Tensor Geometry

The Geometric Viewpoint and its Uses
Author: Christopher T. J. Dodson,Timothy Poston
Publisher: Springer Science & Business Media
ISBN: 3642105149
Category: Mathematics
Page: 434
View: 6514

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This treatment of differential geometry and the mathematics required for general relativity makes the subject accessible, for the first time, to anyone familiar with elementary calculus in one variable and with some knowledge of vector algebra. The emphasis throughout is on the geometry of the mathematics, which is greatly enhanced by the many illustrations presenting figures of three and more dimensions as closely as the book form will allow.

Tensor Geometry

The Geometric Viewpoint and Its Uses
Author: C. T. J. Dodson,T. Poston
Publisher: Springer
ISBN: N.A
Category: Mathematics
Page: 432
View: 467

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Séminaire de Probabilités XLII


Author: Catherine Donati-Martin,Michel Émery,Alain Rouault,Christophe Stricker
Publisher: Springer
ISBN: 3642017630
Category: Mathematics
Page: 449
View: 6260

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This book offers an introduction to rough paths. Coverage also includes the interface between analysis and probability to special processes, Lévy processes and Lévy systems, representation of Gaussian processes, filtrations and quantum probability.

Complexity Dichotomies for Counting Problems: Volume 1, Boolean Domain


Author: Jin-Yi Cai,Xi Chen
Publisher: Cambridge University Press
ISBN: 1108508820
Category: Computers
Page: N.A
View: 2168

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Complexity theory aims to understand and classify computational problems, especially decision problems, according to their inherent complexity. This book uses new techniques to expand the theory for use with counting problems. The authors present dichotomy classifications for broad classes of counting problems in the realm of P and NP. Classifications are proved for partition functions of spin systems, graph homomorphisms, constraint satisfaction problems, and Holant problems. The book assumes minimal prior knowledge of computational complexity theory, developing proof techniques as needed and gradually increasing the generality and abstraction of the theory. This volume presents the theory on the Boolean domain, and includes a thorough presentation of holographic algorithms, culminating in classifications of computational problems studied in exactly solvable models from statistical mechanics.

Symmetry, Representations, and Invariants


Author: Roe Goodman,Nolan R. Wallach
Publisher: Springer Science & Business Media
ISBN: 0387798528
Category: Mathematics
Page: 716
View: 3823

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Symmetry is a key ingredient in many mathematical, physical, and biological theories. Using representation theory and invariant theory to analyze the symmetries that arise from group actions, and with strong emphasis on the geometry and basic theory of Lie groups and Lie algebras, Symmetry, Representations, and Invariants is a significant reworking of an earlier highly-acclaimed work by the authors. The result is a comprehensive introduction to Lie theory, representation theory, invariant theory, and algebraic groups, in a new presentation that is more accessible to students and includes a broader range of applications. The philosophy of the earlier book is retained, i.e., presenting the principal theorems of representation theory for the classical matrix groups as motivation for the general theory of reductive groups. The wealth of examples and discussion prepares the reader for the complete arguments now given in the general case. Key Features of Symmetry, Representations, and Invariants: (1) Early chapters suitable for honors undergraduate or beginning graduate courses, requiring only linear algebra, basic abstract algebra, and advanced calculus; (2) Applications to geometry (curvature tensors), topology (Jones polynomial via symmetry), and combinatorics (symmetric group and Young tableaux); (3) Self-contained chapters, appendices, comprehensive bibliography; (4) More than 350 exercises (most with detailed hints for solutions) further explore main concepts; (5) Serves as an excellent main text for a one-year course in Lie group theory; (6) Benefits physicists as well as mathematicians as a reference work.

Foundations of Differentiable Manifolds and Lie Groups


Author: Frank W. Warner
Publisher: Springer Science & Business Media
ISBN: 1475717997
Category: Mathematics
Page: 276
View: 3617

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Foundations of Differentiable Manifolds and Lie Groups gives a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. Coverage includes differentiable manifolds, tensors and differentiable forms, Lie groups and homogenous spaces, and integration on manifolds. The book also provides a proof of the de Rham theorem via sheaf cohomology theory and develops the local theory of elliptic operators culminating in a proof of the Hodge theorem.

Geometrical Methods of Mathematical Physics


Author: Bernard F. Schutz
Publisher: Cambridge University Press
ISBN: 1107268141
Category: Science
Page: N.A
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In recent years the methods of modern differential geometry have become of considerable importance in theoretical physics and have found application in relativity and cosmology, high-energy physics and field theory, thermodynamics, fluid dynamics and mechanics. This textbook provides an introduction to these methods - in particular Lie derivatives, Lie groups and differential forms - and covers their extensive applications to theoretical physics. The reader is assumed to have some familiarity with advanced calculus, linear algebra and a little elementary operator theory. The advanced physics undergraduate should therefore find the presentation quite accessible. This account will prove valuable for those with backgrounds in physics and applied mathematics who desire an introduction to the subject. Having studied the book, the reader will be able to comprehend research papers that use this mathematics and follow more advanced pure-mathematical expositions.

Manifolds, Tensors and Forms


Author: Paul Renteln
Publisher: Cambridge University Press
ISBN: 1107042194
Category: Science
Page: 340
View: 2903

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Comprehensive treatment of the essentials of modern differential geometry and topology for graduate students in mathematics and the physical sciences.

Differential Geometry and Topology

With a View to Dynamical Systems
Author: Keith Burns,Marian Gidea
Publisher: CRC Press
ISBN: 9781584882534
Category: Mathematics
Page: 400
View: 2417

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Accessible, concise, and self-contained, this book offers an outstanding introduction to three related subjects: differential geometry, differential topology, and dynamical systems. Topics of special interest addressed in the book include Brouwer's fixed point theorem, Morse Theory, and the geodesic flow. Smooth manifolds, Riemannian metrics, affine connections, the curvature tensor, differential forms, and integration on manifolds provide the foundation for many applications in dynamical systems and mechanics. The authors also discuss the Gauss-Bonnet theorem and its implications in non-Euclidean geometry models. The differential topology aspect of the book centers on classical, transversality theory, Sard's theorem, intersection theory, and fixed-point theorems. The construction of the de Rham cohomology builds further arguments for the strong connection between the differential structure and the topological structure. It also furnishes some of the tools necessary for a complete understanding of the Morse theory. These discussions are followed by an introduction to the theory of hyperbolic systems, with emphasis on the quintessential role of the geodesic flow. The integration of geometric theory, topological theory, and concrete applications to dynamical systems set this book apart. With clean, clear prose and effective examples, the authors' intuitive approach creates a treatment that is comprehensible to relative beginners, yet rigorous enough for those with more background and experience in the field.

Differential Geometric Structures


Author: Walter A. Poor
Publisher: Courier Corporation
ISBN: 0486151913
Category: Mathematics
Page: 352
View: 4615

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This introductory text defines geometric structure by specifying parallel transport in an appropriate fiber bundle and focusing on simplest cases of linear parallel transport in a vector bundle. 1981 edition.

Geometrical Properties of Vectors and Convectors

An Introductory Survey of Differentiable Manifolds, Tensors and Forms
Author: Joaquim M. Domingos
Publisher: World Scientific
ISBN: 9812700447
Category: Mathematics
Page: 73
View: 7862

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This is a brief introduction to some geometrical topics including topological spaces, the metric tensor, Euclidean space, manifolds, tensors, r-forms, the orientation of a manifold and the Hodge star operator. It provides the reader who is approaching the subject for the first time with a deeper understanding of the geometrical properties of vectors and covectors. The material prepares the reader for discussions on basic concepts such as the differential of a function as a covector, metric dual, inner product, wedge product and cross product.J M Domingos received his D Phil from the University of Oxford and has now retired from the post of Professor of Physics at the University of Coimbra, Portugal.

Clifford (Geometric) Algebras

with applications to physics, mathematics, and engineering
Author: William Baylis
Publisher: Springer Science & Business Media
ISBN: 1461241049
Category: Science
Page: 517
View: 8385

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This volume is an outgrowth of the 1995 Summer School on Theoretical Physics of the Canadian Association of Physicists (CAP), held in Banff, Alberta, in the Canadian Rockies, from July 30 to August 12,1995. The chapters, based on lectures given at the School, are designed to be tutorial in nature, and many include exercises to assist the learning process. Most lecturers gave three or four fifty-minute lectures aimed at relative novices in the field. More emphasis is therefore placed on pedagogy and establishing comprehension than on erudition and superior scholarship. Of course, new and exciting results are presented in applications of Clifford algebras, but in a coherent and user-friendly way to the nonspecialist. The subject area of the volume is Clifford algebra and its applications. Through the geometric language of the Clifford-algebra approach, many concepts in physics are clarified, united, and extended in new and sometimes surprising directions. In particular, the approach eliminates the formal gaps that traditionally separate clas sical, quantum, and relativistic physics. It thereby makes the study of physics more efficient and the research more penetrating, and it suggests resolutions to a major physics problem of the twentieth century, namely how to unite quantum theory and gravity. The term "geometric algebra" was used by Clifford himself, and David Hestenes has suggested its use in order to emphasize its wide applicability, and b& cause the developments by Clifford were themselves based heavily on previous work by Grassmann, Hamilton, Rodrigues, Gauss, and others.

Geometry: Euclid and Beyond


Author: Robin Hartshorne
Publisher: Springer Science & Business Media
ISBN: 0387226761
Category: Mathematics
Page: 528
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This book offers a unique opportunity to understand the essence of one of the great thinkers of western civilization. A guided reading of Euclid's Elements leads to a critical discussion and rigorous modern treatment of Euclid's geometry and its more recent descendants, with complete proofs. Topics include the introduction of coordinates, the theory of area, history of the parallel postulate, the various non-Euclidean geometries, and the regular and semi-regular polyhedra.

A New Approach to Differential Geometry using Clifford's Geometric Algebra


Author: John Snygg
Publisher: Springer Science & Business Media
ISBN: 081768283X
Category: Mathematics
Page: 465
View: 8327

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Differential geometry is the study of the curvature and calculus of curves and surfaces. A New Approach to Differential Geometry using Clifford's Geometric Algebra simplifies the discussion to an accessible level of differential geometry by introducing Clifford algebra. This presentation is relevant because Clifford algebra is an effective tool for dealing with the rotations intrinsic to the study of curved space. Complete with chapter-by-chapter exercises, an overview of general relativity, and brief biographies of historical figures, this comprehensive textbook presents a valuable introduction to differential geometry. It will serve as a useful resource for upper-level undergraduates, beginning-level graduate students, and researchers in the algebra and physics communities.

Differential Geometry

Connections, Curvature, and Characteristic Classes
Author: Loring W. Tu
Publisher: Springer
ISBN: 3319550845
Category: Mathematics
Page: 347
View: 1576

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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.

Spin Geometry (PMS-38)


Author: H. Blaine Lawson,Marie-Louise Michelsohn
Publisher: Princeton University Press
ISBN: 1400883911
Category: Mathematics
Page: 440
View: 8049

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This book offers a systematic and comprehensive presentation of the concepts of a spin manifold, spinor fields, Dirac operators, and A-genera, which, over the last two decades, have come to play a significant role in many areas of modern mathematics. Since the deeper applications of these ideas require various general forms of the Atiyah-Singer Index Theorem, the theorems and their proofs, together with all prerequisite material, are examined here in detail. The exposition is richly embroidered with examples and applications to a wide spectrum of problems in differential geometry, topology, and mathematical physics. The authors consistently use Clifford algebras and their representations in this exposition. Clifford multiplication and Dirac operator identities are even used in place of the standard tensor calculus. This unique approach unifies all the standard elliptic operators in geometry and brings fresh insights into curvature calculations. The fundamental relationships of Clifford modules to such topics as the theory of Lie groups, K-theory, KR-theory, and Bott Periodicity also receive careful consideration. A special feature of this book is the development of the theory of Cl-linear elliptic operators and the associated index theorem, which connects certain subtle spin-corbordism invariants to classical questions in geometry and has led to some of the most profound relations known between the curvature and topology of manifolds.

Curvature in Mathematics and Physics


Author: Shlomo Sternberg
Publisher: Courier Corporation
ISBN: 0486292711
Category: Mathematics
Page: 416
View: 9661

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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.

TENSORS

MATHEMATICS OF DIFFERENTIAL GEOMETRY AND RELATIVITY
Author: AHSAN, ZAFAR
Publisher: PHI Learning Pvt. Ltd.
ISBN: 812035088X
Category: Mathematics
Page: 240
View: 8547

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The principal aim of analysis of tensors is to investigate those relations which remain valid when we change from one coordinate system to another. This book on Tensors requires only a knowledge of elementary calculus, differential equations and classical mechanics as pre-requisites. It provides the readers with all the information about the tensors along with the derivation of all the tensorial relations/equations in a simple manner. The book also deals in detail with topics of importance to the study of special and general relativity and the geometry of differentiable manifolds with a crystal clear exposition. The concepts dealt within the book are well supported by a number of solved examples. A carefully selected set of unsolved problems is also given at the end of each chapter, and the answers and hints for the solution of these problems are given at the end of the book. The applications of tensors to the fields of differential geometry, relativity, cosmology and electromagnetism is another attraction of the present book. This book is intended to serve as text for postgraduate students of mathematics, physics and engineering. It is ideally suited for both students and teachers who are engaged in research in General Theory of Relativity and Differential Geometry.

Geometric Control of Mechanical Systems

Modeling, Analysis, and Design for Simple Mechanical Control Systems
Author: Francesco Bullo,Andrew D. Lewis
Publisher: Springer Science & Business Media
ISBN: 9780387221953
Category: Science
Page: 727
View: 6503

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The area of analysis and control of mechanical systems using differential geometry is flourishing. This book collects many results over the last decade and provides a comprehensive introduction to the area.