## Global Analysis

Differential Forms in Analysis, Geometry, and Physics
Author: Ilka Agricola,Thomas Friedrich
Publisher: American Mathematical Soc.
ISBN: 0821829513
Category: Mathematics
Page: 343
View: 5494

This book introduces the reader to the world of differential forms and their uses in geometry, analysis, and mathematical physics. It begins with a few basic topics, partly as review, then moves on to vector analysis on manifolds and the study of curves and surfaces in \$3\$-space. Lie groups and homogeneous spaces are discussed, providing the appropriate framework for introducing symmetry in both mathematical and physical contexts. The final third of the book applies the mathematical ideas to important areas of physics: Hamiltonian mechanics, statistical mechanics, and electrodynamics. There are many classroom-tested exercises and examples with excellent figures throughout. The book is ideal as a text for a first course in differential geometry, suitable for advanced undergraduates or graduate students in mathematics or physics.

## Introduction to Geometry and Topology

Author: Werner Ballmann
Publisher: Birkhäuser
ISBN: 3034809832
Category: Mathematics
Page: 169
View: 1800

This book provides an introduction to topology, differential topology, and differential geometry. It is based on manuscripts refined through use in a variety of lecture courses. The first chapter covers elementary results and concepts from point-set topology. An exception is the Jordan Curve Theorem, which is proved for polygonal paths and is intended to give students a first glimpse into the nature of deeper topological problems. The second chapter of the book introduces manifolds and Lie groups, and examines a wide assortment of examples. Further discussion explores tangent bundles, vector bundles, differentials, vector fields, and Lie brackets of vector fields. This discussion is deepened and expanded in the third chapter, which introduces the de Rham cohomology and the oriented integral and gives proofs of the Brouwer Fixed-Point Theorem, the Jordan-Brouwer Separation Theorem, and Stokes's integral formula. The fourth and final chapter is devoted to the fundamentals of differential geometry and traces the development of ideas from curves to submanifolds of Euclidean spaces. Along the way, the book discusses connections and curvature--the central concepts of differential geometry. The discussion culminates with the Gauß equations and the version of Gauß's theorema egregium for submanifolds of arbitrary dimension and codimension. This book is primarily aimed at advanced undergraduates in mathematics and physics and is intended as the template for a one- or two-semester bachelor's course.

## Mathematical Physics: Classical Mechanics

Author: Andreas Knauf
Publisher: Springer
ISBN: 3662557746
Category: Science
Page: 683
View: 2298

As a limit theory of quantum mechanics, classical dynamics comprises a large variety of phenomena, from computable (integrable) to chaotic (mixing) behavior. This book presents the KAM (Kolmogorov-Arnold-Moser) theory and asymptotic completeness in classical scattering. Including a wealth of fascinating examples in physics, it offers not only an excellent selection of basic topics, but also an introduction to a number of current areas of research in the field of classical mechanics. Thanks to the didactic structure and concise appendices, the presentation is self-contained and requires only knowledge of the basic courses in mathematics. The book addresses the needs of graduate and senior undergraduate students in mathematics and physics, and of researchers interested in approaching classical mechanics from a modern point of view.

## Globale Analysis

Differentialformen in Analysis, Geometrie und Physik
Author: Ilka Agricola,Thomas Friedrich
Publisher: Springer-Verlag
ISBN: 3322929035
Category: Mathematics
Page: 283
View: 1657

Das Anliegen des Buches ist es, die klassische Vektoranalysis unter Verwendung der Differentialformen darzulegen. Anwendungen der allgemeinen Stokeschen Formel in Analysis, Geometrie und Topologie werden besprochen. In weiteren Teilen des Buches werden die Integrierbarkeit Pfaffscher Systeme, die Flächentheorie in Euklidischen Räumen sowie Elemente der Lie-Gruppen, Mechanik, Thermodynamik und Elektrodynamik unter Verwendung der Differentialformen behandelt.

## Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

Author: Josef Dick,Frances Y. Kuo,Henryk Woźniakowski
Publisher: Springer
ISBN: 3319724568
Category: Mathematics
Page: 1309
View: 8732

This book is a tribute to Professor Ian Hugh Sloan on the occasion of his 80th birthday. It consists of nearly 60 articles written by international leaders in a diverse range of areas in contemporary computational mathematics. These papers highlight the impact and many achievements of Professor Sloan in his distinguished academic career. The book also presents state of the art knowledge in many computational fields such as quasi-Monte Carlo and Monte Carlo methods for multivariate integration, multi-level methods, finite element methods, uncertainty quantification, spherical designs and integration on the sphere, approximation and interpolation of multivariate functions, oscillatory integrals, and in general in information-based complexity and tractability, as well as in a range of other topics. The book also tells the life story of the renowned mathematician, family man, colleague and friend, who has been an inspiration to many of us. The reader may especially enjoy the story from the perspective of his family, his wife, his daughter and son, as well as grandchildren, who share their views of Ian. The clear message of the book is that Ian H. Sloan has been a role model in science and life.

## Hypoelliptic heat kernel inequalities on lie groups

Author: Tai Alexis Melcher
Publisher: N.A
ISBN: N.A
Category:
Page: 240
View: 3120

## Differential Geometry for Physicists

Author: Bo-Yu Hou,Bo-Yuan Hou
Publisher: World Scientific Publishing Company
ISBN: 9813105097
Category: Mathematics
Page: 560
View: 8770

This book is divided into fourteen chapters, with 18 appendices as introduction to prerequisite topological and algebraic knowledge, etc. The first seven chapters focus on local analysis. This part can be used as a fundamental textbook for graduate students of theoretical physics. Chapters 8–10 discuss geometry on fibre bundles, which facilitates further reference for researchers. The last four chapters deal with the Atiyah-Singer index theorem, its generalization and its application, quantum anomaly, cohomology field theory and noncommutative geometry, giving the reader a glimpse of the frontier of current research in theoretical physics.

## Handbook of Global Analysis

Author: Demeter Krupka,David Saunders
Publisher: Elsevier
ISBN: 9780080556734
Category: Mathematics
Page: 1244
View: 734

This is a comprehensive exposition of topics covered by the American Mathematical Society’s classification “Global Analysis , dealing with modern developments in calculus expressed using abstract terminology. It will be invaluable for graduate students and researchers embarking on advanced studies in mathematics and mathematical physics. This book provides a comprehensive coverage of modern global analysis and geometrical mathematical physics, dealing with topics such as; structures on manifolds, pseudogroups, Lie groupoids, and global Finsler geometry; the topology of manifolds and differentiable mappings; differential equations (including ODEs, differential systems and distributions, and spectral theory); variational theory on manifolds, with applications to physics; function spaces on manifolds; jets, natural bundles and generalizations; and non-commutative geometry. - Comprehensive coverage of modern global analysis and geometrical mathematical physics - Written by world-experts in the field - Up-to-date contents

## Perspectives of Complex Analysis, Differential Geometry and Mathematical Physics

Proceedings of the 5th International Workshop on Complex Structures and Vector Fields : St. Konstantin, Bulgaria, 3-9 September 2000
Author: Stancho Dimiev,Kouei Sekigawa
Publisher: World Scientific
ISBN: 9812810145
Category: Mathematics
Page: 220
View: 4304

This workshop brought together specialists in complex analysis, differential geometry, mathematical physics and applications for stimulating cross-disciplinary discussions. The lectures presented ranged over various current topics in those fields. The proceedings will be of value to graduate students and researchers in complex analysis, differential geometry and theoretical physics, and also related fields. Contents: Length Spectrum of Geodesic Spheres in Non-Flat Complex and Quaternionic Space Forms (T Adachi); Canal Hypersurfaces of Second Type (G Ganchev); Weierstrass Formula for Super Minimal J-Holomorphic Curves of a 6-Dimensional Sphere and Its Applications (H Hashimoto); Real Hypersurfaces of Kaehler Manifold (Sixteen Classes) (M Hristov); Almost Hermitian Manifolds of Poinwise Constant Antiholomorphic Sectional Curvature (O Kassabov & G Ganchev); The Quotient Space of the Complex Projective Plane Under Conjugation is a 4-Sphere (K Kikuchi); On a Generalization of CMC OCo 1 Surfaces Theory (M Kokubu); The Deligne-Simpson Problem (V Kostov); and other papers. Readership: Graduate students and researchers in mathematics and mathematical physics."

## Mathematical Reviews

Author: N.A
Publisher: N.A
ISBN: N.A
Category: Mathematics
Page: N.A
View: 6460

## Differential Forms with Applications to the Physical Sciences

Author: Harley Flanders
Publisher: Courier Corporation
ISBN: 0486139611
Category: Mathematics
Page: 240
View: 2389

A graduate-level text utilizing exterior differential forms in the analysis of a variety of mathematical problems in the physical and engineering sciences. Includes 45 illustrations. Index.

## Differential Geometry and the Calculus of Variations by Robert Hermann

Author: N.A
Publisher: Elsevier
ISBN: 9780080955575
Category: Mathematics
Page: 322
View: 371

In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation; methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; and methods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory. As a result, the book represents a blend of new methods in general computational analysis, and specific, but also generic, techniques for study of systems theory ant its particular branches, such as optimal filtering and information compression. - Best operator approximation, - Non-Lagrange interpolation, - Generic Karhunen-Loeve transform - Generalised low-rank matrix approximation - Optimal data compression - Optimal nonlinear filtering

## Advances in Discrete Differential Geometry

Author: Alexander I. Bobenko
Publisher: Springer
ISBN: 3662504472
Category: Mathematics
Page: 439
View: 8699

This is one of the first books on a newly emerging field of discrete differential geometry and an excellent way to access this exciting area. It surveys the fascinating connections between discrete models in differential geometry and complex analysis, integrable systems and applications in computer graphics. The authors take a closer look at discrete models in differential geometry and dynamical systems. Their curves are polygonal, surfaces are made from triangles and quadrilaterals, and time is discrete. Nevertheless, the difference between the corresponding smooth curves, surfaces and classical dynamical systems with continuous time can hardly be seen. This is the paradigm of structure-preserving discretizations. Current advances in this field are stimulated to a large extent by its relevance for computer graphics and mathematical physics. This book is written by specialists working together on a common research project. It is about differential geometry and dynamical systems, smooth and discrete theories, and on pure mathematics and its practical applications. The interaction of these facets is demonstrated by concrete examples, including discrete conformal mappings, discrete complex analysis, discrete curvatures and special surfaces, discrete integrable systems, conformal texture mappings in computer graphics, and free-form architecture. This richly illustrated book will convince readers that this new branch of mathematics is both beautiful and useful. It will appeal to graduate students and researchers in differential geometry, complex analysis, mathematical physics, numerical methods, discrete geometry, as well as computer graphics and geometry processing.

## Graduate Programs in the Physical Sciences, Mathematics, Agricultural Sciences, the Environment, and Natural Resources 2009

Author: Peterson's Guides Staff
Publisher: Petersons
ISBN: 9780768924022
Category: Reference
Page: 751
View: 6103

Offers information on entrance and degree requirements, expenses and financial aid, programs of study, and faculty research specialties.

## Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers

Author: Hung Nguyen-Schäfer,Jan-Philip Schmidt
Publisher: Springer
ISBN: 3662484978
Category: Technology & Engineering
Page: 376
View: 4552

This book presents tensors and differential geometry in a comprehensive and approachable manner, providing a bridge from the place where physics and engineering mathematics end, and the place where tensor analysis begins. Among the topics examined are tensor analysis, elementary differential geometry of moving surfaces, and k-differential forms. The book includes numerous examples with solutions and concrete calculations, which guide readers through these complex topics step by step. Mindful of the practical needs of engineers and physicists, book favors simplicity over a more rigorous, formal approach. The book shows readers how to work with tensors and differential geometry and how to apply them to modeling the physical and engineering world. The authors provide chapter-length treatment of topics at the intersection of advanced mathematics, and physics and engineering: • General Basis and Bra-Ket Notation • Tensor Analysis • Elementary Differential Geometry • Differential Forms • Applications of Tensors and Differential Geometry • Tensors and Bra-Ket Notation in Quantum Mechanics The text reviews methods and applications in computational fluid dynamics; continuum mechanics; electrodynamics in special relativity; cosmology in the Minkowski four-dimensional space time; and relativistic and non-relativistic quantum mechanics. Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers benefits research scientists and practicing engineers in a variety of fields, who use tensor analysis and differential geometry in the context of applied physics, and electrical and mechanical engineering. It will also interest graduate students in applied physics and engineering.

## Modes

Author: Anna B. Romanowska,Jonathan D. H. Smith
Publisher: World Scientific
ISBN: 9789810249427
Category: Science
Page: 623
View: 8129

Introduces the theory and application of modes, covering topics in universal algebra, category theory, and modal theory, and includes exercises to illustrate concepts.

## Geometry, Topology and Physics, Second Edition

Author: Mikio Nakahara
Publisher: CRC Press
ISBN: 9780750306065
Category: Mathematics
Page: 596
View: 3693

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.

## Quasi-Linear Perturbations of Hamiltonian Klein-Gordon Equations on Spheres

Author: J.-M. Delort
Publisher: American Mathematical Soc.
ISBN: 1470409836
Category: Mathematics
Page: 80
View: 8691

The Hamiltonian ∫X(∣∂tu∣2+∣∇u∣2+m2∣u∣2)dx, defined on functions on R×X, where X is a compact manifold, has critical points which are solutions of the linear Klein-Gordon equation. The author considers perturbations of this Hamiltonian, given by polynomial expressions depending on first order derivatives of u. The associated PDE is then a quasi-linear Klein-Gordon equation. The author shows that, when X is the sphere, and when the mass parameter m is outside an exceptional subset of zero measure, smooth Cauchy data of small size ϵ give rise to almost global solutions, i.e. solutions defined on a time interval of length cNϵ−N for any N. Previous results were limited either to the semi-linear case (when the perturbation of the Hamiltonian depends only on u) or to the one dimensional problem. The proof is based on a quasi-linear version of the Birkhoff normal forms method, relying on convenient generalizations of para-differential calculus.

## Fourier Analysis

Analytic and Geometric Aspects
Author: William O. Bray,P. Milojevic,C.V. Stanojevic
Publisher: CRC Press
ISBN: 9780824792084
Category: Mathematics
Page: 472
View: 3743

Providing complete expository and research papers on the geometric and analytic aspects of Fourier analysis, this work discusses new approaches to classical problems in the theory of trigonometric series, singular integrals/pseudo-differential operators, Fourier analysis on various groups, numerical aspects of Fourier analysis and their applications, wavelets and more.