2Preliminary 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 rst deals with the process of dierentiation 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 Dierentiation

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

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

specically 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, dierentiation 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, dierentiation

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

one quantity resulting from an innitesimal change in another.

2.1.1 Dierentiation from rst principles

Let us consider a function f(x) that depends on only one variable x, together with

numerical constants, for example, f(x) = 3x2 or f(x) = sin x or f(x) = 2 + 3/x.

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