Lectures on the Calculus of Variations by Harris Hancock
Lectures on the Calculus of Variations - Table of Contents
1. Presentation of the Principal Problems of the Calculus of Variations
2. Examples of Special Variations of Curves. Applications to the Catenary
3. Properties of the Catenary
4. Properties of the Function F{^X, Y, X, Y}
5. The Variation of Curves Expressed Analytically. The First Variation
6. The Form of the Solution of the Differential Equation G = 0
7. Removal of Certain Limitations that have been Made. Integration of the Differential Equation G = 0 For The Problems of 1
8. The Second Variation; Its Sign Determined By That of the Function F1
9. Conjugate Points
10. The Criteria that have been Derived Under the Assumption of Certain Special Variations are also Sufficient for the Establishment of the Formula Hitherto Employed
11. The Notion of a Field About the Curve Which Offers A Minimum Or A Maximum Value of the Integral. The Geometrical Meaning of the Conjugate Points
12. A Fourth and Final Condition for the Existence of a Maximum Or a Minimum, and a Proof that the Conditions which have been given are Sufficient Relative Maxima and Minima
13. Statement of the Problem. Derivation of the Necessary Conditions
14. The Isoperimetric Problem
15. Restricted Variations. The Theorems of Steiner
16. The Determination of the Curve of Given Length and given endpoints, Whose Center of Gravity Lies Lowest
17. The Sufficient Conditions
18. Proof of two Theorems which have been Assumed in the Previous Section
What You Will Learn in Lectures on the Calculus of Variations
"Calculus of Variations" by Harris Hancock is a classic text that explores the "calculus of variations", a branch of mathematics focused on finding functions that "optimize" certain quantities. Hancock presents the subject with clarity, combining historical context with rigorous analysis. The book introduces fundamental concepts and gradually develops methods to solve extremal problems, making it suitable for students, researchers, and mathematicians interested in applied mathematics.
A key feature of the text is its detailed treatment of the "Euler–Lagrange equation", the cornerstone of variational methods. Hancock carefully explains its derivation and applications, showing how small variations in functions can lead to maxima or minima of integrals. The book also covers conditions for relative extrema, multiple variables, and constrained problems, providing a solid theoretical foundation for variational analysis.
Hancock’s work stands out for its classical approach and thorough explanations. It not only serves as a reference for solving practical problems in mechanics, physics, and geometry but also offers insights into the historical development of the "calculus of variations". By combining rigor with readability, the book remains a valuable resource for understanding the principles, techniques, and applications of "optimization" in mathematics.
Book Details & Specifications
Title:
Lectures on the Calculus of Variations by Harris Hancock
Publisher:
Cincinnati University Press
Year:
1904
Pages:
322
Type:
PDF
Language:
English
ISBN-10 #:
1167038266
ISBN-13 #:
978-1167038266
License:
Public Domain Work
Amazon:
Amazon
About the Author: Harris Hancock
The author Harris Hancock
(1867–1944) was a notable American mathematician and educator, recognized for his contributions to the "calculus of variations" and mathematical analysis. He taught at the University of Cincinnati and focused on making advanced topics clear and accessible for students and researchers alike. Hancock authored classic texts explaining the "Euler–Lagrange equation" and methods of "optimization" in variational problems. His work combined rigorous theory with practical examples, influencing the teaching and study of applied mathematics, mechanics, and physics.
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