A Brief on Tensor Analysis / Edition 2

by James G. Simmonds

Overview

There are three changes in the second edition. First, with the help of readers and colleagues-thanks to all-I have corrected typographical errors and made minor changes in substance and style. Second, I have added a fewmore Exercises,especially at the end ofChapter4.Third, I have appended a section on Differential Geometry, the essential mathematical tool in the study of two-dimensional structural shells and four-dimensional general relativity. JAMES G. SIMMONDS vii Preface to the First Edition When I was an undergraduate, working as a co-op student at North Ameri­ can Aviation, I tried to learn something about tensors. In the Aeronautical Engineering Department at MIT, I had just finished an introductory course in classical mechanics that so impressed me that to this day I cannot watch a plane in flight-especially in a turn-without imaging it bristling with vec­ tors. Near the end of the course the professor showed that, if an airplane is treated as a rigid body, there arises a mysterious collection of rather simple­ looking integrals called the components of the moment of inertia tensor.

Product Details

ISBN-13:	9780387940885
Publisher:	Springer New York
Publication date:	11/19/1993
Series:	Undergraduate Texts in Mathematics
Edition description:	2nd ed. 1994
Pages:	114
Product dimensions:	6.10(w) x 9.25(h) x 0.36(d)

Table of Contents

I Introduction: Vectors and Tensors.- Three-Dimensional Euclidean Space.- Directed Line Segments.- Addition of Two Vectors.- Multiplication of a Vector v by a Scalar—.- Things That Vectors May Represent.- Cartesian Coordinates.- The Dot Product.- Cartesian Base Vectors.- The Interpretation of Vector Addition.- The Cross Product.- Alternative Interpretation of the Dot and Cross Product. Tensors.- Definitions.- The Cartesian Components of a Second Order Tensor.- The Cartesian Basis for Second Order Tensors.- Exercises.- II General Bases and Tensor Notation.- General Bases.- The Jacobian of a Basis Is Nonzero.- The Summation Convention.- Computing the Dot Product in a General Basis.- Reciprocal Base Vectors.- The Roof (Contravariant) and Cellar (Covariant) Components of a Vector.- Simplification of the Component Form of the Dot Product in a General Basis.- Computing the Cross Product in a General Basis.- A Second Order Tensor Has Four Sets of Components in General.- Change of Basis.- Exercises.- III Newton’s Law and Tensor Calculus.- Rigid Bodies.- New Conservation Laws.- Nomenclature.- Newton’s Law in Cartesian Components.- Newton’s Law in Plane Polar Coordinates.- The Physical Components of a Vector.- The Christoffel Symbols.- General Three-Dimensional Coordinates.- Newton’s Law in General Coordinates.- Computation of the Christoffel Symbols.- An Alternative Formula for Computing the Christoffel Symbols.- A Change of Coordinates.- Transformation of the Christoffel Symbols.- Exercises.- IV The Gradient, the Del Operator, Covariant Differentiation, and the Divergence Theorem.- The Gradient.- Linear and Nonlinear Eigenvalue Problems.- The Del Operator.- The Divergence, Curl, and Gradient of a Vector Field.- The Invariance of— · v,— × v, and—v.- The Covariant Derivative.- The Component Forms of— · v,— × v, and—v.- The Kinematics of Continuum Mechanics.- The Divergence Theorem.- Differential Geometry.- Exercises.

Editorial Reviews

Intended for advanced undergraduates in engineering, physics, mathematics, and applied sciences, this material can serve as a springboard for studies in continuum mechanics and general relativity. Topics include an introduction to vectors and tensors, general bases and tensor notation, Newton's law and tensor calculus, the gradient, del operator, covariant differentiation, and the divergence theorem. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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