An Introduction to Tensor Analysis and Its Geometrical Applications

Author: Jan Arnoldus Schouten

Publisher: Springer Science & Business Media

ISBN: 3662129272

Category: Mathematics

Page: 514

View: 5363

This is an entirely new book. The first edition appeared in 1923 and at that time it was up to date. But in 193 5 and 1938 the author and Prof. D. J. STRUIK published a new book, their Einführung I and li, and this book not only gave the first systematic introduction to the kernel index method but also contained many notions that had come into prominence since 1923. For instance densities, quantities of the second kind, pseudo-quantities, normal Coordinates, the symbolism of exterior forms, the LIE derivative, the theory of variation and deformation and the theory of subprojective connexions were included. Now since 1938 there have been many new developments and so a book on RICCI cal culus and its applications has to cover quite different ground from the book of 1923. Though the purpose remains to make the reader acquainted with RICCI's famous instrument in its modern form, the book must have quite a different methodical structure and quite different applica tions have to be chosen. The first chapter contains algebraical preliminaries but the whole text is modernized and there is a section on hybrid quantities (quantities with indices of the first and of the second kind) and one on the many abridged notations that have been developed by several authors. In the second chapter the most important analytical notions that come before the introduction of a connexion aredealt with in full.
Read More

Author: Marcel Berger

Publisher: Springer Science & Business Media

ISBN: 3642182453

Category: Mathematics

Page: 824

View: 483

This book introduces readers to the living topics of Riemannian Geometry and details the main results known to date. The results are stated without detailed proofs but the main ideas involved are described, affording the reader a sweeping panoramic view of almost the entirety of the field. From the reviews "The book has intrinsic value for a student as well as for an experienced geometer. Additionally, it is really a compendium in Riemannian Geometry." --MATHEMATICAL REVIEWS
Read More

A Centennial Perspective

Author: Abhay Ashtekar,Beverly Berger,James Isenberg,Malcolm MacCallum

Publisher: Cambridge University Press

ISBN: 110703731X

Category: Science

Page: 696

View: 7770

Explore spectacular advances in contemporary physics with this unique celebration of the centennial of Einstein's discovery of general relativity.
Read More

Author: George F. R. Ellis,Roy Maartens,Malcolm A. H. MacCallum

Publisher: Cambridge University Press

ISBN: 0521381150

Category: Science

Page: 622

View: 4264

Surveying key developments and open issues in cosmology for graduate students and researchers, this book focuses on the general concepts and relations that underpin the standard model of the Universe. It also examines anisotropic and inhomogeneous models, and deeper issues, such as quantum cosmology and the multiverse proposal.
Read More

Author: Luther Pfahler Eisenhart

Publisher: Princeton University Press

ISBN: 1400884217

Category: Mathematics

Page: 272

View: 1234

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.
Read More

Author: Nazrul Islam

Publisher: New Age International

ISBN: 8122418384

Category: Tensor algebra

Page: 262

View: 6037

The Book Is Written Is In Easy-To-Read Style With Corresponding Examples. The Main Aim Of This Book Is To Precisely Explain The Fundamentals Of Tensors And Their Applications To Mechanics, Elasticity, Theory Of Relativity, Electromagnetic, Riemannian Geometry And Many Other Disciplines Of Science And Engineering, In A Lucid Manner. The Text Has Been Explained Section Wise, Every Concept Has Been Narrated In The Form Of Definition, Examples And Questions Related To The Concept Taught. The Overall Package Of The Book Is Highly Useful And Interesting For The People Associated With The Field.
Read More

Old and New

Author: Cédric Villani

Publisher: Springer Science & Business Media

ISBN: 3540710507

Category: Mathematics

Page: 976

View: 7134

At the close of the 1980s, the independent contributions of Yann Brenier, Mike Cullen and John Mather launched a revolution in the venerable field of optimal transport founded by G. Monge in the 18th century, which has made breathtaking forays into various other domains of mathematics ever since. The author presents a broad overview of this area, supplying complete and self-contained proofs of all the fundamental results of the theory of optimal transport at the appropriate level of generality. Thus, the book encompasses the broad spectrum ranging from basic theory to the most recent research results. PhD students or researchers can read the entire book without any prior knowledge of the field. A comprehensive bibliography with notes that extensively discuss the existing literature underlines the book’s value as a most welcome reference text on this subject.
Read More

Author: Jan Arnoldus Schouten

Publisher: Courier Corporation

ISBN: 0486655822

Category: Science

Page: 277

View: 7263

This rigorous and advanced mathematical explanation of classic tensor analysis was written by one of the founders of tensor calculus. Its concise exposition of the mathematical basis of the discipline is integrated with well-chosen physical examples of the theory, including those involving elasticity, classical dynamics, relativity, and Dirac's matrix calculus. 1954 edition.
Read More

Pamphlets, leaflets, contributions to newspapers or periodicals, etc., maps

Author: N.A

Publisher: N.A

ISBN: N.A

Category: American literature

Page: N.A

View: 8546

Read More

Geometry and Applications

Author: J. M. Landsberg

Publisher: American Mathematical Soc.

ISBN: 0821869078

Category: Mathematics

Page: 439

View: 9903

Tensors are ubiquitous in the sciences. The geometry of tensors is both a powerful tool for extracting information from data sets, and a beautiful subject in its own right. This book has three intended uses: a classroom textbook, a reference work for researchers in the sciences, and an account of classical and modern results in (aspects of) the theory that will be of interest to researchers in geometry. For classroom use, there is a modern introduction to multilinear algebra and to the geometry and representation theory needed to study tensors, including a large number of exercises. For researchers in the sciences, there is information on tensors in table format for easy reference and a summary of the state of the art in elementary language. This is the first book containing many classical results regarding tensors. Particular applications treated in the book include the complexity of matrix multiplication, P versus NP, signal processing, phylogenetics, and algebraic statistics. For geometers, there is material on secant varieties, G-varieties, spaces with finitely many orbits and how these objects arise in applications, discussions of numerous open questions in geometry arising in applications, and expositions of advanced topics such as the proof of the Alexander-Hirschowitz theorem and of the Weyman-Kempf method for computing syzygies.
Read More

An Introduction

Author: V. S. Varadarajan

Publisher: American Mathematical Soc.

ISBN: 0821835742

Category: Mathematics

Page: 300

View: 4667

Supersymmetry has been the object of study by theoretical physicists since the early 1970's. In recent years it has attracted the interest of mathematicians because of its novelty, and because of significance, both in mathematics and physics, of the main issues it raises. This book presents the foundations of supersymmetry to the mathematically minded reader in a cogent and self-contained manner. It begins with a brief introduction to the physical foundations of the theory, especially the classification of relativistic particles and their wave equations, such as the equations of Dirac and Weyl. It then continues the development of the theory of supermanifolds stressing the analogy with the Grothendieck theory of schemes. All the super linear algebra needed for the book is developed here and the basic theorems are established: differential and integral calculus in supermanifolds, Frobenius theorem, foundations of the theory of super Lie groups, and so on. A special feature of the book is the treatment in depth of the theory of spinors in all dimensions and signatures, which is the basis of all developments of supergeometry both in physics and mathematics, especially in quantum field theory and supergravity.
Read More

Author: Hanno Rund

Publisher: Springer Science & Business Media

ISBN: 3642516106

Category: Mathematics

Page: 284

View: 7355

The present monograph is motivated by two distinct aims. Firstly, an endeavour has been made to furnish a reasonably comprehensive account of the theory of Finsler spaces based on the methods of classical differential geometry. Secondly, it is hoped that this monograph may serve also as an introduction to a branch of differential geometry which is closely related to various topics in theoretical physics, notably analytical dynamics and geometrical optics. With this second object in mind, an attempt has been made to describe the basic aspects of the theory in some detail - even at the expense of conciseness - while in the more specialised sections of the later chapters, which might be of interest chiefly to the specialist, a more succinct style has been adopted. The fact that there exist several fundamentally different points of view with regard to Finsler geometry has rendered the task of writing a coherent account a rather difficult one. This remark is relevant not only to the development of the subject on the basis of the tensor calculus, but is applicable in an even wider sense. The extensive work of H. BUSEMANN has opened up new avenues of approach to Finsler geometry which are independent of the methods of classical tensor analysis. In the latter sense, therefore, a full description of this approach does not fall within the scope of this treatise, although its fundamental l significance cannot be doubted.
Read More

Author: Nicola Gigli

Publisher: American Mathematical Soc.

ISBN: 1470427656

Category: Geometry, Differential

Page: 161

View: 2448

The author discusses in which sense general metric measure spaces possess a first order differential structure. Building on this, spaces with Ricci curvature bounded from below a second order calculus can be developed, permitting the author to define Hessian, covariant/exterior derivatives and Ricci curvature.
Read More

From Stratifolds to Exotic Spheres

Author: Matthias Kreck

Publisher: American Mathematical Soc.

ISBN: 0821848984

Category: Mathematics

Page: 218

View: 1256

This book presents a geometric introduction to the homology of topological spaces and the cohomology of smooth manifolds. The author introduces a new class of stratified spaces, so-called stratifolds. He derives basic concepts from differential topology such as Sard's theorem, partitions of unity and transversality. Based on this, homology groups are constructed in the framework of stratifolds and the homology axioms are proved. This implies that for nice spaces these homology groups agree with ordinary singular homology. Besides the standard computations of homology groups using the axioms, straightforward constructions of important homology classes are given. The author also defines stratifold cohomology groups following an idea of Quillen. Again, certain important cohomology classes occur very naturally in this description, for example, the characteristic classes which are constructed in the book and applied later on. One of the most fundamental results, Poincare duality, is almost a triviality in this approach. Some fundamental invariants, such as the Euler characteristic and the signature, are derived from (co)homology groups. These invariants play a significant role in some of the most spectacular results in differential topology. In particular, the author proves a special case of Hirzebruch's signature theorem and presents as a highlight Milnor's exotic 7-spheres. This book is based on courses the author taught in Mainz and Heidelberg. Readers should be familiar with the basic notions of point-set topology and differential topology. The book can be used for a combined introduction to differential and algebraic topology, as well as for a quick presentation of (co)homology in a course about differential geometry.
Read More