Preface

After a year of freshman calculus, the basic mathematics training in science and engineering is usually completed during the second and the third years of the undergraduate curriculum. Students are usually required to take a sequence of three courses on the subjects of advanced calculus, differential equations, complex calculus, and introductory mathematical physics. Today, majority of the science and engineering departments are finding it convenient to use a single book that assures uniform formalism and a topical coverage in tune with their needs. The objective of Essentials of Mathematical Methods in Science and Engineering is to equip students with the basic mathematical skills required by majority of the science and engineering undergraduate programs.

The book gives a coherent treatment of the selected topics with a style that makes the essential mathematical skills easily accessible to a multidisciplinary audience. Since the book is written in modular format, each chapter covers its subject thoroughly and thus can be read independently. This also makes the book very useful for self-study and as reference or refresher for scientists. It is assumed that the reader has been exposed to two semesters of freshman calculus or has acquired an equivalent level of mathematical maturity.

The entire book contains a sufficient amount of material for a three-semester course meeting three to four hours a week. Respecting the disparity of the mathematics courses offered throughout the world, the topical coverage and the modular structure of the book make it versatile enough to be adopted for a number of different mathematics courses and allows instructors the flexibility to individualize their own teaching while maintaining the integrity and the uniformity of the discussions for their students.

About the Second Edition

The main aim of this book is to meet the demands of the majority of the modern undergraduate physics and engineering programs. It also aims to prepare students for a solid graduate program and establishes the groundwork of my graduate textbook Mathematical Methods in Science and Engineering, Wiley, second edition, 2018. The second edition, while maintaining all the successful features of the first edition, includes two new and extensive chapters (Chapters 6 and 7) entitled Practical Linear Algebra and Applications of Linear Algebra, respectively, and a computer file that includes Matlab codes.

The new chapters were developed and used as I taught linear algebra (3 hrs/week) and mathematical methods courses (3 + 1?h/wk) to engineering students. The file including the Matlab codes is self explanatory but assumes familiarity with the text in the book. These codes were used for the lab section of the mathematical methods course I taught to students with no prior Matlab experience. These codes will be available as open source in https://www.wiley.com, or in http://users.metu.edu.tr/bayin/.

In addition to these, numerous changes have been made to assure easy reading and smooth flow of the complex mathematical arguments. Derivations are given with sufficient detail so that the reader will not be distracted by searching for results in other parts of the book or by needing to write down equations. We have shown carefully selected keywords in boldface and framed key results so that the needed information can be located easily as the reader scans through the pages.

Chapter references are given at the end of each chapter with their full titles. Additional resources for the interested reader is listed at the back with respect to their subject matter. Our suggested references is by all means not meant to be complete. Nowadays, readers can locate additional references by a simple internet search. In particular, readers can use the websites: http://en.wikipedia.org and http://scienceworld.wolfram.com/. Of course, https://arxiv.org is an indispensable tool for researchers on any subject.

This book concentrates on analytic techniques. Computer programs like Mathematica® and MapleT are capable of performing symbolic as well as numerical calculations. Even though they are extremely useful to scientists, one cannot stress enough the importance of a full grasp of the basic mathematical techniques with their intricacies and interdisciplinary connections. Only then the underlying unity and the beauty of the universe begins to appear. There are books specifically written for mathematical methods with these programs, some of which are included in our list for further reading at the back.

With their exclusive chapters and uniform level of formalism, this book connects with my graduate textbook Mathematical Methods in Science and Engineering, Wiley, second edition, 2018, thus forming a complete set spanning a wide range of fundamental mathematical techniques for students, instructors, and researchers.

Summary of the Book

Chapter 1. Functional Analysis: This chapter aims to fill the gap between the introductory calculus and the advanced mathematical analysis courses. It introduces the basic techniques that are used throughout mathematics. Limits, derivatives, integrals, extremum of functions, implicit function theorem, inverse functions, and improper integrals are among the topics discussed.

Chapter 2. Vector Analysis: Since most of the classical theories can be introduced in terms of vectors, we present a rather detailed treatment of vectors and their techniques. Vector algebra, vector differentiation, gradient, divergence and curl operators, vector integration, Green's theorem, integral theorems, and the essential elements of the potential theory are among the topics covered.

Chapter 3. Generalized Coordinates and Tensors: Starting with the Cartesian coordinates, we discuss generalized coordinate systems and their transformations. Basis vectors, transformation matrix, line element, reciprocal basis vectors, covariant and contravariant components, differential operators in generalized coordinates, and introduction to Cartesian and general tensors are among the other essential topics of mathematical methods.

Chapter 4. Determinants and Matrices: A systematic treatment of the basic properties and methods of determinants and matrices that are much needed in science and engineering applications are presented here with examples.

Chapter 5. Linear Algebra: This chapter starts with a discussion of abstract linear spaces, also called vector spaces, and continues with systems of linear equations, inner product spaces, eigenvalue problems, quadratic forms, Hermitian matrices, and Dirac's bra and ket vectors.

Chapter 6. Practical Linear Algebra: In the previous chapter, we concentrate on the abstract properties of linear algebra. In this chapter, we introduce linear algebra from the practitioners point of view. In the first part, we start with systems of linear equations and discuss Gauss-Jordan reduction, row-echelon forms, elementary matrices, row space, column space and null space, rank and nullity, etc. In the second part, we introduce the numerical methods of linear algebra. Partial pivoting, LU-factorization, iteration method, interpolation, power method for eigenvalues, numerical integration, etc. are among the interesting topics discussed. In our accompanying website, we also have Matlab codes that the readers can experiment with the methods introduced in this chapter.

Chapter 7. Applications of Linear Algebra: This chapter introduces some of the important applications of linear algebra from different branches of science and engineering. We give examples from chemical engineering, linear programming, economics, geometry, elimination theory, coding theory, cryptography, and graph theory.

Chapter 8. Sequences and Series: This chapter starts with sequences and series of numbers and then introduces absolute convergence and tests for convergence. We then extend our discussion to series of functions and introduce the concept of uniform convergence. Power series and Taylor series are discussed in detail with applications.

Chapter 9. Complex Numbers and Functions: After the complex number system is introduced and their algebra is discussed, complex functions, complex differentiation, Cauchy-Riemann conditions and analytic functions are the main topics of this chapter.

Chapter 10. Complex Analysis: We introduce the complex integral theorems and discuss residues, Taylor series, and Laurent series along with their convergence properties.

Chapter 11. Ordinary Differential Equations: We start with the general properties of differential equations, their solutions, and boundary conditions. The most commonly encountered differential equations in applications are either first- or second-order ordinary differential equations. Hence, we discuss these two cases separately in detail and introduce methods of finding their analytic solutions. We also study linear equations of higher order. We finally conclude with the Frobenius method applied to first- and second-order differential equations with interesting and carefully selected examples.

Chapter 12. Second-Order Differential Equations and Special Functions: In this chapter, we discuss three of the most frequently encountered second-order differential equations of physics and engineering, that is, Legendre, Hermite, and Laguerre equations. We study these equations in...