Variational methods with applications in science and engineering (Q2867007)

This monograph consists of three parts.NEWLINENEWLINEPart I, Variational Methods, pp. 3--113, contains three chapters: Preliminaries; Calculus of variations; Rayleigh-Ritz, Galerkin, and Finite-Element Methods.NEWLINENEWLINEPart II, Physical Applications, pp. 117--299, contains six chapters: Hamilton's Principle; Classical Mechanics; Stability of Dynamical Systems; Optics and Electromagnetics; Modern Physics; Fluid Mechanics.NEWLINENEWLINEPart III, Optimization, pp. 303--402, contains three chapters: Optimization and Control; Image Processing and Data Analysis; Numerical Grid Generation.NEWLINENEWLINEThe bibliography, pp. 403--407, contains mostly references to various books.NEWLINENEWLINEThere is a subject index on pp. 409--413.NEWLINENEWLINEPart I of this book is intended to teach engineering students calculus of variations. It is not suitable for teaching mathematics students: some statements are not formulated in the form they should be used in the calculus of variations, there are no proofs, and there are some inaccuracies. For example, in the discussion (p. 21) of fundamental lemma of calculus of variations, the lemma is formulated for continuous functions and not for integrable functions, and the smoothness of the ``arbitrary'' function is not defined. The adjoint operator (pp. 22--25) is defined not clearly: the domain of the adjoint operator is not discussed, and the difference between symmetric and selfadjoint operators is blurred. The definition (p. 33) of the symbol \(\delta\) in the calculus of variations is also not clear: the ends of the interval are assumed fixed, which is not always the case even in this book (see Example 2.6 on p. 51, and p. 53, Section Variable End Points, where proofs are not given). On p. 77, lines 3 and 4 from the bottom are called Euler's equations, but there are no equations there. Since \(\Lambda\) is differentiated on p. 77, one may ask why \(\Lambda\) is differentiable, but this is not explained. In Part III, p.369, formula (11.13) for the curvature of the plane curve is incorrect: \(|u^{''}|\) should replace \(u^{''}\). In the Section on Rayleigh-Ritz, Galerkin and Finite-Element Method no convergence results are proved or discussed. It is not clear if engineering students will be able to use the presented material in a research work after studying this text.NEWLINENEWLINEPart II deals with applications of calculus of variations to physics. The equations of physics can be written in variational form since they are the Lagrange equations for suitable Lagrangians. The examples considered in this book are standard and there is not much originality in the presentation. In the Chapter on Electromagnetics no wave scattering problems are mentioned and no wave propagation problems are considered. In a discussion of the Schrödinger's equation there is no mentioning of the character of the spectrum of the Schrödinger's operators, the scattering amplitude is not mentioned, etc. In the Chapter on Fluid Mechanics the turbulence is not mentioned. The books by L.Landau and E.Lifshitz, Course of theoretical physics, vol.1--10, Pergamon Press, Oxford, 1984, are not referenced. In their books the variational principles in physics play an important role. The presentation in part II is rather superficial.NEWLINENEWLINEPart III deals with optimization. There is no discussion of the general optimization procedures and their convergence. For the most part, the author considers some explicitly written functionals and solves the corresponding Euler's equations. The Image Processing section is based mostly on minimizing some integral functionals. The variety of denoising methods is not discussed. Only the total variation (TV) denoising method is mentioned. No discussion of the advantages and disadvantages of this method is given. The variety of segmentation methods is not discussed either. Splines are mentioned as a tool for approximation of curves, but the quality of such an approximation is not discussed. On pp. 374--378, Section 11.3, an important question is mentioned: how does one represent a given large set of functions by linear combinations of a few orthonormal basis functions with minimal error? How does one find the optimal in this sense set of basis functions given the set of data functions? The author's explanations are not clear. He proposes to minimize integral (11.18), namely, \(J:=||u-\phi_i||^2=min\), where the minimization is taken over \(\phi_i\). What are the sets \(\{u\}\) and \(\{\phi_i\}\) is not explained. If \(u\) is a given function, the reader may think that \(\phi_i=u\) is the minimizer of \(J\). The functional \(J\) depends on \(i\), but this dependence is not shown.NEWLINENEWLINEThe book is written for the first year graduate students in various engineering areas. The author writes that ``there is little emphasis on mathematical proofs''. The material is presented superficially. The knowledge the students are supposed to acquire from this textbook is not well defined. In the reviewer's opinion, there are much better books on calculus of variations (for example, books by \textit{B. Dacorogna} [Introduction to the calculus of variations. Cahiers Mathématiques de l'École Polytechnique Fédérale de Lausanne. 3. Lausanne: Presses Polytechniques et Universitaires Romandes (1992; Zbl 0757.49001)], not cited by the author, or by \textit{I. M. Gelfand} and \textit{S. V. Fomin} [Calculus of variations. Revised English edition. Translated and edited by Richard A. Silverman. 3rd. printing. Englewood Cliffs, N.J.: Prentice-Hall, Inc. (1965; Zbl 0127.05402)], cited by the author). On the material of Parts II and III there are many books and students can learn the corresponding material in several courses, not only in a course on variational methods.