# Search Results for "tensors-mathematics-of-differential-geometry-and-relativity"

## TENSORS

*MATHEMATICS OF DIFFERENTIAL GEOMETRY AND RELATIVITY*

**Author**: AHSAN, ZAFAR**Publisher:**PHI Learning Pvt. Ltd.**ISBN:**812035088X**Category:**Mathematics**Page:**240**View:**2427

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.

## Differential Geometry And Tensors

**Author**: K.K. Dube**Publisher:**I. K. International Pvt Ltd**ISBN:**9380026587**Category:****Page:**382**View:**6596

The purpose of this book is to give a simple, lucid, rigorous and comprehensive account of fundamental notions of Differential Geometry and Tensors. The book is self-contained and divided in two parts. Section A deals with Differential Geometry and Section B is devoted to the study of Tensors. Section A deals with: " Theory of curves, envelopes and developables. " Curves on surfaces and fundamental magnitudes, curvature of surfaces and lines of curvature. " Fundamental equations of surface theory. " Geodesics. Section B deals with: " Tensor algebra. " Tensor calculus. " Christoffel symbols and their properties. " Riemann symbols and Einstein space, and their properties. " Physical components of contravariant and covariant vectors. " Geodesics and Parallelism of vectors. " Differentiable manifolds, charts, atlases.

## TEXTBOOK OF TENSOR CALCULUS AND DIFFERENTIAL GEOMETRY

**Author**: PRASUN KUMAR NAYAK**Publisher:**PHI Learning Pvt. Ltd.**ISBN:**812034507X**Category:**Mathematics**Page:**552**View:**9933

Primarily intended for the undergraduate and postgraduate students of mathematics, this textbook covers both geometry and tensor in a single volume. This book aims to provide a conceptual exposition of the fundamental results in the theory of tensors. It also illustrates the applications of tensors to differential geometry, mechanics and relativity. Organized in ten chapters, it provides the origin and nature of the tensor along with the scope of the tensor calculus. Besides this, it also discusses N-dimensional Riemannian space, characteristic peculiarity of Riemannian space, intrinsic property of surfaces, and properties and transformation of Christoffel’s symbols. Besides the students of mathematics, this book will be equally useful for the postgraduate students of physics. KEY FEATURES : Contains 250 worked out examples Includes more than 350 unsolved problems Gives thorough foundation in Tensors

## Tensor Calculus

*A Concise Course*

**Author**: Barry Spain**Publisher:**Courier Corporation**ISBN:**0486428311**Category:**Mathematics**Page:**125**View:**3499

A compact exposition of the theory of tensors, this text also illustrates the power of the tensor technique by its applications to differential geometry, elasticity, and relativity. Explores tensor algebra, the line element, covariant differentiation, geodesics and parallelism, and curvature tensor. Also covers Euclidean 3-dimensional differential geometry, Cartesian tensors and elasticity, and the theory of relativity. 1960 edition.

## 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:**3400

## 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:**7033

This book comprehensively presents topics, such as Dirac notation, tensor analysis, elementary differential geometry of moving surfaces, and k-differential forms. Additionally, two new chapters of Cartan differential forms and Dirac and tensor notations in quantum mechanics are added to this second edition. The reader is provided with hands-on calculations and worked-out examples at which he will learn how to handle the bra-ket notation, tensors, differential geometry, and differential forms; and to apply them to the physical and engineering world. Many methods and applications are given in CFD, continuum mechanics, electrodynamics in special relativity, cosmology in the Minkowski four-dimensional spacetime, and relativistic and non-relativistic quantum mechanics. Tensors, differential geometry, differential forms, and Dirac notation are very useful advanced mathematical tools in many fields of modern physics and computational engineering. They are involved in special and general relativity physics, quantum mechanics, cosmology, electrodynamics, computational fluid dynamics (CFD), and continuum mechanics. The target audience of this all-in-one book primarily comprises graduate students in mathematics, physics, engineering, research scientists, and engineers.

## Geometrical Methods of Mathematical Physics

**Author**: Bernard F. Schutz**Publisher:**Cambridge University Press**ISBN:**1107268141**Category:**Science**Page:**N.A**View:**749

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.

## Tensors

*The Mathematics of Relativity Theory and Continuum Mechanics*

**Author**: Anadi Jiban Das**Publisher:**Springer Science & Business Media**ISBN:**0387694692**Category:**Science**Page:**290**View:**3385

Here is a modern introduction to the theory of tensor algebra and tensor analysis. It discusses tensor algebra and introduces differential manifold. Coverage also details tensor analysis, differential forms, connection forms, and curvature tensor. In addition, the book investigates Riemannian and pseudo-Riemannian manifolds in great detail. Throughout, examples and problems are furnished from the theory of relativity and continuum mechanics.

## Differential Geometry and Relativity Theory

*An Introduction*

**Author**: Richard L. Faber**Publisher:**CRC Press**ISBN:**9780824717490**Category:**Mathematics**Page:**272**View:**5845

## 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:**3013

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.

## The Absolute Differential Calculus

*Calculus of Tensors*

**Author**: Tullio Levi-Civita**Publisher:**Dover Publications**ISBN:**9780486446370**Category:**Mathematics**Page:**452**View:**7153

A chief requirement in the study of relativity is knowledge of the absolute differential calculus, the subject that Einstein found necessary for developing his ideas mathematically. Tullio Levi-Civita was one of the founders of this field of mathematics, and he presents a clear, detailed exposition of the subject in this classic book. The first section of the three-part treatment examines functional determinants and matrices; systems of total differential equations; linear partial differential equations in complete systems; and algebraic foundations of the absolute differential calculus, concluding with a geometrical introduction to the theory of differential quadratic forms. Part two, a study of the fundamental quadratic form and the absolute differential calculus, focuses on covariant differentiation, invariants and differential parameters, and locally geodesic coordinates; Riemann's symbols and properties relating to curvature, Ricci's and Einstein's symbols, and geodesic deviation; relations between two different metrics referred to the same parameters, manifolds of constant curvature; and differential quadratic forms of class zero and class one; and some applications of intrinsic geometry. The third and final section explores physical applications, including the evolution of mechanics and geometrical optics and their relation to a four-dimensional world according to Einstein; and gravitational equations and general relativity.

## Mathematical Theory of General Relativity

**Author**: L. N. Katkar**Publisher:**Alpha Science International Limited**ISBN:**9781842658062**Category:**Mathematics**Page:**176**View:**7804

THE MATHEMATICAL THEORY OF GENERAL RELATIVITY is prepared for M. Sc. Students of Mathematics and Physics of Indian Universities. The aim of writing this book is to give the reader a feeling for the necessity and beauty of the laws of general relativity. The contents of the book will attract both mathematicians and physicists which provides motivation and applications of many ideas and powerful mathematical methods of modern analysis and differential geometry. An attempt has been made to make the presentation comprehensive, rigorous and yet simple. I have carried out most calculations and transformations in great detail. One of the features of this book is that in almost all chapters numerous examples have been solved by using the well known mathematical techniques viz., the tensors and the differential forms. In preparing this book, I have followed some standard texts which are cited in the references. I do not claim any originality but have our own way of presentation and hope that the readers will like the approach.

## A Brief on Tensor Analysis

**Author**: James G. Simmonds**Publisher:**Springer Science & Business Media**ISBN:**1441985220**Category:**Mathematics**Page:**114**View:**4155

In this text which gradually develops the tools for formulating and manipulating the field equations of Continuum Mechanics, the mathematics of tensor analysis is introduced in four, well-separated stages, and the physical interpretation and application of vectors and tensors are stressed throughout. This new edition contains more exercises. In addition, the author has appended a section on Differential Geometry.

## Spacetime

*Foundations of General Relativity and Differential Geometry*

**Author**: Marcus Kriele**Publisher:**Springer Science & Business Media**ISBN:**3540483543**Category:**Science**Page:**436**View:**4493

One of the most of exciting aspects is the general relativity pred- tion of black holes and the Such Big Bang. predictions gained weight the theorems through Penrose. singularity pioneered In various by te- books on theorems general relativity singularity are and then presented used to that black holes exist and that the argue universe started with a To date what has big been is bang. a critical of what lacking analysis these theorems predict-’ We of really give a proof a typical singul- theorem and this ity use theorem to illustrate problems arising through the of possibilities violations" and "causality weak "shell very crossing These singularities". add to the problems weight of view that the point theorems alone singularity are not sufficient to the existence of predict physical singularities. The mathematical theme of the book In order to both solid gain a of and intuition understanding good for any mathematical theory, one,should to realise it as model of try a a fam- iar non-mathematical theories have had concept. Physical an especially the important on of and impact development mathematics, conversely various modern theories physical rather require sophisticated mathem- ics for their formulation. both and mathematics Today, physics are so that it is often difficult complex to master the theories in both very s- in the of jects. However, case differential pseudo-Riemannian geometry or the general relativity between and mathematics relationship physics is and it is therefore especially close, to from interd- possible profit an ciplinary approach.

## Tensors, Differential Forms, and Variational Principles

**Author**: David Lovelock,Hanno Rund**Publisher:**Courier Corporation**ISBN:**048613198X**Category:**Mathematics**Page:**400**View:**3882

Incisive, self-contained account of tensor analysis and the calculus of exterior differential forms, interaction between the concept of invariance and the calculus of variations. Emphasis is on analytical techniques. Includes problems.

## Differential Forms and the Geometry of General Relativity

**Author**: Tevian Dray**Publisher:**CRC Press**ISBN:**1466510005**Category:**Mathematics**Page:**321**View:**3512

Differential Forms and the Geometry of General Relativity provides readers with a coherent path to understanding relativity. Requiring little more than calculus and some linear algebra, it helps readers learn just enough differential geometry to grasp the basics of general relativity. The book contains two intertwined but distinct halves. Designed for advanced undergraduate or beginning graduate students in mathematics or physics, most of the text requires little more than familiarity with calculus and linear algebra. The first half presents an introduction to general relativity that describes some of the surprising implications of relativity without introducing more formalism than necessary. This nonstandard approach uses differential forms rather than tensor calculus and minimizes the use of "index gymnastics" as much as possible. The second half of the book takes a more detailed look at the mathematics of differential forms. It covers the theory behind the mathematics used in the first half by emphasizing a conceptual understanding instead of formal proofs. The book provides a language to describe curvature, the key geometric idea in general relativity.

## Tensor Calculus

**Author**: J. L. Synge,A. Schild**Publisher:**Courier Corporation**ISBN:**048614139X**Category:**Mathematics**Page:**336**View:**4829

Fundamental introduction of absolute differential calculus and for those interested in applications of tensor calculus to mathematical physics and engineering. Topics include spaces and tensors; basic operations in Riemannian space, curvature of space, more.

## Tensors and their Applications

**Author**: Nazrul Islam**Publisher:**New Age International**ISBN:**8122418384**Category:**Tensor algebra**Page:**262**View:**3058

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.

## Tensor and Vector Analysis

*With Applications to Differential Geometry*

**Author**: C. E. Springer**Publisher:**Courier Corporation**ISBN:**048632091X**Category:**Mathematics**Page:**256**View:**5712

Assuming only a knowledge of basic calculus, this text's elementary development of tensor theory focuses on concepts related to vector analysis. The book also forms an introduction to metric differential geometry. 1962 edition.

## Tensors, Relativity, and Cosmology

**Author**: Mirjana Dalarsson,Nils Dalarsson**Publisher:**Academic Press**ISBN:**0128034017**Category:**Science**Page:**276**View:**9391

Tensors, Relativity, and Cosmology, Second Edition, combines relativity, astrophysics, and cosmology in a single volume, providing a simplified introduction to each subject that is followed by detailed mathematical derivations. The book includes a section on general relativity that gives the case for a curved space-time, presents the mathematical background (tensor calculus, Riemannian geometry), discusses the Einstein equation and its solutions (including black holes and Penrose processes), and considers the energy-momentum tensor for various solutions. In addition, a section on relativistic astrophysics discusses stellar contraction and collapse, neutron stars and their equations of state, black holes, and accretion onto collapsed objects, with a final section on cosmology discussing cosmological models, observational tests, and scenarios for the early universe. This fully revised and updated second edition includes new material on relativistic effects, such as the behavior of clocks and measuring rods in motion, relativistic addition of velocities, and the twin paradox, as well as new material on gravitational waves, amongst other topics. Clearly combines relativity, astrophysics, and cosmology in a single volume Extensive introductions to each section are followed by relevant examples and numerous exercises Presents topics of interest to those researching and studying tensor calculus, the theory of relativity, gravitation, cosmology, quantum cosmology, Robertson-Walker Metrics, curvature tensors, kinematics, black holes, and more Fully revised and updated with 80 pages of new material on relativistic effects, such as relativity of simultaneity and relativity of the concept of distance, amongst other topics Provides an easy-to-understand approach to this advanced field of mathematics and modern physics by providing highly detailed derivations of all equations and results