Mathematical Methods for Physics and Engineering A Comprehensive Guide

In chapter 2, we discussed functions f of only one variable x, which were usually written f(x). Certain constants and parameters may also have appeared in the definition of f, e.g. f(x) = ax+2 contains the constant 2 and the parameter a, but only x was considered as a variable and only the derivatives f(n)(x) = dnf/dxn were defined.

However, we may equally well consider functions that depend on more than one variable, e.g. the function f(x, y) = x 2 + 3xy, which depends on the two variables x and y. For any pair of values x, y, the function f(x, y) has a well-defined value, e.g. f(2, 3) = 22. This notion can clearly be extended to functions dependent on more than two variables. For the n-variable case, we write f(x 1, x 2, …, xn ) for a function that depends on the variables x 1, x 2, …, xn . When n = 2, x 1 and x 2 correspond to the variables x and y used above.

Functions of one variable, like f(x), can be represented by a graph on a plane sheet of paper, and it is apparent that functions of two variables can, with little effort, be represented by a surface in three-dimensional space. Thus, we may also picture f(x, y) as describing the variation of height with position in a mountainous landscape. Functions of many variables, however, are usually very difficult to visualise and so the preliminary discussion in this chapter will concentrate on functions of just two variables.