INTRODUCTION TO VECTORS AND TENSORS - Texas A&M University

INTRODUCTION TO VECTORS AND TENSORS

Vector and Tensor Analysis

Volume 2

Ray M. Bowen

Mechanical Engineering

Texas A&M University

College Station, Texas

and

C.-C. Wang

Mathematical Sciences

Rice University

Houston, Texas

Copyright Ray M. Bowen and C.-C. Wang

(ISBN 0-306-37509-5 (v. 2))

____________________________________________________________________________

PREFACE

To Volume 2

This is the second volume of a two-volume work on vectors and tensors. Volume 1 is concerned

with the algebra of vectors and tensors, while this volume is concerned with the geometrical

aspects of vectors and tensors. This volume begins with a discussion of Euclidean manifolds. The

principal mathematical entity considered in this volume is a field, which is defined on a domain in a

Euclidean manifold. The values of the field may be vectors or tensors. We investigate results due

to the distribution of the vector or tensor values of the field on its domain. While we do not discuss

general differentiable manifolds, we do include a chapter on vector and tensor fields defined on

hypersurfaces in a Euclidean manifold.

This volume contains frequent references to Volume 1. However, references are limited to

basic algebraic concepts, and a student with a modest background in linear algebra should be able

to utilize this volume as an independent textbook. As indicated in the preface to Volume 1, this

volume is suitable for a one-semester course on vector and tensor analysis. On occasions when we

have taught a one ¨Csemester course, we covered material from Chapters 9, 10, and 11 of this

volume. This course also covered the material in Chapters 0,3,4,5, and 8 from Volume 1.

We wish to thank the U.S. National Science Foundation for its support during the

preparation of this work. We also wish to take this opportunity to thank Dr. Kurt Reinicke for

critically checking the entire manuscript and offering improvements on many points.

Houston, Texas

R.M.B.

C.-C.W.

iii

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CONTENTS

Vol. 2

Vector and Tensor Analysis

Contents of Volume 1¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­

vii

PART III. VECTOR AND TENSOR ANALYSIS

Selected Readings for Part III¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­

296

CHAPTER 9. Euclidean Manifolds¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­..

297

Section 43.

Section 44.

Section 45.

Section 46.

Section 47.

Section 48.

Euclidean Point Spaces¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­..

Coordinate Systems¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­

Transformation Rules for Vector and Tensor Fields¡­.

Anholonomic and Physical Components of Tensors¡­.

Christoffel Symbols and Covariant Differentiation¡­...

Covariant Derivatives along Curves¡­¡­¡­¡­¡­¡­¡­..

297

306

324

332

339

353

CHAPTER 10. Vector Fields and Differential Forms¡­¡­¡­¡­¡­¡­¡­¡­¡­...

359

Section 49.

Section 5O.

Section 51.

Section 52.

Section 53.

Section 54.

CHAPTER 11.

Section 55.

Section 56.

Section 57.

Section 58.

Section 59.

Section 60.

Lie Derivatives¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­..

Frobenius Theorem¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­

Differential Forms and Exterior Derivative¡­¡­¡­¡­..

The Dual Form of Frobenius Theorem: the Poincar¨¦

Lemma¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­..

Vector Fields in a Three-Dimensiona1 Euclidean

Manifold, I. Invariants and Intrinsic Equations¡­¡­..

Vector Fields in a Three-Dimensiona1 Euclidean

Manifold, II. Representations for Special

Class of Vector Fields¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­.

359

368

373

381

389

399

Hypersurfaces in a Euclidean Manifold

Normal Vector, Tangent Plane, and Surface Metric¡­

Surface Covariant Derivatives¡­¡­¡­¡­¡­¡­¡­¡­¡­.

Surface Geodesics and the Exponential Map¡­¡­¡­..

Surface Curvature, I. The Formulas of Weingarten

and Gauss¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­

Surface Curvature, II. The Riemann-Christoffel

Tensor and the Ricci Identities¡­¡­¡­¡­¡­¡­¡­¡­...

Surface Curvature, III. The Equations of Gauss and Codazzi

v

407

416

425

433

443

449

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