Introduction to vectors and tensors, Vol 1: linear and multilinear algebra
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This work represents our effort to present the basic concepts of vector and tensor analysis. Volume I begins with a brief discussion of algebraic structures followed by a rather detailed discussion of the algebra of vectors and tensors. Volume II begins with a discussion of Euclidean Manifolds which leads to a development of the analytical and geometrical aspects of vector and tensor fields. a discussion of general differentiable manifolds. We have not included a discussion of general differentiable manifolds. However, we have included a chapter on vector and tensor fields defined on Hypersurfaces in a Euclidean Manifold. In preparing this two volume work our intention is to present to Engineering and Science students a modern introduction to vectors and tensors. Traditional courses on applied mathematics have emphasized problem solving techniques rather than the systematic development of concepts. As a result, it is possible for such courses to become terminal mathematics courses rather than courses which equip the student to develop his or her understanding further. As Engineering students our courses on vectors and tensors were taught in the traditional way. We learned to identify vectors and tensors by formal transformation rules rather than by their common mathematical structure. The subject seemed to consist of nothing but a collection of mathematical manipulations of long equations decorated by a multitude of subscripts and superscripts. Prior to our applying vector and tensor analysis to our research area of modern continuum mechanics, we almost had to relearn the subject. Therefore, one of our objectives in writing this book is to make available a modern introductory textbook suitable for the first in-depth exposure to vectors and tensors. Because of our interest in applications, it is our hope that this book will aid students in their efforts to use vectors and tensors in applied areas. The presentation of the basic mathematical concepts is, we hope, as clear and brief as possible without being overly abstract. Since we have written an introductory text, no attempt has been made to include every possible topic. The topics we have included tend to reflect our personal bias. We make no claim that there are not other introductory topics which could have been included. Basically the text was designed in order that each volume could be used in a one-semester course. We feel Volume I is suitable for an introductory linear algebra course of one semester. Given this course, or an equivalent, Volume II is suitable for a one semester course on vector and tensor analysis. Many exercises are included in each volume. However, it is likely that teachers will wish to generate additional exercises. Several times during the preparation of this book we taught a one semester course to students with a very limited background in linear algebra and no background in tensor analysis. Typically these students were majoring in Engineering or one of the Physical Sciences. However, we occasionally had students from the Social Sciences. For this one semester course, we covered the material in Chapters 0, 3, 4, 5, 7 and 8 from Volume I and selected topics from Chapters 9, 10, and 11 from Volume 2. As to level, our classes have contained juniors, seniors and graduate students. These students seemed to experience no unusual difficulty with the material. It is a pleasure to acknowledge our indebtedness to our students for their help and forbearance. Also, we wish to thank the U. S. National Science Foundation for its support during the preparation of this work. We especially wish to express our appreciation for the patience and understanding of our wives and children during the extended period this work was in preparation.