A First Course in the Calculus of Variations (Student Mathematical Library) (Student Mathematical Library, 72)

This book is intended for a first course in the calculus of variations, at the senior or beginning graduate level. The reader will learn methods for finding functions that maximize or minimize integrals. The text lays out important necessary and sufficient conditions for extrema in historical order, and it illustrates these conditions with numerous worked-out examples from mechanics, optics, geometry, and other fields. The exposition starts with simple integrals containing a single independent variable, a single dependent variable, and a single derivative, subject to weak variations, but steadily moves on to more advanced topics, including multivariate problems, constrained extrema, homogeneous problems, problems with variable endpoints, broken extremals, strong variations, and sufficiency conditions. Numerous line drawings clarify the mathematics. Each chapter ends with recommended readings that introduce the student to the relevant scientific literature and with exercises that consolidate understanding. eadership: Undergraduate students interested in the calculus of variations. This book is intended for a first course in the calculus of variations, at the senior or beginning graduate level. The reader will learn methods for finding functions that maximize or minimize integrals. The text lays out important necessary and sufficient conditions for extrema in historical order, and it illustrates these conditions with numerous worked-out examples from mechanics, optics, geometry, and other fields. The exposition starts with simple integrals containing a single independent variable, a single dependent variable, and a single derivative, subject to weak variations, but steadily moves on to more advanced topics, including multivariate problems, constrained extrema, homogeneous problems, problems with variable endpoints, broken extremals, strong variations, and sufficiency conditions. Numerous line drawings clarify the mathematics. Each chapter ends with recommended readings that introduce the student to the relevant scientific literature and with exercises that consolidate understanding. Mark Kot teaches at the University of Washington, Seattle. WA Cover Title page 4 Table of contents 6 Preface 10 Free 1. Introduction 12 2. The First Variation 38 3. Cases and Examples 58 4. Basic Generalizations 80 5. Constraints 118 6. The Second Variation 150 7. Review and Preview 182 8. The Homogeneous Problem 188 9. Variable-Endpoint Conditions 202 10. Broken Extremals 226 11. Strong Variations 246 12. Sufficient Conditions 266 Bibliography 288 Index 306 Back Cover