Mathematics for Physicists

3 Differential calculus

The introduction of the infinitesimal calculus, independently by Newton and Leibnitz in the late seventeenth century, was one of the most important events not only in the history of mathematics but also of physics, where it has been an indispensable tool ever since.

In this chapter and the one that follows, we introduce the formalism in the context of functions of a single variable. We start by considering differentiation, the calculation of the instantaneous rate of change of a function as its argument changes. So, for example, given a function x(t), which specifies the position of a particle moving in one dimension as a function of time t, the operation of differentiation yields a function representing the velocity. The inverse operation, called integration, will be discussed in Chapter 4 and enables the position x(t) to be deduced from and the value of x at some time, for example t = 0. These two operations – differentiation and integration – play a crucial role in understanding not only mechanics, but the whole of physical science. Both rest on ideas of limits and continuity, to which we now turn.

3.1 Limits and continuity

In previous chapters, we have used the ideas of limits and continuity in simple cases where their meaning is obvious. In this ...