Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

2


Preliminary calculus


This chapter is concerned with the formalism of probably the most widely used


mathematical technique in the physical sciences, namely the calculus. The chapter


divides into two sections. The first deals with the process of differentiation and the


second with its inverse process, integration. The material covered is essential for


the remainder of the book and serves as a reference. Readers who have previously


studied these topics should ensure familiarity by looking at the worked examples


in the main text and by attempting the exercises at the end of the chapter.


2.1 Differentiation

Differentiation is the process of determining how quickly or slowly a function


varies, as the quantity on which it depends, itsargument, is changed. More


specifically it is the procedure for obtaining an expression (numerical or algebraic)


for the rate of change of the function with respect to its argument. Familiar


examples of rates of change include acceleration (the rate of change of velocity)


and chemical reaction rate (the rate of change of chemical composition). Both


acceleration and reaction rate give a measure of the change of a quantity with


respect to time. However, differentiation may also be applied to changes with


respect to other quantities, for example the change in pressure with respect to a


change in temperature.


Although it will not be apparent from what we have said so far, differentiation

is in fact a limiting process, that is, it deals only with the infinitesimal change in


one quantity resulting from an infinitesimal change in another.


2.1.1 Differentiation from first principles

Let us consider a functionf(x) that depends on only one variablex, together with


numerical constants, for example,f(x)=3x^2 orf(x)=sinxorf(x)=2+3/x.

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