© 2018

Calculus of Variations

An Introduction to the One-Dimensional Theory with Examples and Exercises

  • Provides examples and exercises with solutions, allowing for self-study

  • Includes numerous figures to help aid the reader

  • Includes advances problems and proofs in the appendix


Part of the Texts in Applied Mathematics book series (TAM, volume 67)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Hansjörg Kielhöfer
    Pages 1-56
  3. Hansjörg Kielhöfer
    Pages 57-137
  4. Hansjörg Kielhöfer
    Pages 139-181
  5. Back Matter
    Pages 183-227

About this book


This clear and concise textbook provides a rigorous introduction to the calculus of variations, depending on functions of one variable and their first derivatives. It is based on a translation of a German edition of the book Variationsrechnung (Vieweg+Teubner Verlag, 2010), translated and updated by the author himself.  Topics include: the Euler-Lagrange equation for one-dimensional variational problems, with and without constraints, as well as an introduction to the direct methods. The book targets students who have a solid background in calculus and linear algebra, not necessarily in functional analysis. Some advanced mathematical tools, possibly not familiar to the reader, are given along with proofs in the appendix. Numerous figures, advanced problems and proofs, examples, and exercises with solutions accompany the book, making it suitable for self-study.

The book will be particularly useful for beginning graduate students from the physical, engineering, and mathematical sciences with a rigorous theoretical background. 


Euler-Lagrange Equation Natural Boundary Conditions Variational Problems with Constraints Direct Methods in the Calculus of Variations Application of the Direct Methods Lagrange Multipliers Weierstraß-Erdmann Corner Conditions

Authors and affiliations

  1. 1.RimstingGermany

About the authors

Hansjörg Kielhöfer is a Professor at the University of Augsburg, Germany.

Bibliographic information

  • Book Title Calculus of Variations
  • Book Subtitle An Introduction to the One-Dimensional Theory with Examples and Exercises
  • Authors Hansjörg Kielhöfer
  • Series Title Texts in Applied Mathematics
  • Series Abbreviated Title Texts in Applied Math.
  • DOI
  • Copyright Information Springer International Publishing AG 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-319-71122-5
  • Softcover ISBN 978-3-319-89038-8
  • eBook ISBN 978-3-319-71123-2
  • Series ISSN 0939-2475
  • Series E-ISSN 2196-9949
  • Edition Number 1
  • Number of Pages XVI, 227
  • Number of Illustrations 59 b/w illustrations, 0 illustrations in colour
  • Topics Calculus of Variations and Optimal Control; Optimization
  • Buy this book on publisher's site


“This is a friendly and well-written introduction to a topic of great interest in modern analysis.” (M. Kunzinger, Monatshefte für Mathematik, Vol. 193 (4), 2020)

“This book is an introductory textbook … . The textbook is appropriate for students with solid background in calculus and linear algebra but without preliminary knowledge of variational analysis. … this textbook is useful for beginning graduate students in physical, engineering, and mathematical sciences having a rigorous theoretical background.” (Mihail Voicu, zbMATH 1390.49001, 2018)