Concise Vector Analysis

PREFACE

THIS book aims at presenting concisely an introductory account of the methods and techniques of vector analysis. These methods are now accepted as indispensable tools in mathematics, and also in sciences such as physics or engineering. The first two chapters deal with vector algebra, the next two with vector calculus, and the last with some standard applications. The aim has been to provide a simple presentation, keeping the physical ideas to the forefront, and emphasising ease of understanding rather than mathematical rigour. Each chapter contains illustrative examples, as well as exercises, most of which are taken from old University examination papers. I am grateful to the Universities of Cambridge, Ceylon, London and Oxford for permission to use questions from their examination papers. The following abbreviations have been used :

The book is based on lectures given by the author for many years in the University of Ceylon. The idea of writing a book on this subject originated from the need in the Universities in Ceylon for mathematical books in Sinhala and Tamil. It is a pleasure to make these available in the English language to a wider group of students of mathematics, physics and engineering in universities and technical colleges.

Chapter 1

Vectors and Vector Addition

1.1.Vectors

MATHEMATICS has played an important part in the advance of science, and mathematical language has become essential for the formulation of the laws of science. Numbers, functions, vectors, tensors, spinors, matrices are examples of mathematical entities which occur in various branches of applied mathematics.

The physical quantities with which we are concerned here may be divided into two groups: (a) scalars, (b) vectors. A scalar requires only its magnitude for its specification. For example, temperature, mass, density, volume and energy are scalars. Vectors require both magnitude and direction for their specification. Force, displacement, velocity, acceleration and momentum are vectors.

Vectors may be classified in different ways. In one elementary classification we have three types of vectors distinguished by their effects :

In the early part of this book whenever we speak of a vector without stating its classification we mean a free vector. Two vectors of the same magnitude and direction, but acting along parallel lines (or along the same line) will be considered as equal or identical. Thereafter other types of vectors are considered.

Strictly speaking, every quantity which has a magnitude and a direction is not necessarily a vector. A vector is defined as a quantity with a magnitude and a direction, and which obeys the same addition rule as displacements, that is, the rule known as the parallelogram law of addition. (As an example of a quantity which has magnitude and direction but is not a vector, consider rotations of a rigid body through finite angles. These have magnitudes and directions, but do not obey the parallelogram law of addition, and are not vectors.)

A vector is denoted by symbols in bold type such as a, b, P, Q or by letters with arrows written above such as , .

1.2.Representation of vectors

A vector may be represented by a directed line segment, the direction of the line indicating the direction of the vector and the length of segment representing, in an appropriate scale of units, the magnitude of the vector. Parallel segments of the same length will represent equal or identical vectors. Since the exact position of vectors is not important and parallel segments represent equal vectors, it is convenient to represent vectors in a diagram by line segments starting from the same origin, say O. Thus the segment represents a vector a, where the direction of is the same as the direction of the vector a, and the length in a suitable scale of units is equal to the magnitude of the vector a. We denote the magnitude of the vector a by the symbol a written in italics or by the symbol |a| which is read as modulus of a or briefly as "mod a".

Fig. 1

1.3.Vector addition

Let , represent two vectors a and b respectively. We complete the parallelogram OACB, which has 'OA, OB as adjacent edges.

Fig. 2

= a, = b. Then

. Let = c. The line segment represents the displacement of a particle from O to A, and the displacement from A to C. The effect of these two displacements in succession is the same as that of a single displacement from O to C. Hence we write

that is

where + denotes vectorial addition.

This is the triangle rule for vector addition.

We may state the rule in an alternative way, namely that the sum of two vectors a and b is the vector c, which is represented by the diagonal OC of the parallelogram of which OA and OB are adjacent edges. We write

that is,

we see that

This shows that vector addition is commutative, and the order of addition may be interchanged.

We consider also the associative law of addition. Take three vectors P, Q, R and represent them by respectively. Construct P + Q and Q + R and then (P + Q) + R and P + (Q + R).

Fig. 3

Hence

The brackets which have been used to show the order of the operations may therefore be omitted, and the sum on either side be written as

This result may be extended to define the sum of any number of vectors P, Q, … , W.

Suppose we show in a diagram, Fig. 4(a), the vectors P, Q, … ,. W drawn from an origin O. We draw a polygon AB … K in which the vectors are represented in a suitable scale by directed segments , which are placed in succession, with

. Join AK. Then represents in the same scale the vector sum

If the sum is denoted by R, then . R is called the resultant vector, and P, Q, … are called the components of R.

Fig. 4

1.4.–a, O, λa

The symbol –a is used to indicate a vector which has the same magnitude as a but is opposite in direction. If , then

A vector whose terminal points coincide is called a null vector and is denoted by the symbol O. We see that

Also

The sum of the vectors a and – b, that is a + (–b), may be written a – b. In Fig. 2, , and , by the parallelogram law. Hence represents a – b.

If λ is a positive scalar, and a a vector then λa is defined to be the vector which has the same direction as a but whose magnitude is λ times that of a. If represents a, and represents λa, then the points O, A, A′ lie on a line and their distances are such that OA′/OA = λ.

Fig. 5

This multiplication of a vector by a scalar obeys the distributive law.

That is,

This may be verified by using properties of similar triangles. Let , then . Let A′ and B′ be such that