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Lecture Notes on Calculus of Variations

Author : Kung Ching Chang
Publisher : World Scientific
Release : 2016-09-16
Category : Mathematics
ISBN : 9789813144705

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Book Lecture Notes on Calculus of Variations Description/Summary:

This is based on the course "Calculus of Variations" taught at Peking University from 2006 to 2010 for advanced undergraduate to graduate students majoring in mathematics. The book contains 20 lectures covering both the theoretical background material as well as an abundant collection of applications. Lectures 1–8 focus on the classical theory of calculus of variations. Lectures 9–14 introduce direct methods along with their theoretical foundations. Lectures 15–20 showcase a broad collection of applications. The book offers a panoramic view of the very important topic on calculus of variations. This is a valuable resource not only to mathematicians, but also to those students in engineering, economics, and management, etc.

Calculus of Variations

Author : I. M. Gelfand,S. V. Fomin
Publisher : Courier Corporation
Release : 2012-04-26
Category : Mathematics
ISBN : 9780486135014

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Book Calculus of Variations Description/Summary:

Fresh, lively text serves as a modern introduction to the subject, with applications to the mechanics of systems with a finite number of degrees of freedom. Ideal for math and physics students.

Selected Chapters in the Calculus of Variations

Author : Jürgen Moser
Publisher : Springer Science & Business Media
Release : 2003-05-23
Category : Mathematics
ISBN : 3764321857

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Book Selected Chapters in the Calculus of Variations Description/Summary:

0.1 Introduction These lecture notes describe a new development in the calculus of variations which is called Aubry-Mather-Theory. The starting point for the theoretical physicist Aubry was a model for the descrip tion of the motion of electrons in a two-dimensional crystal. Aubry investigated a related discrete variational problem and the corresponding minimal solutions. On the other hand, Mather started with a specific class of area-preserving annulus mappings, the so-called monotone twist maps. These maps appear in mechanics as Poincare maps. Such maps were studied by Birkhoff during the 1920s in several papers. In 1982, Mather succeeded to make essential progress in this field and to prove the existence of a class of closed invariant subsets which are now called Mather sets. His existence theorem is based again on a variational principle. Although these two investigations have different motivations, they are closely re lated and have the same mathematical foundation. We will not follow those ap proaches but will make a connection to classical results of Jacobi, Legendre, Weier strass and others from the 19th century. Therefore in Chapter I, we will put together the results of the classical theory which are the most important for us. The notion of extremal fields will be most relevant. In Chapter II we will investigate variational problems on the 2-dimensional torus. We will look at the corresponding global minimals as well as at the relation be tween minimals and extremal fields. In this way, we will be led to Mather sets.

Introduction to the Calculus of Variations

Author : Bernard Dacorogna
Publisher : Imperial College Press
Release : 2004
Category : Mathematics
ISBN : 9781860945083

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Book Introduction to the Calculus of Variations Description/Summary:

- Serves as an excellent introduction to the calculus of variations - Useful to researchers in different fields of mathematics who want to get a concise but broad introduction to the subject - Includes more than 70 exercises with solutions

Calculus of Variations and Optimal Control Theory

Author : Daniel Liberzon
Publisher : Princeton University Press
Release : 2012
Category : Mathematics
ISBN : 9780691151878

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Book Calculus of Variations and Optimal Control Theory Description/Summary:

This textbook offers a concise yet rigorous introduction to calculus of variations and optimal control theory, and is a self-contained resource for graduate students in engineering, applied mathematics, and related subjects. Designed specifically for a one-semester course, the book begins with calculus of variations, preparing the ground for optimal control. It then gives a complete proof of the maximum principle and covers key topics such as the Hamilton-Jacobi-Bellman theory of dynamic programming and linear-quadratic optimal control. Calculus of Variations and Optimal Control Theory also traces the historical development of the subject and features numerous exercises, notes and references at the end of each chapter, and suggestions for further study. Offers a concise yet rigorous introduction Requires limited background in control theory or advanced mathematics Provides a complete proof of the maximum principle Uses consistent notation in the exposition of classical and modern topics Traces the historical development of the subject Solutions manual (available only to teachers) Leading universities that have adopted this book include: University of Illinois at Urbana-Champaign ECE 553: Optimum Control Systems Georgia Institute of Technology ECE 6553: Optimal Control and Optimization University of Pennsylvania ESE 680: Optimal Control Theory University of Notre Dame EE 60565: Optimal Control

Lectures on the Calculus of Variations

Author : Oskar Bolza
Publisher : Courier Dover Publications
Release : 2018-03-21
Category : Mathematics
ISBN : 9780486822365

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Book Lectures on the Calculus of Variations Description/Summary:

Pioneering modern treatise studies the development of the subject from Euler to Hilbert, addressing basic problems with sufficient generality and rigor to provide a sound introduction for serious study. 1904 edition.

Lectures on Linear Algebra

Author : I. M. Gelfand
Publisher : Courier Corporation
Release : 1989-01-01
Category : Mathematics
ISBN : 0486660826

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Book Lectures on Linear Algebra Description/Summary:

Prominent Russian mathematician's concise, well-written exposition considers n-dimensional spaces, linear and bilinear forms, linear transformations, canonical form of an arbitrary linear transformation, and an introduction to tensors. While not designed as an introductory text, the book's well-chosen topics, brevity of presentation, and the author's reputation will recommend it to all students, teachers, and mathematicians working in this sector.

Lectures on Optimal Transport

Author : Luigi Ambrosio,Elia Brué,Daniele Semola
Publisher : Springer Nature
Release : 2021-07-22
Category : Mathematics
ISBN : 9783030721626

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Book Lectures on Optimal Transport Description/Summary:

This textbook is addressed to PhD or senior undergraduate students in mathematics, with interests in analysis, calculus of variations, probability and optimal transport. It originated from the teaching experience of the first author in the Scuola Normale Superiore, where a course on optimal transport and its applications has been given many times during the last 20 years. The topics and the tools were chosen at a sufficiently general and advanced level so that the student or scholar interested in a more specific theme would gain from the book the necessary background to explore it. After a large and detailed introduction to classical theory, more specific attention is devoted to applications to geometric and functional inequalities and to partial differential equations.

Lectures on Elliptic Partial Differential Equations

Author : Luigi Ambrosio,Alessandro Carlotto,Annalisa Massaccesi
Publisher : Springer
Release : 2019-01-10
Category : Mathematics
ISBN : 9788876426513

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Book Lectures on Elliptic Partial Differential Equations Description/Summary:

The book originates from the Elliptic PDE course given by the first author at the Scuola Normale Superiore in recent years. It covers the most classical aspects of the theory of Elliptic Partial Differential Equations and Calculus of Variations, including also more recent developments on partial regularity for systems and the theory of viscosity solutions.

A First Course in the Calculus of Variations

Author : Mark Kot
Publisher : American Mathematical Society
Release : 2014-10-06
Category : Mathematics
ISBN : 9781470414955

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Book A First Course in the Calculus of Variations Description/Summary:

This book is intended for a first course in the calculus of variations, at the senior or beginning graduate level. The reader will learn methods for finding functions that maximize or minimize integrals. The text lays out important necessary and sufficient conditions for extrema in historical order, and it illustrates these conditions with numerous worked-out examples from mechanics, optics, geometry, and other fields. The exposition starts with simple integrals containing a single independent variable, a single dependent variable, and a single derivative, subject to weak variations, but steadily moves on to more advanced topics, including multivariate problems, constrained extrema, homogeneous problems, problems with variable endpoints, broken extremals, strong variations, and sufficiency conditions. Numerous line drawings clarify the mathematics. Each chapter ends with recommended readings that introduce the student to the relevant scientific literature and with exercises that consolidate understanding.

The Calculus of Variations and Functional Analysis

Author : L. P. Lebedev,Michael J. Cloud
Publisher : World Scientific
Release : 2003
Category : Mathematics
ISBN : 9789812794994

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Book The Calculus of Variations and Functional Analysis Description/Summary:

This volume is aimed at those who are concerned about Chinese medicine - how it works, what its current state is and, most important, how to make full use of it. The audience therefore includes clinicians who want to serve their patients better and patients who are eager to supplement their own conventional treatment. The authors of the book belong to three different fields, modern medicine, Chinese medicine and pharmacology. They provide information from their areas of expertise and concern, attempting to make it comprehensive for users. The approach is macroscopic and philosophical; readers convinced of the philosophy are to seek specific assistance.

Theorems on Regularity and Singularity of Energy Minimizing Maps

Author : Leon Simon
Publisher : Birkhäuser
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034891936

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Book Theorems on Regularity and Singularity of Energy Minimizing Maps Description/Summary:

The aim of these lecture notes is to give an essentially self-contained introduction to the basic regularity theory for energy minimizing maps, including recent developments concerning the structure of the singular set and asymptotics on approach to the singular set. Specialized knowledge in partial differential equations or the geometric calculus of variations is not required; a good general background in mathematical analysis would be adequate preparation.

Dynamic Optimization, Second Edition

Author : Morton I. Kamien,Nancy L. Schwartz
Publisher : Courier Corporation
Release : 2013-04-17
Category : Mathematics
ISBN : 9780486310282

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Book Dynamic Optimization, Second Edition Description/Summary:

Since its initial publication, this text has defined courses in dynamic optimization taught to economics and management science students. The two-part treatment covers the calculus of variations and optimal control. 1998 edition.

Mathematical Tools for Physicists

Author : Michael Grinfeld
Publisher : John Wiley & Sons
Release : 2015-01-12
Category : Science
ISBN : 9783527411887

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Book Mathematical Tools for Physicists Description/Summary:

The new edition is significantly updated and expanded. This unique collection of review articles, ranging from fundamental concepts up to latest applications, contains individual contributions written by renowned experts in the relevant fields. Much attention is paid to ensuring fast access to the information, with each carefully reviewed article featuring cross-referencing, references to the most relevant publications in the field, and suggestions for further reading, both introductory as well as more specialized. While the chapters on group theory, integral transforms, Monte Carlo methods, numerical analysis, perturbation theory, and special functions are thoroughly rewritten, completely new content includes sections on commutative algebra, computational algebraic topology, differential geometry, dynamical systems, functional analysis, graph and network theory, PDEs of mathematical physics, probability theory, stochastic differential equations, and variational methods.

The Calculus of Variations in the Large

Author : Marston Morse
Publisher : American Mathematical Soc.
Release : 1934-12-31
Category : Mathematics
ISBN : 9780821810187

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Book The Calculus of Variations in the Large Description/Summary:

Morse theory is a study of deep connections between analysis and topology. In its classical form, it provides a relationship between the critical points of certain smooth functions on a manifold and the topology of the manifold. It has been used by geometers, topologists, physicists, and others as a remarkably effective tool to study manifolds. In the 1980s and 1990s, Morse theory was extended to infinite dimensions with great success. This book is Morse's own exposition of his ideas. It has been called one of the most important and influential mathematical works of the twentieth century. Calculus of Variations in the Large is certainly one of the essential references on Morse theory.