How to find stationary values of functions of a single variable f(x), of several variables f(x, y, …) and of constrained variables, where x, y, … are subject to the n constraints gi(x, y, …) = 0, i = 1, 2, …, n will be known to the reader and is summarized in Sections A.3 and A.7 of Appendix A. In all those cases the forms of the functions f and gi were known, and the problem was one of finding the appropriate values of the variables x, y, etc.
We now turn to a different kind of problem in which we are interested in bringing about a particular condition for a given expression (usually maximizing or minimizing it) by varying the functions on which the expression depends. For instance, we might want to know in what shape a fixed length of rope should be arranged so as to enclose the largest possible area, or in what shape it will hang when suspended under gravity from two fixed points. In each case we are concerned with a general maximization or minimization criterion by which the function y(x) that satisfies the given problem may be found.
The calculus of variations provides a method for finding the function y(x). The problem must first be expressed in a mathematical form, and the form most commonly applicable to such problems is an integral.
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