In a previous chapter it was shown how to find stationary values of functions of a single variable f(x), of several variables f(x, y,…) and of constrained variables f(x, y,…) subject to gi(x, y,…) = 0, (i = 1, 2, …, m). In all these cases the forms of the functions f and gi were known and the problem was one of finding suitable values of the variables x, y,…

We now turn to a different kind of problem, one in which there are not free variables which must be chosen in order to bring about a particular condition for a given function, but in which the functions are free and must be chosen to bring about a particular condition for a given expression which depends upon these functions.

To give a more concrete example of the type of question to be answered, we may ask the following. ‘Why does a uniform rope suspended between two points take up the shape it does? Why doesn't it hang in an arc of a circle or in the form of three sides of a rectangle? Is it possible to predict the shape in which it will hang, that is to find a function, y = y(x), that gives the vertical height of the rope as a function of horizontal position?’