Introduction to the Mathematics of Variation

Book Preface

This book is about the calculus of variations which is a subject concerned mainly with optimization of functionals. However, because part of it is based on using ordinary calculus in solving optimization problems, “Calculus of Variations” in its original title is modified to become “Mathematics of Variation”. In fact, the book is essentially a collection of solved problems with rather modest theoretical background and hence it is based on the method of “learning by example and practice” which in our view is the most e’ective way for learning mathematics and overcoming the diÿculties of its abstraction. The main merit of the book is its clarity, intuitive structure and rather inclusiveness as it includes the main topics and applications of this subject. The structure of the book is that it starts with a preliminary chapter which provides a general theoretical background about this subject with many solved problems related to this background. In the remaining chapters we present a number of common topics and applications related to the mathematical methods of variation. So, the remaining chapters consist essentially of solved problems classified according to certain mathematical and physical criteria with some introductory and general background.

Because the book is about the mathematical methods of variation, we do not explain things like inte-gration or partial di˙erentiation or how to obtain solutions of di˙erential equations although we usually make short explanatory remarks or put the results in a format that is easy to understand and verify. We also avoid going through many theoretical details and technicalities of the calculus of variations to avoid unnecessary distraction from our main practical objectives and to save space. For example, we do not go through the derivation of the formulae of the calculus of variations because such details can be found in almost every book on this subject. Similarly, we avoid going through technicalities related for example to the nature of the arguments of real-valued functions (such as the square root or natural logarithm) and how and when these arguments should be non-negative. So, in brief we take things rather easy in presenting the subject using general understanding and common sense in dealing with the varia-tional problems. Accordingly, the materials in this book have essentially practical objectives rather than pedantic mathematical purposes. So, the target of this book is essentially scientists, engineers and applied mathematicians rather than mathematicians.

The materials in this book require decent background in general mathematics (mostly in single-variable and multi-variable di˙erential and integral calculus). Some problems and applications also require rea-sonable background in physics which is the main field of application of the calculus (and mathematics) of variations. The book can be used as a text or as a reference for an introductory course on this subject as part of an undergraduate curriculum in physics or engineering or applied mathematics. The book can also be used as a source of supplementary pedagogical materials used in tutorial sessions associated with such a course.
Taha Sochi