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VARIATIONAL METHODS IN OPTIMIZATIONSMITH D.wydawnictwo: DOVER , rok wydania 1998, wydanie Icena netto: 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.
Functionals Introduction; Examples of Optimizational Problems Vector Spaces Functionals Normed Vector Spaces Continuous Functionals Linear Functionals A Fundamental Necessary Condition for an Extremum Introduction A Fundamental Necessary Condition for an Extremum Some Remarks on the Gâteaux Variation Examples on the Calculation of Gâteaux Variations An Optimization Problem in Production Planning Some Remarks on the Fréchet Differential The Euler-Lagrange Necessary Condition for an Extremum with Constraints Extremum Problems with a Single Constraint Weak Continuity of Variations Statement of the Euler-Lagrange Multiplier Theorem for a Single Constraint Three Examples, and Some Remarks on the Geometrical Significance of the Multiplier Theorem Proof of the Euler-Lagrange Multiplier Theorem The Euler-Lagrange Multiplier Theorem for Many Constraints An Optimum Consumption Policy with Terminal Savings Constraint During a Period of Inflation The Meaning of the Euler-Lagrange Multipliers Chaplygin's Problem, or a Modern Version of Queen Dido's Problem The John Multiplier Theorem Applications of the Euler-Lagrange Multiplier Theorem in the Calculus of Variations Problems with Fixed End Points John Bernoulli's Brachistochrone Problem, and Brachistochrones Through the Earth Geodesic Curves Problems with Variable End Points How to Design a Thrilling Chute-the-Chute Functionals Involving Several Unknown Functions Fermat's Principle in Geometrical Optics Hamilton's Principle of Stationary Action; an Example on Small Vibrations The McShane-Blankinship Curtain Rod Problem; Functionals Involving Higher-Order Derivatives Functionals Involving Several Independent Variables; the Minimal Surface Problem The Vibrating String Applications of the Euler-Lagrange Multiplier Theorem to Problems with Global Pointwise Inequality Constraints Slack Functions and Composite Curves An Optimum Consumption Policy with Terminal Savings Constraint Without Extreme Hardship A Problem in Production Planning with Inequality Constraints Applications of the Euler-Lagrange Multiplier Theorem in Elementary Control Theory Introduction A Rocket Control Problem: Minimum Time A Rocket Control Problem: Minimum Fuel A More General Control Problem A Simple Bang-Bang Problem Some Remarks on the Maximum Principle and Dynamic Programming The Variational Description of Sturm-Liouville Eigenvalues Introduction to Sturm-Liouville Problems The Relation Between the Lowest Eigenvalue and the Rayleigh Quotient The Rayleigh-Ritz Method for the Lowest Eigenvalue Higher Eigenvalues and the Rayleigh Quotient The Courant Minimax Principle Some Implications of the Courant Minimax Principle Further Extensions of the Theory Some General Remarks on the Ritz Method of Approximate Minimization Some Remarks on the Use of the Second Variation in Extremum Problems Higher-Order Variations A Necessary Condition Involving the Second Variation at an Extremum Sufficient Conditions for a Local Extremum The Cauchy and Schwarz Inequalities An Example on Normed Vector Spaces An Integral Inequality A Fundamental Lemma of the Calculus of Variations Du Bois-Reymond's Derivation of the Euler-Lagrange Equation A Useful Result from Calculus The Construction of a Certain Function The Fundamental Lemma for the Case of Several Independent Variables The Kinetic Energy for a Certain Model of an Elastic String The Variation of an Initial Value Problem with Respect to a Parameter Subject Index; Author Index 394 pages
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