Concise Vector Analysis

C. J. Eliezer

• Publisher: Dover Publications
• Year: 2016
• ISBN:
  9780486809236
• Topic: Matematica
• Subtopic: Analisi vettoriale

This concise introduction to the methods and techniques of vector analysis is suitable for college undergraduates in mathematics as well as students of physics and engineering. Rich in exercises and examples, the straightforward presentation focuses on physical ideas rather than mathematical rigor.
The treatment begins with a chapter on vectors and vector addition, followed by a chapter on products of vector. Two succeeding chapters on vector calculus cover a variety of topics, including functions of a vector; line, surface, and volume integrals; the Laplacian operator, and more. The text concludes with a survey of standard applications, including Poinsot's central axis, Gauss's theorem, gravitational potential, Green's theorems, and other subjects.

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 :

(i)	Unlocalized or free vector which has magnitude and direction but no particular position associated with it, e.g. the moment of a couple.
(ii)	Sliding vector or vector localized along a straight line, e.g. force acting on a rigid body (though we speak of a force as acting at a point, by the principle of transmissibility of force, it is the line of action and not the point of application that is needed to specify the force).
(iii)	Tied vector or vector localized at a point, e.g. electric field.

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 consi...