Physics : Vector & Scalar

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Cambridge University Press 0521652278 - Mathematical Methods for Physicists: A Concise Introduction - Tai L. Chow Excerpt More information

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Vector and tensor analysis

Vectors and scalars Vector methods have become standard tools for the physicists. In this chapter we discuss the properties of the vectors and vector ®elds that occur in classical physics. We will do so in a way, and in a notation, that leads to the formation of abstract linear vector spaces in Chapter 5. A physical quantity that is completely speci®ed, in appropriate units, by a single number (called its magnitude) such as volume, mass, and temperature is called a scalar. Scalar quantities are treated as ordinary real numbers. They obey all the regular rules of algebraic addition, subtraction, multiplication, division, and so on. There are also physical quantities which require a magnitude and a direction for their complete speci®cation. These are called vectors if their combination with each other is commutative (that is the order of addition may be changed without a€ecting the result). Thus not all quantities possessing magnitude and direction are vectors. Angular displacement, for example, may be characterised by magnitude and direction but is not a vector, for the addition of two or more angular displacements is not, in general, commutative (Fig. 1.1). In print, we shall denote vectors by boldface letters (such as A) and use ordinary italic letters (such as A) for their magnitudes; in writing, vectors are usually ~ ~ represented by a letter with an arrow above it such as A. A given vector A (or A) can be written as

^ A ˆ AA;

…1:1†

^ where A is the magnitude of vector A and so it has unit and dimension, and A is a dimensionless unit vector with a unity magnitude having the direction of A. Thus ^ A ˆ A=A.
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