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Advanced Engineering Analysis is a textbook on modern engineering analysis, covering the calculus of variations, functional analysis, and control theory, as well as applications of these disciplines to mechanics. The book offers a brief and concise, yet complete explanation of essential theory and applications. It contains exercises with hints and solutions, ideal for self-study.

Preface	p. v
Basic Calculus of Variations	p. 1
Introduction	p. 1
Euler's Equation for the Simplest Problem	p. 15
Properties of Extremals of the Simplest Functional	p. 21
Ritz's Method	p. 23
Natural Boundary Conditions	p. 31
Extensions to More General Functionals	p. 34
Functionals Depending on Functions in Many Variables	p. 43
A Functional with Integrand Depending on Partial Derivatives of Higher Order	p. 49
The First Variation	p. 54
Isoperimetric Problems	p. 65
General Form of the First Variation	p. 72
Movable Ends of Extremals	p. 76
Broken Extremals: Weierstrass-Erdmann Conditions and Related Problems	p. 80
Sufficient Conditions for Minimum	p. 85
Exercises	p. 94
Applications of the Calculus of Variations in Mechanics	p. 99
Elementary Problems for Elastic Structures	p. 99
Some Extremal Principles of Mechanics	p. 108
Conservation Laws	p. 127
Conservation Laws and Noether's Theorem	p. 131
Functionals Depending on Higher Derivatives of y	p. 139
Noether's Theorem, General Case	p. 143
Generalizations	p. 147
Exercises	p. 153
Elements of Optimal Control Theory	p. 159
A Variational Problem as an Optimal Control Problem	p. 159
General Problem of Optimal Control	p. 161
Simplest Problem of Optimal Control	p. 164
Fundamental Solution of a Linear Ordinary Differential Equation	p. 170
The Simplest Problem, Continued	p. 171
Pontryagin's Maximum Principle for the Simplest Problem	p. 173
Some Mathematical Preliminaries	p. 177
General Terminal Control Problem	p. 189
Pontragin's Maximum Principle for the Terminal Optimal Problem	p. 195
Generalization of the Terminal Control Problem	p. 198
Small Variations of Control Function for Terminal Control Problem	p. 202
A Discrete Version of Small Variations of Control Function for Generalized Terminal Control Problems	p. 205
Optimal Time Control Problems	p. 208
Final Remarks on Control Problems	p. 212
Exercises	p. 214
Functional Analysis	p. 215
A Normed Space as a Metric Space	p. 217
Dimension of a Linear Space and Separability	p. 223
Cauchy Sequences and Banach Spaces	p. 227
The Completion Theorem	p. 238
L^p Spaces and the Lebesgue Integral	p. 242
Sobolev Spaces	p. 248
Compactness	p. 250
Inner Product Spaces, Hilbert Spaces	p. 260
Operators and Functionals	p. 264
Contraction Mapping Principle	p. 269
Some Approximation Theory	p. 276
Orthogonal Decomposition of a Hilbert Space and the Riesz Representation Theorem	p. 280
Basis, Gram-Schmidt Procedure, and Fourier Series in Hilbert Space	p. 284
Weak Convergence	p. 291
Adjoint and Self-Adjoint Operators	p. 298
Compact Operators	p. 304
Closed Operators	p. 311
On the Sobolev Imbedding Theorem	p. 315
Some Energy Spaces in Mechanics	p. 320
Introduction to Spectral Concepts	p. 337
The Fredholm Theory in Hilbert Spaces	p. 343
Exercises	p. 352
Applications of Functional Analysis in Mechanics	p. 359
Some Mechanics Problems from the Standpoint of the Calculus of Variations; the Virtual Work Principle	p. 359
Generalized Solution of the Equilibrium Problem for a Clamped Rod with Springs	p. 364
Equilibrium Problem for a Clamped Membrane and its Generalized Solution	p. 367
Equilibrium of a Free Membrane	p. 369
Some Other Equilibrium Problems of Linear Mechanics	p. 371
The Ritz and Bubnov-Galerkin Methods	p. 379
The Hamilton-Ostrogradski Principle and Generalized Setup of Dynamical Problems in Classical Mechanics	p. 381
Generalized Setup of Dynamic Problem for Membrane	p. 383
Other Dynamic Problems of Linear Mechanics	p. 397
The Fourier Method	p. 399
An Eigenfrequency Boundary Value Problem Arising in Linear Mechanics	p. 400
The Spectral Theorem	p. 404
The Fourier Method, Continued	p. 410
Equilibrium of a von Kármán Plate	p. 415
A Unilateral Problem	p. 425
Exercises	p. 431
Hints for Selected Exercises	p. 433
Bibliography	p. 483
Index	p. 485
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