Curvature in Mathematics and Physics

Author: Shlomo Sternberg

Publisher: Courier Corporation

ISBN: 0486292711

Category: Mathematics

Page: 416

View: 2467

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Expert treatment introduces semi-Riemannian geometry and its principal physical application, Einstein's theory of general relativity, using the Cartan exterior calculus as a principal tool. Prerequisites include linear algebra and advanced calculus. 2012 edition.

Curvature in Mathematics and Physics

Author: Shlomo Sternberg

Publisher: Courier Corporation

ISBN: 0486478556

Category: Mathematics

Page: 405

View: 3722

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As astronaut Donald K. Slayton notes in his Foreword, this chronicle emphasizes the cooperation of "humans on space and on the ground. It realistically balances the role of the highly visible astronaut with the mammoth supporting team." An official NASA publication, Suddenly, Tomorrow Came is profusely illustrated with forty-four figures and tables, plus sixty-three photographs. Historian Paul Dickson brings the narrative up to date with an informative new Introduction.

Gravitational Curvature

An Introduction to Einstein's Theory

Author: Theodore Frankel

Publisher: Courier Corporation

ISBN: 0486481212

Category: Science

Page: 172

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This classic text and reference monograph applies modern differential geometry to general relativity. A brief mathematical introduction to gravitational curvature, it emphasizes the subject's geometric essence and stresses the global aspects of cosmology. Suitable for independent study as well as for courses in differential geometry, relativity, and cosmology. 1979 edition.

Differential Geometry

Bundles, Connections, Metrics and Curvature

Author: Clifford Henry Taubes

Publisher: OUP Oxford

ISBN: 0191621226

Category: Mathematics

Page: 312

View: 3443

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Bundles, connections, metrics and curvature are the 'lingua franca' of modern differential geometry and theoretical physics. This book will supply a graduate student in mathematics or theoretical physics with the fundamentals of these objects. Many of the tools used in differential topology are introduced and the basic results about differentiable manifolds, smooth maps, differential forms, vector fields, Lie groups, and Grassmanians are all presented here. Other material covered includes the basic theorems about geodesics and Jacobi fields, the classification theorem for flat connections, the definition of characteristic classes, and also an introduction to complex and Kähler geometry. Differential Geometry uses many of the classical examples from, and applications of, the subjects it covers, in particular those where closed form expressions are available, to bring abstract ideas to life. Helpfully, proofs are offered for almost all assertions throughout. All of the introductory material is presented in full and this is the only such source with the classical examples presented in detail.

The Geometry of Kerr Black Holes

Author: Barrett O'Neill

Publisher: Courier Corporation

ISBN: 0486783111

Category: Science

Page: 400

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Suitable for advanced undergraduates and graduate students of mathematics as well as for physicists, this unique monograph and self-contained treatment constitutes an introduction to modern techniques in differential geometry. 1995 edition.

Differential Geometry and Mathematical Physics

Part II. Fibre Bundles, Topology and Gauge Fields

Author: Gerd Rudolph,Matthias Schmidt

Publisher: Springer

ISBN: 9402409599

Category: Science

Page: 830

View: 3156

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The book is devoted to the study of the geometrical and topological structure of gauge theories. It consists of the following three building blocks:- Geometry and topology of fibre bundles,- Clifford algebras, spin structures and Dirac operators,- Gauge theory.Written in the style of a mathematical textbook, it combines a comprehensive presentation of the mathematical foundations with a discussion of a variety of advanced topics in gauge theory.The first building block includes a number of specific topics, like invariant connections, universal connections, H-structures and the Postnikov approximation of classifying spaces.Given the great importance of Dirac operators in gauge theory, a complete proof of the Atiyah-Singer Index Theorem is presented. The gauge theory part contains the study of Yang-Mills equations (including the theory of instantons and the classical stability analysis), the discussion of various models with matter fields (including magnetic monopoles, the Seiberg-Witten model and dimensional reduction) and the investigation of the structure of the gauge orbit space. The final chapter is devoted to elements of quantum gauge theory including the discussion of the Gribov problem, anomalies and the implementation of the non-generic gauge orbit strata in the framework of Hamiltonian lattice gauge theory.The book is addressed both to physicists and mathematicians. It is intended to be accessible to students starting from a graduate level.

Topology and Geometry for Physicists

Author: Charles Nash,Siddhartha Sen

Publisher: Courier Corporation

ISBN: 0486318362

Category: Mathematics

Page: 320

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Differential geometry and topology are essential tools for many theoretical physicists, particularly in the study of condensed matter physics, gravity, and particle physics. Written by physicists for physics students, this text introduces geometrical and topological methods in theoretical physics and applied mathematics. It assumes no detailed background in topology or geometry, and it emphasizes physical motivations, enabling students to apply the techniques to their physics formulas and research. "Thoroughly recommended" by The Physics Bulletin, this volume's physics applications range from condensed matter physics and statistical mechanics to elementary particle theory. Its main mathematical topics include differential forms, homotopy, homology, cohomology, fiber bundles, connection and covariant derivatives, and Morse theory.

Explorations in Mathematical Physics

The Concepts Behind an Elegant Language

Author: Don Koks

Publisher: Springer Science & Business Media

ISBN: 0387309438

Category: Science

Page: 544

View: 8615

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Have you ever wondered why the language of modern physics centres on geometry? Or how quantum operators and Dirac brackets work? What a convolution really is? What tensors are all about? Or what field theory and lagrangians are, and why gravity is described as curvature? This book takes you on a tour of the main ideas forming the language of modern mathematical physics. Here you will meet novel approaches to concepts such as determinants and geometry, wave function evolution, statistics, signal processing, and three-dimensional rotations. You will see how the accelerated frames of special relativity tell us about gravity. On the journey, you will discover how tensor notation relates to vector calculus, how differential geometry is built on intuitive concepts, and how variational calculus leads to field theory. You will meet quantum measurement theory, along with Green functions and the art of complex integration, and finally general relativity and cosmology. The book takes a fresh approach to tensor analysis built solely on the metric and vectors, with no need for one-forms. This gives a much more geometrical and intuitive insight into vector and tensor calculus, together with general relativity, than do traditional, more abstract methods. Don Koks is a physicist at the Defence Science and Technology Organisation in Adelaide, Australia. His doctorate in quantum cosmology was obtained from the Department of Physics and Mathematical Physics at Adelaide University. Prior work at the University of Auckland specialised in applied accelerator physics, along with pure and applied mathematics.

Dynamical Systems

Author: Shlomo Sternberg

Publisher: Courier Corporation

ISBN: 0486135144

Category: Mathematics

Page: 272

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A pioneer in the field of dynamical systems discusses one-dimensional dynamics, differential equations, random walks, iterated function systems, symbolic dynamics, and Markov chains. Supplementary materials include PowerPoint slides and MATLAB exercises. 2010 edition.

Differential Forms and Connections

Author: R. W. R. Darling

Publisher: Cambridge University Press

ISBN: 9780521468008

Category: Mathematics

Page: 256

View: 4133

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This book introduces the tools of modern differential geometry--exterior calculus, manifolds, vector bundles, connections--and covers both classical surface theory, the modern theory of connections, and curvature. Also included is a chapter on applications to theoretical physics. The author uses the powerful and concise calculus of differential forms throughout. Through the use of numerous concrete examples, the author develops computational skills in the familiar Euclidean context before exposing the reader to the more abstract setting of manifolds. The only prerequisites are multivariate calculus and linear algebra; no knowledge of topology is assumed. Nearly 200 exercises make the book ideal for both classroom use and self-study for advanced undergraduate and beginning graduate students in mathematics, physics, and engineering.

The Geometry of Physics

An Introduction

Author: Theodore Frankel

Publisher: Cambridge University Press

ISBN: 1139505610

Category: Mathematics

Page: N.A

View: 5073

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This book provides a working knowledge of those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms that are essential for a deeper understanding of both classical and modern physics and engineering. Included are discussions of analytical and fluid dynamics, electromagnetism (in flat and curved space), thermodynamics, the Dirac operator and spinors, and gauge fields, including Yang–Mills, the Aharonov–Bohm effect, Berry phase and instanton winding numbers, quarks and quark model for mesons. Before discussing abstract notions of differential geometry, geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space. The book is ideal for graduate and advanced undergraduate students of physics, engineering or mathematics as a course text or for self study. This third edition includes an overview of Cartan's exterior differential forms, which previews many of the geometric concepts developed in the text.

Topology and Geometry for Physics

Author: Helmut Eschrig

Publisher: Springer

ISBN: 3642147003

Category: Science

Page: 390

View: 2828

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A concise but self-contained introduction of the central concepts of modern topology and differential geometry on a mathematical level is given specifically with applications in physics in mind. All basic concepts are systematically provided including sketches of the proofs of most statements. Smooth finite-dimensional manifolds, tensor and exterior calculus operating on them, homotopy, (co)homology theory including Morse theory of critical points, as well as the theory of fiber bundles and Riemannian geometry, are treated. Examples from physics comprise topological charges, the topology of periodic boundary conditions for solids, gauge fields, geometric phases in quantum physics and gravitation.

Differential Geometry

Connections, Curvature, and Characteristic Classes

Author: Loring W. Tu

Publisher: Springer

ISBN: 3319550845

Category: Mathematics

Page: 347

View: 9674

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This text presents a graduate-level introduction to differential geometry for mathematics and physics students. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. Along the way we encounter some of the high points in the history of differential geometry, for example, Gauss' Theorema Egregium and the Gauss–Bonnet theorem. Exercises throughout the book test the reader’s understanding of the material and sometimes illustrate extensions of the theory. Initially, the prerequisites for the reader include a passing familiarity with manifolds. After the first chapter, it becomes necessary to understand and manipulate differential forms. A knowledge of de Rham cohomology is required for the last third of the text. Prerequisite material is contained in author's text An Introduction to Manifolds, and can be learned in one semester. For the benefit of the reader and to establish common notations, Appendix A recalls the basics of manifold theory. Additionally, in an attempt to make the exposition more self-contained, sections on algebraic constructions such as the tensor product and the exterior power are included. Differential geometry, as its name implies, is the study of geometry using differential calculus. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein's general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is also useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields. The field has even found applications to group theory as in Gromov's work and to probability theory as in Diaconis's work. It is not too far-fetched to argue that differential geometry should be in every mathematician's arsenal.

Mathematics For Physics: An Illustrated Handbook

Author: Marsh Adam

Publisher: World Scientific

ISBN: 9813233931

Category: Science

Page: 300

View: 1393

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This unique book complements traditional textbooks by providing a visual yet rigorous survey of the mathematics used in theoretical physics beyond that typically covered in undergraduate math and physics courses. The exposition is pedagogical but compact, and the emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints. Certain topics which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations, are avoided. The primary physical models targeted are general relativity, spinors, and gauge theories, with notable chapters on Riemannian geometry, Clifford algebras, and fiber bundles. Contents: Mathematical StructuresAbstract AlgebraVector AlgebrasTopological SpacesAlgebraic TopologyManifoldsLie GroupsClifford GroupsRiemannian ManifoldsFiber BundlesCategories and Functors Readership: Students in mathematics and physics who want to explore a level deeper into actual mathematical content. Keywords: Gauge Theory;Spinor;Clifford Algebra;Homology;Homotopy;Differential Geometry;Lie Group;Lie Algebra;Pauli Matrix;Dirac Matrix;Riemannian Geometry;Parallel Transport;Ricci Curvature;Divergence;Ehresmann ConnectionReview: Key Features: The approach taken by this book to the subject material differs from other titles in that it is intuitive and visual yet also mathematically rigorous, allowing concise coverage of a large breadth of material and providing a cross-subject synthesis while at the same time serving as a useful referenceThe book is also unique in that it can be used in three ways: as the basis for a course, as a supporting text for related courses, and as a reference. These uses could apply to both undergraduate and graduate students. In addition, there is a substantial audience for the book among independent researchers, amateur physicists, and readers of popular science who want to explore a level deeper into actual mathematical contentThe treatments of Riemannian geometry, Clifford algebras, and fiber bundles are particularly notable, including detailed figures and geometric viewpoints that would seem to be novel to the literature

The Language of Shape

The Role of Curvature in Condensed Matter: Physics, Chemistry and Biology

Author: S. Hyde,Z. Blum,T. Landh,S. Lidin,B.W. Ninham,S. Andersson,K. Larsson

Publisher: Elsevier

ISBN: 9780080542546

Category: Science

Page: 383

View: 2891

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This book develops the thesis that structure and function in a variety of condensed systems - from the atomic assemblies in inorganic frameworks and organic molecules, through molecular self-assemblies to proteins - can be unified when curvature and surface geometry are taken together with molecular shape and forces. An astonishing variety of synthetic and biological assemblies can be accurately modelled and understood in terms of hyperbolic surfaces, whose richness and beauty are only now being revealed by applied mathematicians, physicists, chemists and crystallographers. These surfaces, often close to periodic minimal surfaces, weave and twist through space, carving out interconnected labyrinths whose range of topologies and symmetries challenge the imaginative powers. The book offers an overview of these structures and structural transformations, convincingly demonstrating their ubiquity in covalent frameworks from zeolites used for cracking oil and pollution control to enzymes and structural proteins, thermotropic and lyotropic bicontinuous mesophases formed by surfactants, detergents and lipids, synthetic block copolymer and protein networks, as well as biological cell assemblies, from muscles to membranes in prokaryotic and eukaryotic cells. The relation between structure and function is analysed in terms of the previously neglected hidden variables of curvature and topology. Thus, the catalytic activity of zeolites and enzymes, the superior material properties of interpenetrating networks in microstructured polymer composites, the transport requirements in cells, the transmission of nerve signals and the folding of DNA can be more easily understood in the light of this. The text is liberally sprinkled with figures and colour plates, making it accessible to both the beginning graduate student and researchers in condensed matter physics and chemistry, mineralogists, crystallographers and biologists.

Geometry, Topology and Physics, Second Edition

Author: Mikio Nakahara

Publisher: CRC Press

ISBN: 9780750306065

Category: Mathematics

Page: 596

View: 2865

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Differential geometry and topology have become essential tools for many theoretical physicists. In particular, they are indispensable in theoretical studies of condensed matter physics, gravity, and particle physics. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields. The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the development of the subject. The book features a considerably expanded first chapter, reviewing aspects of path integral quantization and gauge theories. Chapter 2 introduces the mathematical concepts of maps, vector spaces, and topology. The following chapters focus on more elaborate concepts in geometry and topology and discuss the application of these concepts to liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later chapters unify geometry and topology, exploring fiber bundles, characteristic classes, and index theorems. New to this second edition is the proof of the index theorem in terms of supersymmetric quantum mechanics. The final two chapters are devoted to the most fascinating applications of geometry and topology in contemporary physics, namely the study of anomalies in gauge field theories and the analysis of Polakov's bosonic string theory from the geometrical point of view. Geometry, Topology and Physics, Second Edition is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics.

Geometry, Relativity and the Fourth Dimension

Author: Rudolf Rucker

Publisher: Courier Corporation

ISBN: 0486140334

Category: Science

Page: 160

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Exposition of fourth dimension, concepts of relativity as Flatland characters continue adventures. Topics include curved space time as a higher dimension, special relativity, and shape of space-time. Includes 141 illustrations.

The Penrose Transform

Its Interaction with Representation Theory

Author: Robert J. Baston,Michael G. Eastwood

Publisher: Courier Dover Publications

ISBN: 0486816621

Category: Mathematics

Page: 256

View: 3053

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Geared toward students of physics and mathematics; presupposes no familiarity with twistor theory. "A huge amount of information, well organized and condensed into less than 200 pages." — Mathematical Reviews. 1989 edition.