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The great strength of this book is its exercises. They are lengthy, difficult, and many (described as ¿projects¿) are broken into manageable pieces. The author says (p. iv) about the exercises that ¿I spent at least three items as much effort in preparing them as I did on the main text itself,¿ and it shows. The level of the exercises is somewhere around that of the first sections of Pólya and Szego¿s Problems and Theorems in Analysis, although that book is slanted heavily towards complex function theory and the present book sticks to real functions. The narrative part of the book is well-done too, and contains many interesting things, but it is not such a standout as the exercises. The term ¿classical¿ in the title indicates that the book is slanted towards the concrete and has quite a lot on properties of particular series and integrals. In olden days it might have been titled Advanced Calculus, although it doesn¿t go very far into multi-variable calculus. In modern terms it is a text for a first rigorous course in mathematical analysis. This is a very competitive field, and as the present book was written in 1981, you would expect some better texts to have come out since then. Considering only the narrative part and not the exercises, I think Ross¿s Elementary Analysis would be a better choice for most courses. The present book is more advanced in some aspects, and in particular it develops the Lebesgue integral rather than the Riemann integral (through step functions rather than measure). Despite the slant towards concreteness, it does prove results in more generality when it¿s not much harder to do so. For example, it proves the Stone-Weierstrass theorem in its full generality rather than the Weierstrass approximation theorem.

This book is remarkable in its accessible treatment of interaction effects. Although this concept can be challenging for students (even those with some background in statistics), this book presents the material in a very accessible manner, with plenty of examples to help the reader understand how to interpret their results.¿

Although the calculus of variations has ancient origins in questions of Ar­ istotle and Zenodoros, its mathematical principles first emerged in the post­ calculus investigations of Newton, the Bernoullis, Euler, and Lagrange. Its results now supply fundamental tools of exploration to both mathematicians and those in the applied sciences. (Indeed, the macroscopic statements ob­ tained through variational principles may provide the only valid mathemati­ cal formulations of many physical laws. ) Because of its classical origins, variational calculus retains the spirit of natural philosophy common to most mathematical investigations prior to this century. The original applications, including the Bernoulli problem of finding the brachistochrone, require opti­ mizing (maximizing or minimizing) the mass, force, time, or energy of some physical system under various constraints. The solutions to these problems satisfy related differential equations discovered by Euler and Lagrange, and the variational principles of mechanics (especially that of Hamilton from the last century) show the importance of also considering solutions that just provide stationary behavior for some measure of performance of the system. However, many recent applications do involve optimization, in particular, those concerned with problems in optimal control. Optimal control is the rapidly expanding field developed during the last half-century to analyze optimal behavior of a constrained process that evolves in time according to prescribed laws. Its applications now embrace a variety of new disciplines, including economics and production planning.