Problems and Worked Solutions in Vector Analysis

L.R. Shorter

• Publisher: Dover Publications
• Year: 2014
• ISBN:
  9780486795485
• Topic: Mathematics
• Subtopic: Vector Analysis
• Index: Mathematics

`A handy book like this,` noted The Mathematical Gazette, `will fill a great want.` Devoted to fully worked out examples, this unique text constitutes a self-contained introductory course in vector analysis for undergraduate and graduate students of applied mathematics.
Opening chapters define vector addition and subtraction, show how to resolve and determine the direction of two or more vectors, and explain systems of coordinates, vector equations of a plane and straight line, relative velocity and acceleration, and infinitely small vectors. The following chapters deal with scalar and vector multiplication, axial and polar vectors, areas, differentiation of vector functions, gradient, curl, divergence, and analytical properties of the position vector. Applications of vector analysis to dynamics and physics are the focus of the final chapter, including such topics as moving rigid bodies, energy of a moving rigid system, central forces, equipotential surfaces, Gauss's theorem, and vector flow.

CHAPTER I

ADDITION

[1]. Definition of a Vector.

A vector is a magnitude which can be represented by a straight line of finite length drawn in a definite direction.

In the following pages the term vector will be applied both to the magnitude itself and to its representation by a finite straight line drawn in a parallel direction.

Nothing is said in this definition of the point from which the vector starts—the initial point—so that we may consider the position of this point to be arbitrary.

There are, then, an infinite number of vectors having the common properties of a given finite length and a given direction. Hence all vectors having the same given length and the same direction are to be considered equivalent: thus a vector from any initial point may be replaced by an equivalent vector from any other initial point. The point which terminates a vector is called its terminal point or extremity.

Vectors in the text will be represented by clarendon letters, such as a, b, c, r; but when referred to in diagrams, they will be represented by large capitals in clarendon, thus AB, CD.

Finite algebraical quantities in which direction is not involved are called scalar quantities or scalars: these will be represented in the text in ordinary type.

Such quantities as Force, Acceleration, Velocity (when the direction of motion of the latter is taken into account), Fluid Flow in a definite direction, are examples of vectors.

Among scalars are included: the temperature of a body at a given point in it; the work done by forces on a body when it is displaced by their action from one position to another; the mass of a body; an interval of time.

[2]. Addition of Vectors.

The term “addition” is applied to a method of combining vectors, which is analogous to the ordinary addition of algebraical quantities, and which is familiar in the graphic composition and resolution of forces.

FIG. 1.

Let AB represent a vector, with initial point A and terminal point B (Fig. 1), and let another BC be drawn from the terminal point B of AB. The plus sign in clarendon type, +, will be employed to denote this operation. Calling the vector AB, a and the vector BC, b, the process just described is symbolised by a + b.

Now if we consider the point A as joined to C by the vector AC or c, we see that the process symbolised by a + b leads to the same point as that reached by drawing the vector c from A, so that we may say,

where = may be read as “equivalent to.” (See also next Section.)

This, then, is an example of the addition of vectors, and we conclude from it, that the addition of two vectors amounts to the determination of a third vector having the same initial point as the first, its terminal point coinciding with the final point reached by carrying out the process of addition in the way indicated above.

When a and b have the same direction the numerical or scalar value of their sum is the same as that of the numerical measures of a and b: in all other cases the numerical or scalar value of the vector is algebraically less.

If from the initial point A in Fig. 1, the vector AB′ is drawn equal to BC, and then a vector B′C equal to the vector a is drawn from B′, its terminal point coincides with C, because the lines form a parallelogram and the addition of these two latter vectors gives the same vector diagonal c as in the first ...