Calculus of variations Books

Book SynopsisCalculus Set Free: Infinitesimals to the Rescue is a single-variable calculus textbook that incorporates the use of infinitesimal methods.Trade ReviewCalculus Set Free is a well-written and self-contained text which offers a novel and mathematically rigorous approach to the topics typically present in Calculus 1 and 2. The text is largely successful in what it sets out to accomplish, and teachers interested in offering an introduction to Calculus built on an alternative theoretical approach should consider this text. * John Ross, MAA Reviews *Table of ContentsReview 1: Hyperreals, Limits, and Continuity 2: Derivatives 3: Applications of the Derivative 4: Integration 5: Transcendental Functions 6: Applications of Integration 7: Techniques of Integration 8: Alternate Representations: Parametric and Polar Curves 9: Additional Applications of Integration 10: Sequences and Series

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Book SynopsisCalculus Set Free: Infinitesimals to the Rescue is a single-variable calculus textbook that incorporates the use of infinitesimal methods.Trade ReviewCalculus Set Free is a well-written and self-contained text which offers a novel and mathematically rigorous approach to the topics typically present in Calculus 1 and 2. The text is largely successful in what it sets out to accomplish, and teachers interested in offering an introduction to Calculus built on an alternative theoretical approach should consider this text. * John Ross, MAA Reviews *Table of ContentsReview 1: Hyperreals, Limits, and Continuity 2: Derivatives 3: Applications of the Derivative 4: Integration 5: Transcendental Functions 6: Applications of Integration 7: Techniques of Integration 8: Alternate Representations: Parametric and Polar Curves 9: Additional Applications of Integration 10: Sequences and Series

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Book SynopsisThe theory of Gamma-convergence is commonly recognized as an ideal and flexible tool for the description of the asymptotic behaviour of variational problems. Its applications range from the mathematical analysis of composites to the theory of phase transitions, from Image Processing to Fracture Mechanics. This text, written by an expert in the field, provides a brief and simple introduction to this subject, based on the treatment of a series of fundamental problems that illustrate the main features and techniques of Gamma-convergence and at the same time provide a stimulating starting point for further studies. The main part is set in a one-dimensional framework that highlights the main issues of Gamma-convergence without the burden of higher-dimensional technicalities. The text deals in sequence with increasingly complex problems, first treating integral functionals, then homogenisation, segmentation problems, phase transitions, free-discontinuity problems and their discrete and contiTrade ReviewThe presentation is overall quite clear, and the style is often captivating. Many figures, examples and exercises complete the monograph. Finally, it is worth adding a mention on the bibiography, which is at present a truly complete account of papers in this area. * Mathematical Reviews *Table of ContentsA. SOME QUICK RECALLS

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Book SynopsisThis short guide to modern error analysis is primarily intended to be used in undergraduate laboratories in the physical sciences. No prior knowledge of statistics is assumed. The necessary concepts are introduced where needed and illustrated graphically. The book emphasises the use of computers for error calculations and data fitting.Trade ReviewWith the shift from analytic methods to spreadsheet-based techniques, this book will enable students simultaneously to (a) become fluent in the choice and application of appropriate methods (b) understand the underlying principles. * David Saxon, University of Glasgow *This is a rather beautiful little book. * David J. Hand, International Statistical Review *Table of Contents1. Errors in the physical sciences ; 2. Random errors in measurement ; 3. Uncertainties as probabilities ; 4. Error propagation ; 5. Data visualisation and reduction ; 6. Least-squares fitting of complex functions ; 7. Computer minimisation and the error matrix ; 8. Hypothesis testing - how good are our models ; 9. Topics for further summary

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Book SynopsisFundamentals of analytic function theory â plus lucid exposition of 5 important applications: potential theory, ordinary differential equations, Fourier transforms, Laplace transforms, and asymptotic expansions. Includes 66 figures.

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Book SynopsisThis book is meant to give an account of recent developments in the theory of Plateau's problem for parametric minimal surfaces and surfaces of prescribed constant mean curvature ("H-surfaces") and its analytical framework. A comprehensive overview of the classical existence and regularity theory for disc-type minimal and H-surfaces is given and reTable of Contents*FrontMatter, pg. i*Contents, pg. vii*Preface, pg. ix*I. Existence of a solution, pg. 5*II. Unstable minimal surfaces, pg. 33*III. The existence of surfaces of prescribed constant mean curvature spanning a Jordan curve in IR3, pg. 91*IV. Unstable H-surfaces, pg. 111*References, pg. 141

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Book SynopsisThe study of natural and social phemomena indicates that the future development of many processes depends not only on their present state, but also on their history. Such processes can be described mathematically by using the machinery of equations with aftereffect. This book presents control theory for hereditary systems of various types.Table of ContentsElements of the theory of systems with aftereffect The dynamic programming method Optimality conditions for deterministic systems with aftereffect Investigation of self-adjusting systems with reference model Optimal control of stochastic systems Optimal control of systems defined by stochastic integro-functional equations Optimal estimation Optimal control with incomplete data Bibliography.

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Book SynopsisThis text presents a complete treatment of variational problems on discrete sets with an overall behavior driven by surface energies. Covering both applications and perspectives, it can be used as an advanced graduate course text, as well as a reference for mathematical analysts and applied mathematicians working in related fields.Table of Contents1. Introduction; 2. Preliminaries; 3. Homogenization of pairwise systems with positive coefficients; 4. Compactness and integral representation; 5. Random lattices; 6. Extensions; 7. Frustrated systems; 8. Perspectives towards dense graphs; A. Multiscale analysis; B. Spin systems as limits of elastic interactions; References; Index.

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Book SynopsisThis engaging graduate-level introduction to geometric measure theory bridges analysis and geometry, taking readers from basic theory to some of the most celebrated results in modern analysis. Explanatory pictures, detailed proofs, exercises and helpful remarks make it suitable for self-study and also a useful reference for researchers.Trade Review'The book is a clear exposition of the theory of sets of finite perimeter, that introduces this topic in a very elegant and original way, and shows some deep and important results and applications … Although most of the results contained in this book are classical, some of them appear in this volume for the first time in book form, and even the more classical topics which one may find in several other books are presented here with a strong touch of originality which makes this book pretty unique … I strongly recommend this excellent book to every researcher or graduate student in the field of calculus of variations and geometric measure theory.' Alessio Figalli, Canadian Mathematical Society Notes'The first aim of the book is to provide an introduction for beginners to the theory of sets of finite perimeter, presenting results concerning the existence, symmetry, regularity and structure of singularities in some variational problems involving length and area … The secondary aim … is to provide a multi-leveled introduction to the study of other variational problems … an interested reader is able to enter with relative ease several parts of geometric measure theory and to apply some tools from this theory in the study of other problems from mathematics … This is a well-written book by a specialist in the field … It provides generous guidance to the reader [and] is recommended … not only to beginners who can find an up-to-date source in the field but also to specialists … It is an invitation to understand and to approach some deep and difficult problems from mathematics and physics.' Vasile Oproiu, Zentralblatt MATHTable of ContentsPart I. Radon Measures on Rn: 1. Outer measures; 2. Borel and Radon measures; 3. Hausdorff measures; 4. Radon measures and continuous functions; 5. Differentiation of Radon measures; 6. Two further applications of differentiation theory; 7. Lipschitz functions; 8. Area formula; 9. Gauss–Green theorem; 10. Rectifiable sets and blow-ups of Radon measures; 11. Tangential differentiability and the area formula; Part II. Sets of Finite Perimeter: 12. Sets of finite perimeter and the Direct Method; 13. The coarea formula and the approximation theorem; 14. The Euclidean isoperimetric problem; 15. Reduced boundary and De Giorgi's structure theorem; 16. Federer's theorem and comparison sets; 17. First and second variation of perimeter; 18. Slicing boundaries of sets of finite perimeter; 19. Equilibrium shapes of liquids and sessile drops; 20. Anisotropic surface energies; Part III. Regularity Theory and Analysis of Singularities: 21. (Λ, r0)-perimeter minimizers; 22. Excess and the height bound; 23. The Lipschitz approximation theorem; 24. The reverse Poincaré inequality; 25. Harmonic approximation and excess improvement; 26. Iteration, partial regularity, and singular sets; 27. Higher regularity theorems; 28. Analysis of singularities; Part IV. Minimizing Clusters: 29. Existence of minimizing clusters; 30. Regularity of minimizing clusters; References; Index.

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Book SynopsisA comprehensive guide to using energy principles and variational methods for solving problems in solid mechanics This book provides a systematic, highly practical introduction to the use of energy principles, traditional variational methods, and the finite element method for the solution of engineering problems involving bars, beams, torsion, plane elasticity, trusses, and plates. It begins with a review of the basic equations of mechanics, the concepts of work and energy, and key topics from variational calculus. It presents virtual work and energy principles, energy methods of solid and structural mechanics, Hamilton's principle for dynamical systems, and classical variational methods of approximation. And it takes a more unified approach than that found in most solid mechanics books, to introduce the finite element method. Featuring more than 200 illustrations and tables, this Third Edition has been extensively reorganized and contains much new Table of ContentsAbout the Author xvii About the Companion Website xix Preface to the Third Edition xxi Preface to the Second Edition xxiii Preface to the First Edition xxv 1. Introduction and Mathematical Preliminaries 1 1.1 Introduction 1 1.1.1 Preliminary Comments 1 1.1.2 The Role of Energy Methods and Variational Principles 1 1.1.3 A Brief Review of Historical Developments 2 1.1.4 Preview 4 1.2 Vectors 5 1.2.1 Introduction 5 1.2.2 Definition of a Vector 6 1.2.3 Scalar and Vector Products 8 1.2.4 Components of a Vector 12 1.2.5 Summation Convention 13 1.2.6 Vector Calculus 17 1.2.7 Gradient, Divergence, and Curl Theorems 22 1.3 Tensors 26 1.3.1 Second-Order Tensors 26 1.3.2 General Properties of a Dyadic 29 1.3.3 Nonion Form and Matrix Representation of a Dyad 30 1.3.4 Eigenvectors Associated with Dyads 34 1.4 Summary 39 Problems 40 2. Review of Equations of Solid Mechanics 47 2.1 Introduction 47 2.1.1 Classification of Equations 47 2.1.2 Descriptions of Motion 48 2.2 Balance of Linear and Angular Momenta 50 2.2.1 Equations of Motion 50 2.2.2 Symmetry of Stress Tensors 54 2.3 Kinematics of Deformation 56 2.3.1 Green-Lagrange Strain Tensor 56 2.3.2 Strain Compatibility Equations 62 2.4 Constitutive Equations 65 2.4.1 Introduction 65 2.4.2 Generalized Hooke's Law 66 2.4.3 Plane Stress-Reduced Constitutive Relations 68 2.4.4 Thermoelastic Constitutive Relations 70 2.5 Theories of Straight Beams 71 2.5.1 Introduction 71 2.5.2 The Bernoulli-Euler Beam Theory 73 2.5.3 The Timoshenko Beam Theory 76 2.5.4 The von Ka’rma’n Theory of Beams 81 2.5.4.1 Preliminary Discussion 81 2.5.4.2 The Bernoulli-Euler Beam Theory 82 2.5.4.3 The Timoshenko Beam Theory 84 2.6 Summary 85 Problems 88 3. Work, Energy, and Variational Calculus 97 3.1 Concepts of Work and Energy 97 3.1.1 Preliminary Comments 97 3.1.2 External and Internal Work Done 98 3.2 Strain Energy and Complementary Strain Energy 102 3.2.1 General Development 102 3.2.2 Expressions for Strain Energy and Complementary Strain Energy Densities of Isotropic Linear Elastic Solids 107 3.2.2.1 Stain energy density 107 3.2.2.2 Complementary stain energy density 108 3.2.3 Strain Energy and Complementary Strain Energy for Trusses 109 3.2.4 Strain Energy and Complementary Strain Energy for Torsional Members 114 3.2.5 Strain Energy and Complementary Strain Energy for Beams 117 3.2.5.1 The Bernoulli-Euler Beam Theory 117 3.2.5.2 The Timoshenko Beam Theory 119 3.3 Total Potential Energy and Total Complementary Energy 123 3.3.1 Introduction 123 3.3.2 Total Potential Energy of Beams 124 3.3.3 Total Complementary Energy of Beams 125 3.4 Virtual Work 126 3.4.1 Virtual Displacements 126 3.4.2 Virtual Forces 131 3.5 Calculus of Variations 135 3.5.1 The Variational Operator 135 3.5.2 Functionals 138 3.5.3 The First Variation of a Functional 139 3.5.4 Fundamental Lemma of Variational Calculus 140 3.5.5 Extremum of a Functional 141 3.5.6 The Euler Equations 143 3.5.7 Natural and Essential Boundary Conditions 146 3.5.8 Minimization of Functionals with Equality Constraints 151 3.5.8.1 The Lagrange Multiplier Method 151 3.5.8.2 The Penalty Function Method 153 3.6 Summary 156 Problems 159 4. Virtual Work and Energy Principles of Mechanics 167 4.1 Introduction 167 4.2 The Principle of Virtual Displacements 167 4.2.1 Rigid Bodies 167 4.2.2 Deformable Solids 168 4.2.3 Unit Dummy-Displacement Method 172 4.3 The Principle of Minimum Total Potential Energy and Castigliano's Theorem I 179 4.3.1 The Principle of Minimum Total Potential Energy179 4.3.2 Castigliano's Theorem I 188 4.4 The Principle of Virtual Forces 196 4.4.1 Deformable Solids 196 4.4.2 Unit Dummy-Load Method 198 4.5 Principle of Minimum Total Complementary Potential Energy and Castigliano's Theorem II 204 4.5.1 The Principle of the Minimum total Complementary Potential Energy 204 4.5.2 Castigliano's Theorem II 206 4.6 Clapeyron's, Betti's, and Maxwell's Theorems 217 4.6.1 Principle of Superposition for Linear Problems 217 4.6.2 Clapeyron's Theorem 220 4.6.3 Types of Elasticity Problems and Uniqueness of Solutions 224 4.6.4 Betti's Reciprocity Theorem 226 4.6.5 Maxwell's Reciprocity Theorem 230 4.7 Summary 232 Problems 235 5. Dynamical Systems: Hamilton's Principle 243 5.1 Introduction 243 5.2 Hamilton's Principle for Discrete Systems 243 5.3 Hamilton's Principle for a Continuum 249 5.4 Hamilton's Principle for Constrained Systems 255 5.5 Rayleigh's Method 260 5.6 Summary 262 Problems 263 6. Direct Variational Methods 269 6.1 Introduction 269 6.2 Concepts from Functional Analysis 270 6.2.1 General Introduction 270 6.2.2 Linear Vector Spaces 271 6.2.3 Normed and Inner Product Spaces 276 6.2.3.1 Norm 276 6.2.3.2 Inner product 279 6.2.3.3 Orthogonality 280 6.2.4 Transformations, and Linear and Bilinear Forms 281 6.2.5 Minimum of a Quadratic Functional 282 6.3 The Ritz Method 287 6.3.1 Introduction 287 6.3.2 Description of the Method 288 6.3.3 Properties of Approximation Functions 293 6.3.3.1 Preliminary Comments 293 6.3.3.2 Boundary Conditions 293 6.3.3.3 Convergence 294 6.3.3.4 Completeness 294 6.3.3.5 Requirements on ɸ0 and ɸi 295 6.3.4 General Features of the Ritz Method 299 6.3.5 Examples 300 6.3.6 The Ritz Method for General Boundary-Value Problems 323 6.3.6.1 Preliminary Comments 323 6.3.6.2 Weak Forms 323 6.3.6.3 Model Equation 1 324 6.3.6.4 Model Equation 2 328 6.3.6.5 Model Equation 3 330 6.3.6.6 Ritz Approximations 332 6.4 Weighted-Residual Methods 337 6.4.1 Introduction 337 6.4.2 The General Method of Weighted Residuals 339 6.4.3 The Galerkin Method 44 6.4.4 The Least-Squares Method 349 6.4.5 The Collocation Method 356 6.4.6 The Subdomain Method 359 6.4.7 Eigenvalue and Time-Dependent Problems 361 6.4.7.1 Eigenvalue Problems 361 6.4.7.2 Time-Dependent Problems 362 6.5 Summary 381 Problems 383 7. Theory and Analysis of Plates 391 7.1 Introduction 391 7.1.1 General Comments 391 7.1.2 An Overview of Plate Theories 393 7.1.2.1 The Classical Plate Theory 394 7.1.2.2 The First-Order Plate Theory 395 7.1.2.3 The Third-Order Plate Theory 396 7.1.2.4 Stress-Based Theories 397 7.2 The Classical Plate Theory 398 7.2.1 Governing Equations of Circular Plates 398 7.2.2 Analysis of Circular Plates 405 7.2.2.1 Analytical Solutions For Bending 405 7.2.2.2 Analytical Solutions For Buckling 411 7.2.2.3 Variational Solutions 414 7.2.3 Governing Equations in Rectangular Coordinates 427 7.2.4 Navier Solutions of Rectangular Plates 435 7.2.4.1 Bending 438 7.2.4.2 Natural Vibration 443 7.2.4.3 Buckling Analysis 445 7.2.4.4 Transient Analysis 447 7.2.5 Lévy Solutions of Rectangular Plates 449 7.2.6 Variational Solutions: Bending 454 7.2.7 Variational Solutions: Natural Vibration 470 7.2.8 Variational Solutions: Buckling 475 7.2.8.1 Rectangular Plates Simply Supported along Two Opposite Sides and Compressed in the Direction Perpendicular to Those Sides 475 7.2.8.2 Formulation for Rectangular Plates with Arbitrary Boundary Conditions 478 7.3 The First-Order Shear Deformation Plate Theory 486 7.3.1 Equations of Circular Plates 486 7.3.2 Exact Solutions of Axisymmetric Circular Plates 488 7.3.3 Equations of Plates in Rectangular Coordinates 492 7.3.4 Exact Solutions of Rectangular Plates 496 7.3.4.1 Bending Analysis 498 7.3.4.2 Natural Vibration 501 7.3.4.3 Buckling Analysis 502 7.3.5 Variational Solutions of Circular and Rectangular Plates 503 7.3.5.1 Axisymmetric Circular Plates 503 7.3.5.2 Rectangular Plates 505 7.4 Relationships Between Bending Solutions of Classical and Shear Deformation Theories 507 7.4.1 Beams 507 7.4.1.1 Governing Equations 508 7.4.1.2 Relationships Between BET and TBT 508 7.4.2 Circular Plates 512 7.4.3 Rectangular Plates 516 7.5 Summary 521 Problems 521 8. The Finite Element Method 527 8.1 Introduction 527 8.2 Finite Element Analysis of Straight Bars 529 8.2.1 Governing Equation 529 8.2.2 Representation of the Domain by Finite Elements 530 8.2.3 Weak Form over an Element 531 8.2.4 Approximation over an Element 532 8.2.5 Finite Element Equations 537 8.2.5.1 Linear Element 538 8.2.5.2 Quadratic Element 539 8.2.6 Assembly (Connectivity) of Elements 539 8.2.7 Imposition of Boundary Conditions 542 8.2.8 Postprocessing 543 8.3 Finite Element Analysis of the Bernoulli-Euler Beam Theory 549 8.3.1 Governing Equation 549 8.3.2 Weak Form over an Element 549 8.3.3 Derivation of the Approximation Functions 550 8.3.4 Finite Element Model 552 8.3.5 Assembly of Element Equations 553 8.3.6 Imposition of Boundary Conditions 555 8.4 Finite Element Analysis of the Timoshenko Beam Theory 558 8.4.1 Governing Equations 558 8.4.2 Weak Forms 558 8.4.3 Finite Element Models 559 8.4.4 Reduced Integration Element (RIE) 559 8.4.5 Consistent Interpolation Element (CIE) 561 8.4.6 Superconvergent Element (SCE) 562 8.5 Finite Element Analysis of the Classical Plate Theory 565 8.5.1 Introduction 565 8.5.2 General Formulation 566 8.5.3 Conforming and Nonconforming Plate Elements 568 8.5.4 Fully Discretized Finite Element Models 569 8.5.4.1 Static Bending 569 8.5.4.2 Buckling 569 8.5.4.3 Natural Vibration 570 8.5.4.4 Transient Response 570 8.6 Finite Element Analysis of the First-Order Shear Deformation Plate Theory 574 8.6.1 Governing Equations and Weak Forms 574 8.6.2 Finite Element Approximations 576 8.6.3 Finite Element Model 577 8.6.4 Numerical Integration 579 8.6.5 Numerical Examples 582 8.6.5.1 Isotropic Plates 582 8.6.5.2 Laminated Plates 584 8.7 Summary 587 Problems 588 9. Mixed Variational and Finite Element Formulations 595 9.1 Introduction 595 9.1.1 General Comments 595 9.1.2 Mixed Variational Principles 595 9.1.3 Extremum and Stationary Behavior of Functionals 597 9.2 Stationary Variational Principles 599 9.2.1 Minimum Total Potential Energy 599 9.2.2 The Hellinger-Reissner Variational Principle 601 9.2.3 The Reissner Variational Principle 605 9.3 Variational Solutions Based on Mixed Formulations 606 9.4 Mixed Finite Element Models of Beams 610 9.4.1 The Bernoulli-Euler Beam Theory 610 9.4.1.1 Governing Equations And Weak Forms 610 9.4.1.2 Weak-Form Mixed Finite Element Model 610 9.4.1.3 Weighted-Residual Finite Element Models 613 9.4.2 The Timoshenko Beam Theory 615 9.4.2.1 Governing Equations 615 9.4.2.2 General Finite Element Model 615 9.4.2.3 ASD-LLCC Element 617 9.4.2.4 ASD-QLCC Element 617 9.4.2.5 ASD-HQLC Element 618 9.5 Mixed Finite Element Analysis of the Classical Plate Theory 620 9.5.1 Preliminary Comments 620 9.5.2 Mixed Model I 620 9.5.2.1 Governing Equations 620 9.5.2.2 Weak Forms 621 9.5.2.3 Finite Element Model 622 9.5.3 Mixed Model II 625 9.5.3.1 Governing Equations 625 9.5.3.2 Weak Forms 625 9.5.3.3 Finite Element Model 626 9.6 Summary 630 Problems 631 10. Analysis of Functionally Graded Beams and Plates 635 10.1 Introduction 635 10.2 Functionally Graded Beams 638 10.2.1 The Bernoulli-Euler Beam Theory 638 10.2.1.1 Displacement and strain fields 638 10.2.1.2 Equations of motion and boundary conditions 638 10.2.2 The Timoshenko Beam Theory 639 10.2.2.1 Displacement and strain fields 639 10.2.2.2 Equations of motion and boundary conditions 640 10.2.3 Equations of Motion in terms of Generalized Displacements 641 10.2.3.1 Constitutive Equations 641 10.2.3.2 Stress Resultants of BET 641 10.2.3.3 Stress Resultants of TBT 642 10.2.3.4 Equations of Motion of the BET 642 10.2.3.5 Equations of Motion of the TBT 642 10.2.4 Stiffiness Coefficients643 10.3 Functionally Graded Circular Plates 645 10.3.1 Introduction 645 10.3.2 Classical Plate Theory 646 10.3.2.1 Displacement and Strain Fields 646 10.3.2.2 Equations of Motion 646 10.3.3 First-Order Shear Deformation Theory 647 10.3.3.1 Displacement and Strain Fields 647 10.3.3.2 Equations of Motion 648 10.3.4 Plate Constitutive Relations 649 10.3.4.1 Classical Plate Theory 649 10.3.4.2 First-Order Plate Theory 649 10.4 A General Third-Order Plate Theory 650 10.4.1 Introduction 650 10.4.2 Displacements and Strains 651 10.4.3 Equations of Motion 653 10.4.4 Constitutive Relations 657 10.4.5 Specialization to Other Theories 658 10.4.5.1 A General Third-Order Plate Theory with Traction-Free Top and Bottom Surfaces 658 10.4.5.2 The Reddy Third-Order Plate Theory 661 10.4.5.3 The First-Order Plate Theory 663 10.4.5.4 The Classical Plate Theory 664 10.5 Navier's Solutions 664 10.5.1 Preliminary Comments 664 10.5.2 Analysis of Beams 665 10.5.2.1 Bernoulli-Euler Beams 665 10.5.2.2 Timoshenko Beams 667 10.5.2.3 Numerical Results 669 10.5.3 Analysis of Plates 671 10.5.3.1 Boundary Conditions 672 10.5.3.2 Expansions of Generalized Displacements 672 10.5.3.3 Bending Analysis 673 10.5.3.4 Free Vibration Analysis 676 10.5.3.5 Buckling Analysis 677 10.5.3.6 Numerical Results 679 10.6 Finite Element Models 681 10.6.1 Bending of Beams 681 10.6.1.1 Bernoulli-Euler Beam Theory 681 10.6.1.2 Timoshenko Beam Theory 683 10.6.2 Axisymmetric Bending of Circular Plates 684 10.6.2.1 Classical Plate Theory 681 10.6.2.2 First-Order Shear Deformation Plate Theory 686 10.6.3 Solution of Nonlinear Equations 688 10.6.3.1 Times approximation 688 10.6.3.2 Newton's Iteration Approach 688 10.6.3.3 Tangent Stiffiness Coefficients for the BET 690 10.6.3.4 Tangent Stiffiness Coefficients for the TBT 692 10.6.3.5 Tangent Stiffiness Coefficients for the CPT 693 10.6.3.6 Tangent Stiffiness Coefficients for the FSDT 693 10.6.4 Numerical Results for Beams and Circular Plates 694 10.6.4.1 Beams 694 10.6.4.2 Circular Plates 697 10.7 Summary 699 Problems 700 References 701 Answers to Most Problems 711 Index 723

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Book SynopsisThis volume on mathematical control theory contains high quality articles covering the broad range of this field. The internationally renowned authors provide an overview of many different aspects of control theory, offering a historical perspective while bringing the reader up to the very forefront of current research.Table of Contents1 Path Integrals and Stability.- 1.1 Introduction.- 1.2 Path Independence.- 1.3 Positivity of Quadratic Differential Forms.- 1.4 Lyapunov Theory for High-Order Differential Equations.- 1.5 The Bezoutian.- 1.5.1 The Routh Test.- 1.5.2 The Kharitonov Theorem.- 1.6 Dissipative Systems.- 1.7 Stability of Nonautonomous Systems.- 1.8 Conclusions.- 1.9 Appendixes.- 1.9.1 Appendix A: Notation.- 1.9.2 Appendix B: Linear Differential Systems.- 1.9.3 Appendix C: Proofs.- 2 The Estimation Algebra of Nonlinear Filtering Systems.- 2.1 Introduction.- 2.2 The Filtering Model and Background.- 2.3 Starting from the Beginning.- 2.4 Early Results on the Homomorphism Principle.- 2.5 Automorphisms that Preserve Estimation Algebras.- 2.6 BM Estimation Algebra.- 2.7 Structure of Exact Estimation Algebra.- 2.8 Structure of BM Estimation Algebras.- 2.9 Connection with Metaplectic Groups.- 2.10 Wei-Norman Representation of Filters.- 2.11 Perturbation Algebra and Estimation Algebra.- 2.12 Lie-Algebraic Classification of Maximal Rank Estimation Algebras.- 2.13 Complete Characterization of Finite-Dimensional Estimation Algebras.- 2.14 Estimation Algebra of the Identification Problem.- 2.15 Solutions to the Riccati P.D.E.- 2.16 Filters with Non-Gaussian Initial Conditions.- 2.17 Back to the Beginning.- 2.18 Acknowledgement.- 3 Feedback Linearization.- 3.1 Introduction.- 3.2 Linearization of a Smooth Vector Field.- 3.3 Linearization of a Smooth Control System by Change-of-State Coordinates.- 3.4 Feedback Linearization.- 3.5 Input-Output Linearization.- 3.6 Approximate Feedback Linearization.- 3.7 Normal Forms of Control Systems.- 3.8 Observers with Linearizable Error Dynamics.- 3.9 Nonlinear Regulation and Model Matching.- 3.10 Backstepping.- 3.11 Feedback Linearization and System Inversion.- 3.12 Conclusion.- 4 On the Global Analysis of Linear Systems.- 4.1 Introduction.- 4.2 The Geometry of Rational Functions.- 4.2.1 Spaces of Scalar-Input/Scalar-Output Linear Systems.- 4.2.2 The Deterministic Partial Realization Problem.- 4.3 Group Actions and the Geometry of Linear Systems.- 4.3.1 The Geometry of Matrix-Valued Rational Functions.- 4.3.2 Applications to Canonical Forms.- 4.3.3 A Signature Formula for the Maslov Index.- 4.4 The Geometry of Inverse Eigenvalue Problems.- 4.4.1 Inverse Eigenvalue Problems and the Hopf Degree.- 4.4.2 Pole Assignment by Output Feedback.- 4.5 Nonlinear Optimization on Spaces of Systems.- 4.5.1 A Classical Example: Uniqueness of Maximum Likelihood Estimates.- 4.5.2 The Partial Realization Problem Revisited.- 4.5.3 A Geometric Parameterization of Positive Rational Covariance Extensions.- 4.5.4 A Convex Optimization Scheme for Rational Covariance Extensions.- 5 Geometry and Optimal Control.- 5.1 Introduction.- 5.2 From Queen Dido to the Maximum Principle.- 5.3 Invariance, Covariance, and Lie Brackets.- 5.4 The Maximum Principle.- 5.5 The Maximum Principle as a Necessary Condition for Set Separation.- 5.6 Weakly Approximating Cones and Transversality.- 5.7 A Streamlined Version of the Classical Maximum Principle.- 5.8 Clarke’s Nonsmooth Version and the ?ojasiewicz Improvement.- 5.9 Multidifferentials, Flows, and a General Version of the Maximum Principle.- 5.10 Three Ways to Make the Maximum Principle Intrinsic on Manifolds.- 5.11 Conclusion.- 6 Languages, Behaviors, Hybrid Architectures, and Motion Control.- 6.1 Introduction.- 6.2 MDLe: A Language for Motion Control.- 6.2.1 Performance Measure of a Plan.- 6.3 Hybrid Architecture.- 6.4 Application of MDLe to Path Planning with Nonholonomic Robots.- 6.4.1 Nonholonomic Constraints.- 6.5 PNMR: Path Planner for Nonholonomic Mobile Robots.- 6.5.1 Planning in the Obstacle-Free Disk.- 6.5.2 Tracing Boundaries.- 6.5.3 World Model Update.- 6.6 Conclusions.- 7 Optimal Control, Geometry, and Mechanics.- 7.1 Introduction.- 7.2 Variational Problems with Constraints and Optimal Control.- 7.3 Invariant Optimal Problems on Lie Groups.- 7.4 Sub-Riemannian Spheres—The Contact Case.- 7.5 Sub-Riemannian Systems on Lie Groups.- 7.6 Heavy Top and the Elastic Problem.- 7.7 Conclusion.- 8 Optimal Control, Optimization, and Analytical Mechanics.- 8.1 Introduction.- 8.2 Modeling Variational Problems in Mechanics and Control.- 8.2.1 Introduction.- 8.2.2 Variational Systems without External Forces.- 8.2.3 Mechanical Systems with External Forces.- 8.2.4 Relation to Optimal Control.- 8.2.5 Reduction.- 8.2.6 A Special Case.- 8.3 Optimization.- 8.4 Optimal Control Problems and Integrable Systems.- 8.4.1 Introduction.- 8.4.2 Optimal Control on Adjoint Orbits.- 8.4.3 Optimal Control on Symmetric Spaces.- 8.4.4 Optimal Control and the Toda Flow.- 9 The Geometry of Controlled Mechanical Systems.- 9.1 Introduction.- 9.2 Second-Order Generalized Control Systems.- 9.3 Flat Systems and Systems with Flat Inputs.- 9.4 Averaging Lagrangian and Hamiltonian Systems with Oscillatory Inputs.- 9.5 Stability and Flatness in Mechanical Systems with Oscillatory Inputs.- 9.6 Concluding Remarks.

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Book SynopsisComposites have been studied for more than 150 years, and interest in their properties has been growing. This classic volume provides the foundations for understanding a broad range of composite properties, including electrical, magnetic, electromagnetic, elastic and viscoelastic, piezoelectric, thermal, fluid flow through porous materials, thermoelectric, pyroelectric, magnetoelectric, and conduction in the presence of a magnetic field (Hall effect). Exact solutions of the PDEs in model geometries provide one avenue of understanding composites; other avenues include microstructure-independent exact relations satisfied by effective moduli, for which the general theory is reviewed; approximation formulae for effective moduli; and series expansions for the fields and effective moduli that are the basis of numerical methods for computing these fields and moduli. The range of properties that composites can exhibit can be explored either through the model geometries or through microstructure-independent bounds on the properties. These bounds are obtained through variational principles, analytic methods, and Hilbert space approaches. Most interesting is when the properties of the composite are unlike those of the constituent materials, and there has been an explosion of interest in such composites, now known as metamaterials. The Theory of Composites surveys these aspects, among others, and complements the new body of literature that has emerged since the book was written. It remains relevant today by providing historical background, a compendium of numerous results, and through elucidating many of the tools still used today in the analysis of composite properties. This book is intended for applied mathematicians, physicists, and electrical and mechanical engineers. It will also be of interest to graduate students.

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Book SynopsisThe book Sequences and Series in Calculus is designed as the first college/university calculus course for students who take and do well on the AP AB exam in high school and who are interested in a more proof-oriented treatment of calculus. The text begins with an ε-ℕ treatment of sequence convergence, then builds on this to discuss convergence of series—first series of real numbers, then series of functions. The difference between uniform and pointwise convergence is discussed in some detail. This is followed by a discussion of calculus on power series and Taylor series. Finally improper integrals, integration by parts and partial fractions integration all are introduced. This book is designed both to teach calculus, and to give the readers and students a taste of analysis to help them determine if they wish to study this material even more deeply. It might be used by colleges and universities who teach special versions of calculus courses for their most mathematically advanced entering first-year students, as might its older sibling text Multivariable and Vector Calculus which appeared in 2020 and is intended for students who take and do well on the AP BC exam.

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Book SynopsisThis comprehensive treatment of multivariable calculus focuses on the numerous tools that MATLAB® brings to the subject, as it presents introductions to geometry, mathematical physics, and kinematics. Covering simple calculations with MATLAB®, relevant plots, integration, and optimization, the numerous problem sets encourage practice with newly learned skills that cultivate the reader’s understanding of the material. Significant examples illustrate each topic, and fundamental physical applications such as Kepler’s Law, electromagnetism, fluid flow, and energy estimation are brought to prominent position. Perfect for use as a supplement to any standard multivariable calculus text, a “mathematical methods in physics or engineering” class, for independent study, or even as the class text in an “honors” multivariable calculus course, this textbook will appeal to mathematics, engineering, and physical science students. MATLAB® is tightly integrated into every portion of this book, and its graphical capabilities are used to present vibrant pictures of curves and surfaces. Readers benefit from the deep connections made between mathematics and science while learning more about the intrinsic geometry of curves and surfaces. With serious yet elementary explanation of various numerical algorithms, this textbook enlivens the teaching of multivariable calculus and mathematical methods courses for scientists and engineers.Trade Review“The book is addressed to students as well as to instructors of calculus. It helps to understand multivariable analysis utilysing visualization of such geometric structures like domains, curves and surfaces. It also develops the skill of students to use a powerful software for solving modern problems.” (Ivan Podvigin, zbMATH 1400.26001, 2019)Table of Contents1. Introduction.- 2. Vectors and Graphics.- 3. Geometry of Curves.- 4. Kinematics.- 5. Directional Derivatives.- 6. Geometry of Surfaces.- 7. Optimization in Several Variables.- 8. Multiple Integrals.- 9. Multidimensional Calculus.- 10. Physical Applications of Vector Calculus.- 11. MATLAB Tips.- Sample Solutions.- Index.

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Book SynopsisThis book presents a series of lectures on three of the best known examples of free discontinuity problems: the Mumford-Shah model for image segmentation, a variational model for the epitaxial growth of thin films, and the sharp interface limit of the Ohta-Kawasaki model for pattern formation in dyblock copolymers.Table of ContentsIntroduction.- Fine regularity results for Mumford-Shah minimizers: porosity, higher integrability and the Mumford-Shah conjecture.- Variational models for epitaxial growth.- Local and global minimality results for an isoperimetric problem with long-range interactions.

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Book SynopsisThese lecture notes are devoted to the analysis of a nonlocal equation in the whole of Euclidean space. In studying this equation, all the necessary material is introduced in the most self-contained way possible, giving precise references to the literature when necessary. The results presented are original, but no particular prerequisite or knowledge of the previous literature is needed to read this text. The work is accessible to a wide audience and can also serve as introductory research material on the topic of nonlocal nonlinear equations.Table of ContentsIntroduction.- The problem studied in this monograph.- Functional analytical setting.- Existence of a minimal solution and proof of Theorem 2.2.2.- Regularity and positivity of the solution.- Existence of a second solution and proof of Theorem 2.2.4.

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