Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

PRELIMINARY ALGEBRA


be obvious, but they are both formed in the same way in terms of recurrence


relations. Whatever the sign ofn, the series of coefficientsnCkcan be generated


by starting withnC 0 = 1 and using the recurrence relation


nC
k+1=

n−k
k+1

nC
k. (1.59)

The difference is that for positive integernthe series terminates whenk=n,


whereas for negativenthere is no such termination – in line with the infinite


series of terms in the corresponding expansion.


Finally we note that, in fact, equation (1.59) generates the appropriate coef-

ficients for all values ofn, positive or negative, integer or non-integer, with the


obvious exception of the case in whichx=−yandnis negative. For non-integer


nthe expansion does not terminate, even ifnis positive.


1.7 Some particular methods of proof

Much of the mathematics used by physicists and engineers is concerned with


obtaining a particular value, formula or function from a given set of data and


stated conditions. However, just as it is essential in physics to formulate the basic


laws and so be able to set boundaries on what can or cannot happen, so it


is important in mathematics to be able to state general propositions about the


outcomes that are or are not possible. To this end one attempts to establish


theorems that state in as general a way as possible mathematical results that


apply to particular types of situation. We conclude this introductory chapter by


describing two methods that can sometimes be used to prove particular classes


of theorems.


The two general methods of proof are known as proof by induction (which

has already been met in this chapter) and proof by contradiction. They share


the common characteristic that at an early stage in the proof an assumption


is made that a particular (unproven) statement is true; the consequences of


that assumption are then explored. In an inductive proof the conclusion is


reached that the assumption is self-consistent and has other equally consistent


but broader implications, which are then applied to establish the general validity


of the assumption. A proof by contradiction, however, establishes an internal


inconsistency and thus shows that the assumption is unsustainable; the natural


consequence of this is that the negative of the assumption is established as true.


Later in this book use will be made of these methods of proof to explore new

territory, e.g. to examine the properties of vector spaces, matrices and groups.


However, at this stage we will draw our illustrative and test examples from earlier


sections of this chapter and other topics in elementary algebra and number theory.

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