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Emil De Souza Sбnchez Filho Emil de Souza Sánchez Filho Tensor Calculus for Engineers and Physicists Tensor Calculus for Engineers and Physicists Emil de Souza Sa´nchez Filho Tensor Calculus for Engineers and Physicists Emil de Souza Sa´nchez Filho Fluminense Federal University Rio de Janeiro, Rio de Janeiro Brazil ISBN 978-3-319-31519-5 ISBN 978-3-319-31520-1 (eBook) DOI 10.1007/978-3-319-31520-1 Library of Congress Control Number: 2016938417 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland To Sandra, Yuri, Natalia and Lara Preface The Tensor Calculus for Engineers and Physicist provides a rigorous approach to tensor manifolds and their role in several issues of these professions. With a thorough, complete, and unified presentation, this book affords insights into several topics of tensor analysis, which covers all aspects of N-dimensional spaces. Although no emphasis is placed on special and particular problems of Engineer- ing or Physics, the text covers the fundamental and complete study of the aim of these fields of the science. The book makes a brief introduction to the basic concept of the tensorial formalism so as to allow the reader to make a quick and easy review of the essential topics that enable having a dominium over the subsequent themes, without needing to resort to other bibliographical sources on tensors. This book did not have the framework of a math book, which is a work that seeks, above all else, to organize ideas and concepts in a didactic manner so as to allow the familiarity with the tensorial approach and its application of the practical cases of Physics and the areas of Engineering. The development of the various chapters does not cling to any particular field of knowledge, and the concepts and the deductions of the equations are presented so as to permit engineers and physicists to read the text without being experts in any branch of science to which a specific topic applies. The chapters treat the various themes in a sequential manner and the deductions are performed without omission of the intermediary steps, the subjects being treated in a didactic manner and supplemented with various examples in the form of solved exercises with the exception of Chap. 3 that broaches review topics. A few problems with answers are presented at the end of each chapter, seeking to allow the reader to improve his practice in solving exercises on the themes that were broached. Chapter 1 is a brief introduction to the basic concepts of tensorial formalism so as to permit the reader to make a quick and easy review of the essential topics that make possible the knowledge of the subsequent themes that come later, without needing to resort to other bibliographic sources on tensors. vii viii Preface The concepts of covariant, absolute, and contravariant derivatives, with the detailed development of all the expressions concerning these parameters, as well as the deductions of the Christoffel symbols of the first and second kind, are the essence of Chap. 2. Chapter 3 presents the Green, Stokes, and Gauss–Ostrogradsky theorems using a vectorial formulation. The expansion of the concepts of the differential operators studied in Differential Calculus is performed in Chap. 4. The scalar, vectorial, and tensorial fields are defined, and the concepts and expressions for gradient, divergence, and curl are formulated. With the definition of the nabla operator, successive applications of this linear differential operator are carried out and various fundamental relations between the differential operators are deducted, defining the Laplace operator. All the formulas are deducted by means of tensorial approach. The definition of metric spaces with several dimensions, with the introduction of Riemann curvature concept, and the Ricci tensor formulations, the scalar curvature, and the Einstein tensor are the subjects studied in detail in Chap. 5. Various particular cases of Riemann spaces are analyzed, such as the bidimensional spaces, the spaces with constant curvature, the Minkowski space, and the conformal spaces, with the definition of the Weyl tensor. Chapter 6 broaches metric spaces provided with curvature with the introduction of the concepts of the geodesics and the geodesics and Riemann coordinate systems. The geodesics deviation and the parallelism of vectors in curved spaces are studied, with the definition of the torsion tensor concept. The purpose of this book is to give a simple, correct, and comprehensive mathematical explanation of Tensor Calculus, and it is self-contained. Postgraduate and advanced undergraduate students and professionals will find clarity and insight into the subject of this textbook. The preparation of a book is a hard and long work that requires the participation of other people besides the author, which are of fundamental importance in the preparation of the originals and in the tiresome task of reviewing the typing, chiefly in a text such as the one in this book. So, our sincere thanks to all those who helped in the preparation and editing of these pages. In relation to the errors in this text which were not corrected by a more diligent review, it is stressed that they are the author’s responsibility and the author apologizes for them. Rio de Janeiro, Brazil Emil de Souza Sa´nchez Filho December 17, 2015 Historical Introduction This brief history of Tensor Calculus broaches the development of the idea of vector and the advent of the concept of tensor in a synthetic way. The following paragraphs aim to show the history of the development of these themes in the course of time, highlighting the main stages that took place in this evolution of the mathematical knowledge. A few items of bibliographic data of the mathematicians and scientists who participated on this epic journey in a more striking manner are described. The perception of Nature under a purely philosophical focus led Plato in 360 BC to the study of geometry. This philosopher classified the geometric figures into triangles, rectangles, and circles, and with this system, he grounded the basic concepts of geometry. Later Euclid systemized geometry in axiomatic form, starting from the fundamental concepts of points and lines. The wise men of ancient Greece also concerned themselves with the study of the movement of bodies by means of geometric concepts. The texts of Aristotle (384– 322 BC) in Mechanics show that he had the notion of composition of movements. In this work, Aristotle enounced in an axiomatic form that the force that moves a body is collinear with the direction of the body’s movement. In a segment of Mechanics, he describes the velocity of two bodies in linear movement with constant pro- portions between each other, explaining that “When a body moves with a certain proportion, the body needs to move in a straight line, and this is the diameter of the figure formed with the straight lines which have known proportions.” This state- ment deals with the displacements of two bodies—the Greek sage acknowleding that the resultant of these displacements would be the diagonal of the rectangle (the text talks about the diameter) from the composition of the speeds. In the Renaissance, the prominent figure of Leonardo da Vinci (1452–1519) also stood out in the field of sciences. In his writings, he reports that “Mechanics is the paradise of mathematical science, because all the fruits of mathematics are picked here.” Da Vinci conceived concepts on the composition of forces for maintaining the balance of the simple structures, but enunciated them in an erroneous and contradictory manner in view of the present-day knowledge. ix x Historical Introduction The awakening of a new manner of facing the uniform was already blossoming in the 1600s. The ideas about the conception and study of the world were no longer conceived from the scholastic point of view, for reason more than faith had become the way to new discoveries and interpretations of the outside world. In the Nether- lands, where liberal ideas were admitted and free thought could be exercised in full, the Dutch mathematician Simon Stevin (1548–1620), or Stevinus in a Latinized spelling, was the one who demonstrated in a clear manner the rule for the compo- sition of forces, when analyzing the balance of a body located in an inclined plane and supported by weights, one hanging at the end of a lever, and the other hanging from a pulley attached to the vertical cathetus of the inclined plane. This rule is a part of the writings of Galileo Galilei (1564–1642) on the balance of bodies in a tilted plane. However, it became necessary to conceive mathematical formalism that translated these experimental verifications.
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