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Variational Methods in Optimization (Dover Books on Mathematics)by Donald Smith
Synopses & ReviewsPublisher 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 wellillustrated 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 standalone 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 PrenticeHall, 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 Contents1. 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 EulerLagrange 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 EulerLagrange 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 EulerLagrange Multiplier Theorem 3.6 The EulerLagrange 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 EulerLagrange 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 EulerLagrange 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 ChutetheChute 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 McShaneBlankinship Curtain Rod Problem; Functionals Involving HigherOrder Derivatives 4.10 Functionals Involving Several Independent Variables; the Minimal Surface Problem 4.11 The Vibrating String 5. Applications of the EulerLagrange 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 EulerLagrange 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 BangBang Problem 6.6 Some Remarks on the Maximum Principle and Dynamic Programming 7. The Variational Description of SturmLiouville Eigenvalues 7.1 Introduction to SturmLiouville Problems 7.2 The Relation Between the Lowest Eigenvalue and the Rayleigh Quotient 7.3 The RayleighRitz 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 HigherOrder 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 BoisReymond's Derivation of the EulerLagrange 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 What Our Readers Are SayingBe the first to add a comment for a chance to win!Product Details
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