Part I, the basic portion of the text, consists of a thorough treatment of vector algebra and the vector calculus. Part II presents the illustrative matter, demonstrating applications to kinematics, mechanics, and electromagnetic theory. The text stresses geometrical and physical aspects, but it also casts the material in such a way that the logical structure of the subject is made plain. Serious students of mathematics can rigorize the treatment to their own satisfaction. Although intended primarily as a college text, this volume may be used as a reference in vector techniques or as a guide to self-education.
Because the concept of a vector has been greatly generalized in geometry and mathematical physics, this text concludes with a brief introduction to abstract vector spaces, together with the ideas of linear dependence, basis, and dimension. The exposition of these abstract concepts is kept simple and clear. Numerous figures appear throughout the text.
Among the text's outstanding features are numbered definitions and theorems in the development of vector algebra, which appear in italics for easy reference. Most of the theorems include proofs, and coordinate position vectors receive an in-depth treatment. Key concepts for generalized vector spaces are clearly presented and developed, and 57 worked-out illustrative examples aid students in mastering the concepts. A total of 258 exercise problems offer supplements to theories or provide the opportunity to reinforce the understanding of applications, and answers to odd-numbered exercises appear at the end of the book.
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This concise introduction to a basic branch of applied mathematics is indispensable to mathematicians, physicists and engineers. Eminently readable, it covers the elements of vector and tensor analysis, with applications of the theory to specific physics and engineering problems. It lays particular stress on the applications of the theory to fluid dynamics.
The authors begin with a definition of vectors and a discussion of algebraic operations on vectors. The vector concept is then generalized in a natural way, leading to the concept of a tensor. Chapter Three considers algebraic operations on tensors. Next, the authors turn to a systematic study of the differential and integral calculus of vector and tensor functions of space and time. Finally, vector and tensor analysis is considered from both a rudimentary standpoint, and in its fuller ramifications, concluding the volume.
The strength of the book lies in the completely worked out problems and solutions at the end of each chapter. In addition, each chapter incorporates abundant exercise material. Intended primarily for advanced undergraduates and graduate students of math, physics and engineering, the work is self-contained and accessible to any student with a good background in calculus.
Vector and Tensor Analysis With Applications is one of a series of SELECTED RUSSIAN PUBLICATIONS IN THE MATHEMATICAL SCIENCES, several of which have already been published by Dover. The authors are distinguished Russian mathematicians and specialists in gas dynamics and numerical analysis. Richard A. Silverman, editor of the series as well as editor and translator of this volume, has revised and improved the original edition and added a bibliography.
" … a concise, clear and comprehensive treatment … " Prof. Henry G. Booker, University of California, San Diego
Beginning with simple physical examples, the theory of tensors and forms is developed by a process of successive abstractions. This enables the reader to infer generalized principles from concrete situations — departing from the traditional approach to tensors and forms in terms of purely differential-geometric concepts.
The treatment of the calculus of variations of single and multiple integrals is based ab initio on Carathéodory's method of equivalent integrals. Subsequent material explores the effects of invariance postulates on variational principles, focusing ultimately on relativistic field theories. Other discussions include:
• integral invariants
• simple and direct derivations of Noether's theorems
• Riemannian spaces with indefinite metrics
The emphasis in this book is on analytical techniques, with abundant problems, ranging from routine manipulative exercises to technically difficult problems encountered by those using tensor techniques in research activities. A special effort has been made to collect many useful results of a technical nature, not generally discussed in the standard literature. The Appendix, newly revised and enlarged for the Dover edition, presents a reformulation of the principal concepts of the main text within the terminology of current global differential geometry, thus bridging the gap between classical tensor analysis and the fundamentals of more recent global theories.
The many and various topics covered include: the algebra of vectors — linear dependence and independence, transformation equations, the inner product, the cross product, and the algebra of matrixes; the differentiation of vectors — geometry of space curves, kinematics, moving frames of reference, Newtonian orbits and special relativity theory; partial differentiation of vectors — geometry of space curves, kinematics, moving frames of reference, Newtonian orbits and special relativity theory; partial differentiation and associated concepts — surface representations, bases in general coordinate systems, and maxima and minima of functions of two variables; the integration of vectors — line integrals, surface integrals, surface tensors and volume integrals; tensor algebra and analysis — fundamental notions of n-space, transformations and tensors, Riemannian geometry, tensor processes of differentiation, geodesics, the curvature tensor and its algebraic properties, and general relativity theory.
Throughout, Professor Wrede stresses the interrelationships between algebra and geometry, and moves frequently from one to the other. As he points out, vector and tensor analysis provides a kind of bridge between elementary aspects of linear algebra, geometry and analysis. He uses the classical notation for vector analysis, but introduces a more appropriate new notation for tensors, which he correlates with the common vector notation. He stresses proofs and concludes each section with a set of problems designed to help the student get a solid grasp of the ideas, and explore them more thoroughly on his own. His approach features a combination of important historical material with up-to-date developments in both fields. The knowledge of vector and tensor analysis gained in this way is excellent preparation for further studies in differential geometry, applied mathematics, and theoretical physics.
Despite its success as a mathematical tool in the general theory of relativity and its adaptability to a wide range of mathematical and physical problems, tensor analysis has always had a rather restricted level of use, with an emphasis on notation and the manipulation of indices. This book is an attempt to broaden this point of view at the stage where the student first encounters the subject. The authors have treated tensor analysis as a continuation of advanced calculus, striking just the right balance between the formal and abstract approaches to the subject.
The material proceeds from the general to the special. An introductory chapter establishes notation and explains various topics in set theory and topology. Chapters 1 and 2 develop tensor analysis in its function-theoretical and algebraic aspects, respectively. The next two chapters take up vector analysis on manifolds and integration theory. In the last two chapters (5 and 6) several important special structures are studied, those in Chapter 6 illustrating how the previous material can be adapted to clarify the ideas of classical mechanics. The text as a whole offers numerous examples and problems.
A student with a background of advanced calculus and elementary differential equation could readily undertake the study of this book. The more mature the reader is in terms of other mathematical knowledge and experience, the more he will learn from this presentation.
Partial contents: algebraic preliminaries (notation, definitions, determinants, tensor analysis); algebraic geometry (rectilinear coordinates, the plane, the straight line, the quadric cone and the conic, systems of cones and conics, central quadrics, the general quadric, affine transformations); differential geometry (curvilinear coordinates, covariant differentiation, curves in a space, intrinsic geometry of a surface, fundamental formulae of a surface, curves on a surface); applied mathematics (dynamics of a particles, dynamics of rigid bodies, electricity and magnetism, mechanics of continuous media, special theory of relativity).
Motivated examples introduce each idea, with interpretations of physical, algebraic, and geometric contexts, in addition to generalizations to theorems that reflect the essential structure. A combination of matrix and vector methods reinforces both techniques. "Applied" students learn that general theory is a natural and useful culmination of their computations, and "pure" students learn that concrete problems from the physical world have traditionally keynoted abstract intellectual pursuits.
Geared toward upper-level undergraduates, the text features approximately 50 provocative problems at each chapter's end that test students' choice of techniques. Each chapter is also followed by about 25 mental exercises that stimulate imaginative reflection. Answers are given to selected questions.
In this massive graduate-level study, Emeritus Professor Edwards (Australian National University, Canberra) presents a balanced account of both the abstract theory and the applications of linear functional analysis. Written for readers with a basic knowledge of set theory, general topology, and vector spaces, the book includes an abundance of carefully chosen illustrative examples and excellent exercises at the end of each chapter.
Beginning with a chapter of preliminaries on set theory and topology, Dr. Edwards then presents detailed, in-depth discussions of vector spaces and topological vector spaces, the Hahn-Banach theorem (including applications to potential theory, approximation theory, game theory, and other fields) and fixed-point theorems. Subsequent chapters focus on topological duals of certain spaces: radon measures, distribution and linear partial differential equations, open mapping and closed graph theorems, boundedness principles, duality theory, the theory of compact operators and the Krein-Milman theorem and its applications to commutative harmonic analysis.
Clearly and concisely written, Dr. Edwards's book offers rewarding reading to mathematicians and physicists with an interest in the important field of functional analysis. Because of the broad scope of its coverage, this volume will be especially valuable to the reader with a basic knowledge of functional analysis who wishes to learn about parts of the subject other than his own specialties. A comprehensive 32-page bibliography supplies a rich source of references to the basic literature.
These proceedings include the main invited talks and contributed papers given during the workshop. The aim of these lectures was to present a selection of results of the latest research in the field. In addition to covering topics in approximation by solutions of partial differential equations and quadrature formulae, this volume is also concerned with related areas, such as Gaussian quadratures, the Pompelu problem, rational approximation to the Fresnel integral, boundary correspondence of univalent harmonic mappings, the application of the Hilbert transform in two dimensional aerodynamics, finely open sets in the limit set of a finitely generated Kleinian group, scattering theory, harmonic and maximal measures for rational functions and the solution of the classical Dirichlet problem. In addition, this volume includes some problems in potential theory which were presented in the Problem Session at Hanstholm.
Part one opens with considerations of functional determinants and matrices, advancing to systems of total differential equations, linear partial differential equations, algebraic foundations, and a geometrical introduction to theory. The second part addresses covariant differentiation, curvature-related Riemann's symbols and properties, differential quadratic forms of classes zero and one, and intrinsic geometry. The final section focuses on physical applications, covering gravitational equations and general relativity.
Opening chapters define vector addition and subtraction, show how to resolve and determine the direction of two or more vectors, and explain systems of coordinates, vector equations of a plane and straight line, relative velocity and acceleration, and infinitely small vectors. The following chapters deal with scalar and vector multiplication, axial and polar vectors, areas, differentiation of vector functions, gradient, curl, divergence, and analytical properties of the position vector. Applications of vector analysis to dynamics and physics are the focus of the final chapter, including such topics as moving rigid bodies, energy of a moving rigid system, central forces, equipotential surfaces, Gauss's theorem, and vector flow.
The presentation aims at converting the noble art of constructing an entire function with prescribed asymptotic behaviour to a handicraft. For this one should only construct the limit set that describes the asymptotic behaviour of the entire function.
All necessary material is developed within the book, hence it will be most useful as a reference book for the construction of entire functions.
The treatment begins with a chapter on vectors and vector addition, followed by a chapter on products of vector. Two succeeding chapters on vector calculus cover a variety of topics, including functions of a vector; line, surface, and volume integrals; the Laplacian operator, and more. The text concludes with a survey of standard applications, including Poinsot's central axis, Gauss's theorem, gravitational potential, Green's theorems, and other subjects.
The author adroitly weaves these varied topics to reveal a number of delightful interactions. Perhaps more importantly, the topics are presented with an understanding and explanation of their interrelations with other important parts of mathematics: harmonic analysis, differential geometry, partial differential equations, potential theory, abstract algebra, and invariant theory. Although the book examines complex analysis from many different points of view, it uses geometric analysis as its unifying theme.
This methodically designed book contains a rich collection of exercises, examples, and illustrations within each individual chapter, concluding with an extensive bibliography of monographs, research papers, and a thorough index. Seeking to capture the imagination of advanced undergraduate and graduate students with a basic background in complex analysis—and also to spark the interest of seasoned workers in the field—the book imparts a solid education both in complex analysis and in how modern mathematics works.
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Quaternions are one of the most documented inventions in the history of mathematics, and this book is about their invention, and how they are used to rotate vectors about an arbitrary axis. Apart from introducing the reader to the features of quaternions and their associated algebra, the book provides valuable historical facts that bring the subject alive.
Quaternions for Computer Graphics introduces the reader to quaternion algebra by describing concepts of sets, groups, fields and rings. It also includes chapters on imaginary quantities, complex numbers and the complex plane, which are essential to understanding quaternions. The book contains many illustrations and worked examples, which make it essential reading for students, academics, researchers and professional practitioners.