# Variational Calculus with Elementary Convexity

Textbook

Part of the Undergraduate Texts in Mathematics book series (UTM)

1. Front Matter
Pages i-xiv
2. ### Review of Optimization in ℝ d

1. John L. Troutman
Pages 1-9
3. ### Basic Theory

1. Front Matter
Pages 11-11
2. John L. Troutman
Pages 13-34
3. John L. Troutman
Pages 35-51
4. John L. Troutman
Pages 52-92
5. John L. Troutman
Pages 93-98
6. John L. Troutman
Pages 99-141
7. John L. Troutman
Pages 142-190

1. Front Matter
Pages 191-192
2. John L. Troutman
Pages 193-229
3. John L. Troutman
Pages 230-270
4. John L. Troutman
Pages 271-325
5. Back Matter
Pages 327-365

### Introduction

The calculus of variations, whose origins can be traced to the works of Aristotle and Zenodoros, is now Ii vast repository supplying fundamental tools of exploration not only to the mathematician, but-as evidenced by current literature-also to those in most branches of science in which mathematics is applied. (Indeed, the macroscopic statements afforded by variational principles may provide the only valid mathematical formulation of many physical laws. ) As such, it retains the spirit of natural philosophy common to most mathematical investigations prior to this century. How­ ever, it is a discipline in which a single symbol (b) has at times been assigned almost mystical powers of operation and discernment, not readily subsumed into the formal structures of modern mathematics. And it is a field for which it is generally supposed that most questions motivating interest in the subject will probably not be answerable at the introductory level of their formulation. In earlier articles,1,2 it was shown through several examples that a complete characterization of the solution of optimization problems may be available by elementary methods, and it is the purpose of this work to explore further the convexity which underlay these individual successes in the context of a full introductory treatment of the theory of the variational calculus. The required convexity is that determined through Gateaux variations, which can be defined in any real linear space and which provide an unambiguous foundation for the theory.

### Keywords

Convexity Konvexe Funktion Variationsrechnung calculus differential equation

#### Authors and affiliations

1. 1.Department of MathematicsSyracuse UniversitySyracuseUSA

### Bibliographic information

• Book Title Variational Calculus with Elementary Convexity
• Authors J.L. Troutman
• Series Title Undergraduate Texts in Mathematics
• DOI https://doi.org/10.1007/978-1-4684-0158-5
• Copyright Information Springer-Verlag New York 1983
• Publisher Name Springer, New York, NY
• eBook Packages
• Hardcover ISBN 978-0-387-90771-0
• Softcover ISBN 978-1-4684-0160-8
• eBook ISBN 978-1-4684-0158-5
• Series ISSN 0172-6056
• Edition Number 1
• Number of Pages XIV, 365
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site
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