Elements Of Advanced Mathematical Analysis For Physics And Engineering

Deep comprehension of applied sciences requires a solid knowledge of Mathematical Analysis. For most of high level scientific research, the good understanding of Functional Analysis and weak solutions to differential equations is essential. This book aims to deal with the main topics that are necessary to achieve such a knowledge. Still, this is the goal of many other texts in advanced analysis; and then, what would be a good reason to read or to consult this book? In order to answer this question, let us introduce the three Authors. Alberto Ferrero got his degree in Mathematics in 2000 and presently he is researcher in Mathematical Analysis at the Università del Piemonte Orientale. Filippo Gazzola got his degree in Mathematics in 1987 and he is now full professor in Mathematical Analysis at the Politecnico di Milano. Maurizio Zanotti got his degree in Mechanical Engineering in 2004 and presently he is structural and machine designer and lecturer professor in Mathematical Analysis at the Politecnico di Milano. The three Authors, for the variety of their skills, decided to join their expertises to write this book. One of the reasons that should encourage its reading is that the presentation turns out to be a reasonable compromise among the essential mathematical rigor, the importance of the applications and the clearness, which is necessary to make the reference work pleasant to the readers, even to the inexperienced ones. The range of treated topics is quite wide and covers the main basic notions of the scientific research which is based upon mathematical models. We start from vector spaces and Lebesgue integral to reach the frontier of theoretical research such as the study of critical exponents for semilinear elliptic equations and recent problems in fluid dynamics. This long route passes through the theory of Banach and Hilbert spaces, Sobolev spaces, differential equations, Fourier and Laplace transforms, before which we recall some appropriate tools of Complex Analysis. We give all the proofs that have some didactic or applicative interest, while we omit the ones which are too technical or require too high level knowledge. This book has the ambitious purpose to be useful to a broad variety of readers. The first possible beneficiaries are of course the second or third year students of a scientific course of degree: in what follows they will find the topics that are necessary to approach more advanced studies in Mathematics and in other fields, especially Physics and Engineering. This text could be also useful to graduate students who want to start a Ph.D. course: indeed it contains the matter of a multidisciplinary Ph.D. course given by Filippo Gazzola for several years at Politecnico di Milano. Finally, this book could be addressed also to the ones who have already left education far-back but occasionally need to use mathematical tools: we refer both to university professors and their research, and to professionals and designers who want to model a certain phenomenon, but also to the nostalgics of the good old days when they were students. It is precisely for this last type of reader that we have also reported some elementary topics, such as the properties of numerical sets and of the integrals; moreover, every chapter is provided with examples and specific exercises aimed at the involvement of the reader.

This book provides a comprehensive yet informal introduction to differentiating and integrating real functions with one variable. It also covers basic first-order differential equations and introduces higher-dimensional differentiation and integration. The focus is on significant theoretical proofs, accompanied by illustrative examples for clarity. A comprehensive bibliography aids deeper understanding. The concept of a function's differential is a central theme, relating to the "differential" within integrals. The discussion of indefinite integrals (collections of antiderivatives) precedes definite integrals, naturally connecting the two. The Appendix offers essential math formulas, exercise properties, and an in-depth exploration of continuity and differentiability. Select exercise solutions are provided. This book suits short introductory math courses for novice physics/engineering students. It equips them with vital differential and integral calculus tools for real-world applications. It is also useful for first-year undergraduates, reinforcing advanced calculus foundations for better Physics comprehension.

The goal of this book is to publish the latest mathematical techniques, research, and developments in engineering. This book includes a comprehensive range of mathematics applied in engineering areas for different tasks. Various mathematical tools, techniques, strategies, and methods in engineering applications are covered in each chapter. Mathematical techniques are the strength of engineering sciences and form the common foundation of all novel disciplines within the field. Advanced Mathematical Techniques in Engineering Sciences provides an ample range of mathematical tools and techniques applied across various fields of engineering sciences. Using this book, engineers will gain a greater understanding of the practical applications of mathematics in engineering sciences. Features Covers the mathematical techniques applied in engineering sciences Focuses on the latest research in the field of engineering applications Provides insights on an international and transnational scale Offers new studies and research in modeling and simulation

The book comprises ten chapters, Each chapter contains serveral soved problems clarifying the introduced concepts. Some of the examples are taken from the recent literature and serve to illustrate the applications in various fields of engineering and science. At the end of each chapter, there are assignment problems with two levels of difficulty. A list of references is provided at the end of the book. This book is the product of a close collaboration between two mathematicians and an engineer. The engineer has been helpful in pinpointing the problems which engineering students encounter in books written by mathematicians. Contents: Review of Calculus and Ordinary Differential Equations; Series Solutions and Special Functions; Complex Variables; Vector and Tensor Analysis; Partial Differential Equations I; Partial Differential Equations II; Numerical Methods; Numerical Solution of Partial Differential Equations; Calculus of Variations; Special Topics. Readership: Upper level undergraduates, graduate students and researchers in mathematical modeling, mathematical physics and numerical &computational mathematics.

Due to the rapid expansion of the frontiers of physics and engineering, the demand for higher-level mathematics is increasing yearly. This book is designed to provide accessible knowledge of higher-level mathematics demanded in contemporary physics and engineering. Rigorous mathematical structures of important subjects in these fields are fully covered, which will be helpful for readers to become acquainted with certain abstract mathematical concepts. The selected topics are: - Real analysis, Complex analysis, Functional analysis, Lebesgue integration theory, Fourier analysis, Laplace analysis, Wavelet analysis, Differential equations, and Tensor analysis. This book is essentially self-contained, and assumes only standard undergraduate preparation such as elementary calculus and linear algebra. It is thus well suited for graduate students in physics and engineering who are interested in theoretical backgrounds of their own fields. Further, it will also be useful for mathematics students who want to understand how certain abstract concepts in mathematics are applied in a practical situation. The readers will not only acquire basic knowledge toward higher-level mathematics, but also imbibe mathematical skills necessary for contemporary studies of their own fields.

This is a concise reference book on analysis and mathematical physics, leading readers from a foundation to advanced level understanding of the topic. This is the perfect text for graduate or PhD mathematical-science students looking for support in topics such as distributions, Fourier transforms and microlocal analysis, C* Algebras, value distribution of meromorphic functions, noncommutative differential geometry, differential geometry and mathematical physics, mathematical problems of general relativity, and special functions of mathematical physics. Analysis and Mathematical Physics is the sixth volume of the LTCC Advanced Mathematics Series. This series is the first to provide advanced introductions to mathematical science topics to advanced students of mathematics. Editor the three joint heads of the London Taught Course Centre for PhD Students in the Mathematical Sciences (LTCC), each book supports readers in broadening their mathematical knowledge outside of their immediate research disciplines while also covering specialized key areas.

A paperback edition of successful and well reviewed 1995 graduate text on applied mathematics for engineers.

This graduate-level textbook is a detailed exposition of key mathematical tools in analysis aimed at students, researchers, and practitioners across science and engineering. Every topic covered has been specifically chosen because it plays a key role outside the field of pure mathematics. Although the treatment of each topic is mathematical in nature, and concrete applications are not delineated, the principles and tools presented are fundamental to exploring the computational aspects of physics and engineering. Readers are expected to have a solid understanding of linear algebra, in Rn and in general vector spaces. Familiarity with the basic concepts of calculus and real analysis, including Riemann integrals and infinite series of real or complex numbers, is also required.