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.

For reasons that are explained in the preface to Essential Mathematical Methods for the Physical Sciences the text of the third edition of Mathematical Methods for Physics and Engineering (MMPE) (Cambridge: Cambridge University Press, 2006) by Riley, Hobson and Bence, after a number of additions and omissions, has been republished as two slightly overlapping texts. Essential Mathematical Methods for the Physical Sciences (EMMPS) contains most of the more advanced material, and specifically develops mathematical methods that can be applied throughout the physical sciences; an augmented version of the more introductory material, principally concerned with mathematical tools rather than methods, is available as Foundation Mathematics for the Physical Sciences. The full text of MMPE, including all of the more specialized and advanced topics, is still available under its original title.

As in the third edition of MMPE, the penultimate subsection of each chapter of EMMPS consists of a significant number of problems, nearly all of which are based on topics drawn from several sections of that chapter. Also as in the third edition, hints and outline answers are given in the final subsection, but only to the odd-numbered problems, leaving all even-numbered problems free to be set as unaided homework.

This book is the solutions manual for the problems in EMMPS. For the 230 plus oddnumbered problems it contains, complete solutions are available, to both students and their teachers, in the form of this manual; these are in addition to the hints and outline answers given in the main text.