Riemannian Geometry

Author: Luther Pfahler Eisenhart

Publisher: Princeton University Press

ISBN: 1400884217

Category: Mathematics

Page: 272

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In his classic work of geometry, Euclid focused on the properties of flat surfaces. In the age of exploration, mapmakers such as Mercator had to concern themselves with the properties of spherical surfaces. The study of curved surfaces, or non-Euclidean geometry, flowered in the late nineteenth century, as mathematicians such as Riemann increasingly questioned Euclid's parallel postulate, and by relaxing this constraint derived a wealth of new results. These seemingly abstract properties found immediate application in physics upon Einstein's introduction of the general theory of relativity. In this book, Eisenhart succinctly surveys the key concepts of Riemannian geometry, addressing mathematicians and theoretical physicists alike.

Real and Complex Singularities

Ninth International Workshop on Real and Complex Singularities, July 23-28, 2006, ICMC-USP, São Carlos, S.P. Brazil

Author: Marcelo José Saia,José Seade

Publisher: American Mathematical Soc.

ISBN: 0821844970

Category: Mathematics

Page: 261

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This book offers a selection of papers based on talks presented at the Ninth International Workshop on Real and Complex Singularities, a series of biennial workshops organized by the Singularity Theory group at Sao Carlos, S.P., Brazil. The papers deal with all the different topics in singularity theory and its applications, from pure singularity theory related to commutative algebra and algebraic geometry to those topics associated with various aspects of geometry to homotopy theory. Among the topics on pure singularity theory discussed are invariants of singularities, integral closure and equisingularity, classification theory, contact structures and vector fields, and Thom polynomials. Geometric aspects deal with relations of singularity theory with topology, differential geometry and physics. Here topics discussed include the index of quadratic differential forms, obstructions in fundamental groups of plane curve complements, conjugate vectors of immersed manifolds, exotic moduli of Goursat distributions in codimension three, cobordisms of fold maps, etc. The book concludes with notes from the course on the residue theoretical approach to intersection theory.

Topology from the Differentiable Viewpoint

Author: John Willard Milnor

Publisher: Princeton University Press

ISBN: 9780691048338

Category: Mathematics

Page: 64

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This elegant book by distinguished mathematician John Milnor, provides a clear and succinct introduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem.

Lectures on Differential Geometry

Author: S S Chern,W H Chen,K S Lam

Publisher: World Scientific Publishing Company

ISBN: 9813102985

Category: Mathematics

Page: 368

View: 1939

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This book is a translation of an authoritative introductory text based on a lecture series delivered by the renowned differential geometer, Professor S S Chern in Beijing University in 1980. The original Chinese text, authored by Professor Chern and Professor Wei-Huan Chen, was a unique contribution to the mathematics literature, combining simplicity and economy of approach with depth of contents. The present translation is aimed at a wide audience, including (but not limited to) advanced undergraduate and graduate students in mathematics, as well as physicists interested in the diverse applications of differential geometry to physics. In addition to a thorough treatment of the fundamentals of manifold theory, exterior algebra, the exterior calculus, connections on fiber bundles, Riemannian geometry, Lie groups and moving frames, and complex manifolds (with a succinct introduction to the theory of Chern classes), and an appendix on the relationship between differential geometry and theoretical physics, this book includes a new chapter on Finsler geometry and a new appendix on the history and recent developments of differential geometry, the latter prepared specially for this edition by Professor Chern to bring the text into perspectives.

The Topology of Fibre Bundles. (PMS-14)

Author: Norman Steenrod

Publisher: Princeton University Press

ISBN: 1400883873

Category: Mathematics

Page: 224

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Fibre bundles, now an integral part of differential geometry, are also of great importance in modern physics--such as in gauge theory. This book, a succinct introduction to the subject by renown mathematician Norman Steenrod, was the first to present the subject systematically. It begins with a general introduction to bundles, including such topics as differentiable manifolds and covering spaces. The author then provides brief surveys of advanced topics, such as homotopy theory and cohomology theory, before using them to study further properties of fibre bundles. The result is a classic and timeless work of great utility that will appeal to serious mathematicians and theoretical physicists alike.

Morse Theory. (AM-51)

Author: John Milnor

Publisher: Princeton University Press

ISBN: 1400881803

Category: Mathematics

Page: 160

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One of the most cited books in mathematics, John Milnor's exposition of Morse theory has been the most important book on the subject for more than forty years. Morse theory was developed in the 1920s by mathematician Marston Morse. (Morse was on the faculty of the Institute for Advanced Study, and Princeton published his Topological Methods in the Theory of Functions of a Complex Variable in the Annals of Mathematics Studies series in 1947.) One classical application of Morse theory includes the attempt to understand, with only limited information, the large-scale structure of an object. This kind of problem occurs in mathematical physics, dynamic systems, and mechanical engineering. Morse theory has received much attention in the last two decades as a result of a famous paper in which theoretical physicist Edward Witten relates Morse theory to quantum field theory. Milnor was awarded the Fields Medal (the mathematical equivalent of a Nobel Prize) in 1962 for his work in differential topology. He has since received the National Medal of Science (1967) and the Steele Prize from the American Mathematical Society twice (1982 and 2004) in recognition of his explanations of mathematical concepts across a wide range of scienti.c disciplines. The citation reads, "The phrase sublime elegance is rarely associated with mathematical exposition, but it applies to all of Milnor's writings. Reading his books, one is struck with the ease with which the subject is unfolding and it only becomes apparent after re.ection that this ease is the mark of a master.? Milnor has published five books with Princeton University Press.

The Concept of a Riemann Surface

Author: Hermann Weyl

Publisher: Courier Corporation

ISBN: 048613167X

Category: Mathematics

Page: 208

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This classic on the general history of functions combines function theory and geometry, forming the basis of the modern approach to analysis, geometry, and topology. 1955 edition.

General Theory of Relativity

Author: P. A.M. Dirac

Publisher: Princeton University Press

ISBN: 1400884195

Category: Science

Page: 80

View: 9885

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Einstein's general theory of relativity requires a curved space for the description of the physical world. If one wishes to go beyond superficial discussions of the physical relations involved, one needs to set up precise equations for handling curved space. The well-established mathematical technique that accomplishes this is clearly described in this classic book by Nobel Laureate P.A.M. Dirac. Based on a series of lectures given by Dirac at Florida State University, and intended for the advanced undergraduate, General Theory of Relativity comprises thirty-five compact chapters that take the reader point-by-point through the necessary steps for understanding general relativity.

Theory of Lie Groups

Author: Claude Chevalley

Publisher: Courier Dover Publications

ISBN: 0486829669

Category: Mathematics

Page: 224

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The standard text on the subject for many years, this introductory treatment covers classical linear groups, topological groups, manifolds, analytic groups, differential calculus of Cartan, and compact Lie groups and their representations. 1946 edition.

Introduction to Smooth Manifolds

Author: John M. Lee

Publisher: Springer Science & Business Media

ISBN: 0387217525

Category: Mathematics

Page: 631

View: 3417

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Author has written several excellent Springer books.; This book is a sequel to Introduction to Topological Manifolds; Careful and illuminating explanations, excellent diagrams and exemplary motivation; Includes short preliminary sections before each section explaining what is ahead and why

Gradient Flows

In Metric Spaces and in the Space of Probability Measures

Author: Luigi Ambrosio,Nicola Gigli,Giuseppe Savare

Publisher: Springer Science & Business Media

ISBN: 9783764387228

Category: Mathematics

Page: 334

View: 7252

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The book is devoted to the theory of gradient flows in the general framework of metric spaces, and in the more specific setting of the space of probability measures, which provide a surprising link between optimal transportation theory and many evolutionary PDE's related to (non)linear diffusion. Particular emphasis is given to the convergence of the implicit time discretization method and to the error estimates for this discretization, extending the well established theory in Hilbert spaces. The book is split in two main parts that can be read independently of each other.

Operator Techniques in Atomic Spectroscopy

Author: Brian R. Judd

Publisher: Princeton University Press

ISBN: 1400864771

Category: Mathematics

Page: 256

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In the 1920s, when quantum mechanics was in its infancy, chemists and solid state physicists had little choice but to manipulate unwieldy equations to determine the properties of even the simplest molecules. When mathematicians turned their attention to the equations of quantum mechanics, they discovered that these could be expressed in terms of group theory, and from group theory it was a short step to operator methods. This important development lay largely dormant until this book was originally published in 1963. In this pathbreaking publication, Brian Judd made the operator techniques of mathematicians comprehensible to physicists and chemists. He extended the existing methods so that they could handle heavier, more complex molecules and calculate their energy levels, and from there, it was another short step to the mathematical analysis of spectra. This book provides a first-class introduction to continuous groups for physicists and chemists. Although first written from the perspective of atomic spectroscopy, its major topics and methods will appeal to anyone who has an interest in understanding particle theories of nuclear physics. Originally published in 1998. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Modern Classical Physics

Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics

Author: Kip S. Thorne,Roger D. Blandford

Publisher: Princeton University Press

ISBN: 140084889X

Category: Science

Page: 1552

View: 6241

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This first-year, graduate-level text and reference book covers the fundamental concepts and twenty-first-century applications of six major areas of classical physics that every masters- or PhD-level physicist should be exposed to, but often isn't: statistical physics, optics (waves of all sorts), elastodynamics, fluid mechanics, plasma physics, and special and general relativity and cosmology. Growing out of a full-year course that the eminent researchers Kip Thorne and Roger Blandford taught at Caltech for almost three decades, this book is designed to broaden the training of physicists. Its six main topical sections are also designed so they can be used in separate courses, and the book provides an invaluable reference for researchers. Presents all the major fields of classical physics except three prerequisites: classical mechanics, electromagnetism, and elementary thermodynamics Elucidates the interconnections between diverse fields and explains their shared concepts and tools Focuses on fundamental concepts and modern, real-world applications Takes applications from fundamental, experimental, and applied physics; astrophysics and cosmology; geophysics, oceanography, and meteorology; biophysics and chemical physics; engineering and optical science and technology; and information science and technology Emphasizes the quantum roots of classical physics and how to use quantum techniques to elucidate classical concepts or simplify classical calculations Features hundreds of color figures, some five hundred exercises, extensive cross-references, and a detailed index An online illustration package is available to professors

Convex Analysis

Author: R. Tyrrell Rockafellar

Publisher: Princeton University Press

ISBN: 9780691015866

Category: Mathematics

Page: 451

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Available for the first time in paperback, R. Tyrrell Rockafellar's classic study presents readers with a coherent branch of nonlinear mathematical analysis that is especially suited to the study of optimization problems. Rockafellar's theory differs from classical analysis in that differentiability assumptions are replaced by convexity assumptions. The topics treated in this volume include: systems of inequalities, the minimum or maximum of a convex function over a convex set, Lagrange multipliers, minimax theorems and duality, as well as basic results about the structure of convex sets and the continuity and differentiability of convex functions and saddle- functions.This book has firmly established a new and vital area not only for pure mathematics but also for applications to economics and engineering. A sound knowledge of linear algebra and introductory real analysis should provide readers with sufficient background for this book. There is also a guide for the reader who may be using the book as an introduction, indicating which parts are essential and which may be skipped on a first reading."This book should remain for some years as the standard reference for anyone interested in convex analysis." J. D. Pryce, Edinburgh Mathematical Society

Differential Topology

Author: Morris W. Hirsch

Publisher: Springer Science & Business Media

ISBN: 146849449X

Category: Mathematics

Page: 222

View: 9454

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"A very valuable book. In little over 200 pages, it presents a well-organized and surprisingly comprehensive treatment of most of the basic material in differential topology, as far as is accessible without the methods of algebraic topology....There is an abundance of exercises, which supply many beautiful examples and much interesting additional information, and help the reader to become thoroughly familiar with the material of the main text." —MATHEMATICAL REVIEWS

Positive Definite Matrices

Author: Rajendra Bhatia

Publisher: Princeton University Press

ISBN: 9781400827787

Category: Mathematics

Page: 240

View: 9526

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This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.

Differential Topology

Author: Victor Guillemin,Alan Pollack

Publisher: American Mathematical Soc.

ISBN: 0821851934

Category: Mathematics

Page: 222

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Differential Topology provides an elementary and intuitive introduction to the study of smooth manifolds. In the years since its first publication, Guillemin and Pollack's book has become a standard text on the subject. It is a jewel of mathematical exposition, judiciously picking exactly the right mixture of detail and generality to display the richness within. The text is mostly self-contained, requiring only undergraduate analysis and linear algebra. By relying on a unifying idea--transversality--the authors are able to avoid the use of big machinery or ad hoc techniques to establish the main results. In this way, they present intelligent treatments of important theorems, such as the Lefschetz fixed-point theorem, the Poincaré-Hopf index theorem, and Stokes theorem. The book has a wealth of exercises of various types. Some are routine explorations of the main material. In others, the students are guided step-by-step through proofs of fundamental results, such as the Jordan-Brouwer separation theorem. An exercise section in Chapter 4 leads the student through a construction of de Rham cohomology and a proof of its homotopy invariance. The book is suitable for either an introductory graduate course or an advanced undergraduate course.

Differential Forms in Algebraic Topology

Author: Raoul Bott,Loring W. Tu

Publisher: Springer Science & Business Media

ISBN: 1475739516

Category: Mathematics

Page: 338

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Developed from a first-year graduate course in algebraic topology, this text is an informal introduction to some of the main ideas of contemporary homotopy and cohomology theory. The materials are structured around four core areas: de Rham theory, the Cech-de Rham complex, spectral sequences, and characteristic classes. By using the de Rham theory of differential forms as a prototype of cohomology, the machineries of algebraic topology are made easier to assimilate. With its stress on concreteness, motivation, and readability, this book is equally suitable for self-study and as a one-semester course in topology.