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Variational Methods in Optimization (74 Edition)

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Variational Methods in Optimization (74 Edition) Cover

 

Synopses & Reviews

Please note that used books may not include additional media (study guides, CDs, DVDs, solutions manuals, etc.) as described in the publisher comments.

Publisher Comments:

Fostering an increased appreciation of the fundamental theorem of calculus, this highly readable text elucidates applications of the chain rule of differentiation, integration by parts, parametric curves, line integrals, double integrals, and elementary differential equations. A clear and well-illustrated treatment of techniques for solving a wide variety of optimization problems arising in a diverse array of fields, this volume requires only an elementary knowledge of calculus and can be used either by itself or as a supplementary text in a variety of courses. 1974 edition.

Book News Annotation:

Gives an elementary exposition of an extension of the standard differentiation method which leads to an increase in both the number and type of problems that can be solved. Covers problems with fixed endpoints, variable endpoints, isoperimetric constraints, global inequality constraints, the inverse function theorem, and the chain rule of differentiation. Intended for use as a stand-alone text or as a supplement to courses in applied economics, the calculus of variations, optimal control theory, or terminal calculus. An unabridged republication of the work originally published in 1974 by Prentice-Hall, Inc. Annotation c. by Book News, Inc., Portland, OR (booknews.com)

Synopsis:

Highly readable text elucidates applications of the chain rule of differentiation, integration by parts, parametric curves, line integrals, double integrals, and elementary differential equations. 1974 edition.

Synopsis:

Highly readable text elucidates applications of the chain rule of differentiation, integration by parts, parametric curves, line integrals, double integrals, and elementary differential equations. 1974 edition.

Synopsis:

Gives an elementary exposition of an extension of the standard differentiation method which leads to an increase in both the number and type of problems that can be solved. Covers problems with fixed endpoints, variable endpoints, isoperimetric constraints, global inequality constraints, the inverse

Table of Contents

1. Functionals

  1.1 Introduction; Examples of Optimizational Problems

  1.2 Vector Spaces

  1.3 Functionals

  1.4 Normed Vector Spaces

  1.5 Continuous Functionals

  1.6 Linear Functionals

2. A Fundamental Necessary Condition for an Extremum

  2.1 Introduction

  2.2 A Fundamental Necessary Condition for an Extremum

  2.3 Some Remarks on the Gâteaux Variation

  2.4 Examples on the Calculation of Gâteaux Variations

  2.5 An Optimization Problem in Production Planning

  2.6 Some Remarks on the Fréchet Differential

3. The Euler-Lagrange Necessary Condition for an Extremum with Constraints

  3.1 Extremum Problems with a Single Constraint

  3.2 Weak Continuity of Variations

  3.3 Statement of the Euler-Lagrange Multiplier Theorem for a Single Constraint

  3.4 Three Examples, and Some Remarks on the Geometrical Significance of the Multiplier Theorem

  3.5 Proof of the Euler-Lagrange Multiplier Theorem

  3.6 The Euler-Lagrange Multiplier Theorem for Many Constraints

  3.7 An Optimum Consumption Policy with Terminal Savings Constraint During a Period of Inflation

  3.8 The Meaning of the Euler-Lagrange Multipliers

  3.9 Chaplygin's Problem, or a Modern Version of Queen Dido's Problem

  3.10 The John Multiplier Theorem

4. Applications of the Euler-Lagrange Multiplier Theorem in the Calculus of Variations

  4.1 Problems with Fixed End Points

  4.2 John Bernoulli's Brachistochrone Problem, and Brachistochrones Through the Earth

  4.3 Geodesic Curves

  4.4 Problems with Variable End Points

  4.5 How to Design a Thrilling Chute-the-Chute

  4.6 Functionals Involving Several Unknown Functions

  4.7 Fermat's Principle in Geometrical Optics

  4.8 Hamilton's Principle of Stationary Action; an Example on Small Vibrations

  4.9 The McShane-Blankinship Curtain Rod Problem; Functionals Involving Higher-Order Derivatives

  4.10 Functionals Involving Several Independent Variables; the Minimal Surface Problem

  4.11 The Vibrating String

5. Applications of the Euler-Lagrange Multiplier Theorem to Problems with Global Pointwise Inequality Constraints

  5.1 Slack Functions and Composite Curves

  5.2 An Optimum Consumption Policy with Terminal Savings Constraint Without Extreme Hardship

  5.3 A Problem in Production Planning with Inequality Constraints

6. Applications of the Euler-Lagrange Multiplier Theorem in Elementary Control Theory

  6.1 Introduction

  6.2 A Rocket Control Problem: Minimum Time

  6.3 A Rocket Control Problem: Minimum Fuel

  6.4 A More General Control Problem

  6.5 A Simple Bang-Bang Problem

  6.6 Some Remarks on the Maximum Principle and Dynamic Programming

7. The Variational Description of Sturm-Liouville Eigenvalues

  7.1 Introduction to Sturm-Liouville Problems

  7.2 The Relation Between the Lowest Eigenvalue and the Rayleigh Quotient

  7.3 The Rayleigh-Ritz Method for the Lowest Eigenvalue

  7.4 Higher Eigenvalues and the Rayleigh Quotient

  7.5 The Courant Minimax Principle

  7.6 Some Implications of the Courant Minimax Principle

  7.7 Further Extensions of the Theory

  7.8 Some General Remarks on the Ritz Method of Approximate Minimization

8. Some Remarks on the Use of the Second Variation in Extremum Problems

  8.1 Higher-Order Variations

  8.2 A Necessary Condition Involving the Second Variation at an Extremum

  8.3 Sufficient Conditions for a Local Extremum

Appendix 1. The Cauchy and Schwarz Inequalities

Appendix 2. An Example on Normed Vector Spaces

Appendix 3. An Integral Inequality

Appendix 4. A Fundamental Lemma of the Calculus of Variations

Appendix 5. Du Bois-Reymond's Derivation of the Euler-Lagrange Equation

Appendix 6. A Useful Result from Calculus

Appendix 7. The Construction of a Certain Function

Appendix 8. The Fundamental Lemma for the Case of Several Independent Variables

Appendix 9. The Kinetic Energy for a Certain Model of an Elastic String

Appendix 10. The Variation of an Initial Value Problem with Respect to a Parameter

  Subject Index; Author Index

Product Details

ISBN:
9780486404554
Author:
Smith, Donald R.
Publisher:
Dover Publications
Author:
Mathematics
Location:
Mineola, N.Y. :
Subject:
Calculus
Subject:
Mathematical Analysis
Subject:
Applied
Subject:
Mathematical optimization
Subject:
Calculus of variations
Subject:
General Mathematics
Subject:
Mathematics-Applied
Edition Description:
Trade Paper
Series:
Dover Books on Mathematics
Publication Date:
19980631
Binding:
TRADE PAPER
Language:
English
Illustrations:
Yes
Pages:
394
Dimensions:
8.25 x 5.38 in 0.95 lb

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Related Subjects

Arts and Entertainment » Art » General
Science and Mathematics » Mathematics » Advanced
Science and Mathematics » Mathematics » Applied
Science and Mathematics » Mathematics » Calculus » General
Science and Mathematics » Mathematics » Combinatorics
Science and Mathematics » Mathematics » General
Science and Mathematics » Mathematics » Numeric Analysis
Young Adult » General

Variational Methods in Optimization (74 Edition) Used Trade Paper
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Product details 394 pages Dover Publications - English 9780486404554 Reviews:
"Synopsis" by ,
Highly readable text elucidates applications of the chain rule of differentiation, integration by parts, parametric curves, line integrals, double integrals, and elementary differential equations. 1974 edition.

"Synopsis" by ,
Highly readable text elucidates applications of the chain rule of differentiation, integration by parts, parametric curves, line integrals, double integrals, and elementary differential equations. 1974 edition.

"Synopsis" by , Gives an elementary exposition of an extension of the standard differentiation method which leads to an increase in both the number and type of problems that can be solved. Covers problems with fixed endpoints, variable endpoints, isoperimetric constraints, global inequality constraints, the inverse
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