A Brief on Tensor Analysis

verfasst von: James G. Simmonds

Verlag: Springer New York

Buchreihe : Undergraduate Texts in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter I. Introduction: Vectors and Tensors

Abstract

This little book is about tensor analysis, as Einstein’s philosophers’ stone, the absolute differential calculus, is called nowadays. I have written it, though, with an eye not toward general relativity, but to continuum mechanics, a more modest theory that attempts to predict the gross behavior of “the masses of matter we see and use from day to day: air, water, earth, flesh, wood, stone, steel, concrete, glass, rubber, ….”²

James G. Simmonds

Chapter II. General Bases and Tensor Notation

Abstract

While the laws of mechanics can be written in coordinate-free form, they can be solved, in most cases, only if expressed in component form. This requires that we introduce a basis. Though the standard Cartesian basis is often the simplest, the physics and geometry of a problem, and especially the so-called boundary conditions, may dictate another. For example, if we wished to study the temperature distribution in a body the shape of a parallelepiped, we would choose most likely a basis consisting of vectors lying along three co-terminal edges of the body. An aim of tensor analysis is to embrace arbitrary coordinate systems and their associated bases, yet to produce formulas for computing invariants, such as the dot product, that are as simple as the Cartesian forms.

James G. Simmonds

Chapter III. Newton’s Law and Tensor Calculus

Abstract

Newton’s Law of Motion is studied in introductory courses in calculus, physics, and dynamics. Being familiar, fundamental, and simple, Newton’s Law is an ideal vehicle for introducing many of the key ideas in tensor calculus.

James G. Simmonds

Chapter IV. The Gradient, the Del Operator, Covariant Differentiation, and the Divergence Theorem

Abstract

Suppose that you had a topographical map of a piece of land and wanted to indicate at a spot P on the map the slope m of the land in a direction t. This could be done by drawing a vector mt from P, as indicated in Fig. 4.1. Obviously, if the terrain is smooth but not level, there is one direction from P in which the slope is a maximum. This is called the direction of steepest ascent.¹ The associated vector is called the gradient of the elevation at P. If you draw a contour line through P, you will realize that the gradient at P must be ⊥ to this contour.

James G. Simmonds

Backmatter

Titel: A Brief on Tensor Analysis
verfasst von: James G. Simmonds
Verlag: Springer New York
Electronic ISBN: 978-1-4419-8522-4
Print ISBN: 978-1-4612-6424-8
DOI: https://doi.org/10.1007/978-1-4419-8522-4

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter I. Introduction: Vectors and Tensors

Chapter II. General Bases and Tensor Notation

Chapter III. Newton’s Law and Tensor Calculus

Chapter IV. The Gradient, the Del Operator, Covariant Differentiation, and the Divergence Theorem

Backmatter