A First Course in the Calculus of Variations
About this Title
Mark Kot, University of Washington, Seattle, WA
Publication: The Student Mathematical Library
Publication Year 2014: Volume 72
ISBNs: 978-1-4704-1495-5 (print); 978-1-4704-1961-5 (online)
MathSciNet review: MR3241749
MSC: Primary 49-01
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.
Undergraduate students interested in the calculus of variations.
Table of Contents
- Chapter 1. Introduction
- Chapter 2. The first variation
- Chapter 3. Cases and examples
- Chapter 4. Basic generalizations
- Chapter 5. Constraints
- Chapter 6. The second variation
- Chapter 7. Review and preview
- Chapter 8. The homogeneous problem
- Chapter 9. Variable-endpoint conditions
- Chapter 10. Broken extremals
- Chapter 11. Strong variations
- Chapter 12. Sufficient conditions