Vector and Tensor Analysis

Eutiquio C. Young

• Publisher: CRC Press
• Year: 2017
• Print ISBN: 9781138064560
• eBook ISBN: 9781351990523
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Revised and updated throughout, this book presents the fundamental concepts of vector and tensor analysis with their corresponding physical and geometric applications - emphasizing the development of computational skills and basic procedures, and exploring highly complex and technical topics in simplified settings.;This text: incorporates transformation of rectangular cartesian coordinate systems and the invariance of the gradient, divergence and the curl into the discussion of tensors; combines the test for independence of path and the path independence sections; offers new examples and figures that demonstrate computational methods, as well as carify concepts; introduces subtitles in each section to highlight the appearance of new topics; provides definitions and theorems in boldface type for easy identification. It also contains numerical exercises of varying levels of difficulty and many problems solved.

1

VECTOR ALGEBRA

1.1 Introduction

In the study of physics and engineering, we encounter many important quantities which can be described by the specification of their magnitude alone in terms of some appropriate units. For example, the volume of a cube can be described by the number of cubic inches or cubic centimeters, and the temperature at a particular time of a day can be described by giving the number of degrees on a Fahrenheit scale or a Centigrade scale. Such quantities characterized by the fact that they have magnitude only are called scalar quantities. Scalar quantities are represented by real numbers, and they are also called scalars. On the other hand, there are many other physical quantities such as displacement, force, velocity, and acceleration which cannot be described by the specification of their magnitude alone. These quantities possess not only magnitude but also direction so that a complete description of any such quantity must specify these two pieces of information. Thus when a weatherman reports the wind velocity on a particular day, he specifies not only the speed of the wind (magnitude of the wind velocity), but also the direction from which the wind is blowing. Such quantities characterized by having magnitude and direction are called vector quantities.

Just as we use real numbers or scalars to represent and manipulate scalar quantities, so we use the mathematical entities called vectors to represent and manipulate vector quantities. Thus, in a sense, vectors can be thought of as generalized numbers. The study of the representation of vectors, the algebra and calculus of vectors, together with some of their various applications constitute the subject matter of vector analysis.

Scalars and vectors are hard...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface to the Second Edition
• Preface to the First Edition
• Chapter 1 Vector Algebra
• Chapter 2 Differential Calculus of Vector Functions of one Variable
• Chapter 3 Differential Calculus of Scalar and Vector Fields
• Chapter 4 Integral Calculus of Scalar and Vector Fields
• Chapter 5 Tensors in Rectangular Cartesian Coordinate Systems
• Chapter 6 Tensors in General Coordinates
• Solutions to Selected Problems
• Index