Time Consequences of the Lorentz Transformations Four-Vector Formalism and Covariant Equations Relativistic Kinematics Relativistic Dynamics Electrodynamics: Maxwell's Equations Electromagnetic Field Dynamics Maxwell's Equations with Stationary Sources Electromagnetic Radiation Classical Field Theory: Lagrangian and Hamiltonian Formulations Symmetry Transformations Relativistic Fields

Classical Mechanics and Relativity (2版)

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【簡介】 The text covers the entire domain of basic classical mechanics and relativity theory (special and general) and has been revised mainly for the purpose of adding exercises without worked solutions that were missing in the first edition. To retain the format of a readable, yet advanced introductory text that can serve as the companion text for a course in mechanics, the more than 100 new exercises on diverse topics are of moderate range; answers are given and occasionally hints are provided. As before, the text aims to cover the entire spectrum of theoretical mechanics from Newton to Einstein. The reader can observe how in the course of time, deeper and deeper insights were achieved with the development of the basic equations of Newton to those of Euler and Lagrange, and to the geodesic equations of space-time and Einstein's relativity. To include diverse problems, a small section on this topic has been added. 【目錄】 Chapter 1: Introduction Contents: Preface to Second Edition Preface to First Edition Introduction Recapitulation of Newtonian Mechanics The Lagrange Formalism The Canonical or Hamilton Formalism Symmetries and Transformations Looking Beyond Classical Mechanics Two-Body Central Forces Rigid Body Dynamics Small Oscillations and Stability Motivation of the Theory of Relativity A Simple Look at Phenomenological Consequences Aspects of Special Relativity Equation of Motion of a Particle in a Gravitational Field Tensor Calculus for Riemann Spaces Einstein's Equation of the Gravitational Field The Schwarzschild Solution Appendix A: Schwarzschild Orbit Solution Appendix B: Reissner–Nordstrom Metric Bibliography Index Readership: Undergraduate and graduate students who are interested in mechanics and relativity.