Six Themes On Variation

by Cox, Steven J.; Forman, Robin; Jones, Frank; Keyfitz, Barbara Lee; Morgan, Frank; Wolf, Michael; Hardt, Robert

• ISBN13: 9780821837207
• ISBN10: 0821837206
• Format: Paperback
• Copyright: 2004-12-01
• Publisher: Amer Mathematical Society

Summary

The calculus of variations is a beautiful subject with a rich history and with origins in the minimization problems of calculus. Although it is now at the core of many modern mathematical fields, it does not have a well-defined place in most undergraduate mathematics curricula. This volume should nevertheless give the undergraduate reader a sense of its great character and importance. Interesting functionals, such as area or energy, often give rise to problems whose most natural solution occurs by differentiating a one-parameter family of variations of some function. The critical points of the functional are related to the solutions of the associated Euler-Lagrange equation. These differential equations are at the heart of the calculus of variations and its applications to other subjects. Some of the topics addressed in this book are Morse theory, wave mechanics, minimal surfaces, soap bubbles, and modeling traffic flow. All are readily accessible to advanced undergraduates. This book is derived from a workshop sponsored by Rice University. It is suitable for advanced undergraduates, graduate students and research mathematicians interested in the calculus of variations and its applications to other subjects.

Table of Contents

Preface	vii
List of Contributors	xi
Calculus of Variations: What Does ``Variations'' Mean?	1	(12)
Frank Jones
How Many Equilibria Are There? An Introduction to Morse Theory	13	(24)
Robin Forman
Aye, There's the Rub. An Inquiry into Why a Plucked String Comes to Rest	37	(22)
Steven J. Cox
Proof of the Double Bubble Conjecture	59	(20)
Frank Morgan
Minimal Surfaces, Flat Cone Spheres and Moduli Spaces of Staircases	79	(48)
Michael Wolf
Hold That Light! Modeling of Traffic Flow by Differential Equations	127
Barbara Lee Keyfitz