Advanced Engineering Mathematics 6th Edition

Book Preface

In courses such as calculus or differential equations, the content is fairly standardized but the content of a course entitled engineering mathematics often varies considerably between two different academic institutions. Therefore a text entitled Advanced Engineering

Mathematics is a compendium of many mathematical topics, all of which are loosely related by the expedient of either being needed or useful in courses in science and engineering or in subsequent careers in these areas. There is literally no upper bound to the number of topics that could be included in a text such as this. Consequently, this book represents the author’s opinion of what constitutes engineering mathematics.

Content of the Text

For flexibility in topic selection this text is divided into five major parts. As can be seen from the titles of these various parts it should be obvious that it is my belief that the backbone of science/engineering related mathematics is the theory and applications of ordinary and partial differential equations.

Part 1: Ordinary Differential Equations (Chapters 1–6) The six chapters in Part 1 constitute a complete short course in ordinary differential equations. These chapters, with some modifications, correspond to Chapters 1, 2, 3, 4, 5, 6, 7, and 9 in the text A First Course in Differential Equations with Modeling Applications, Eleventh Edition, by Dennis G. Zill (Cengage Learning). In Chapter 2 the focus is on methods for solving first-order differential equations and their applications. Chapter 3 deals mainly with linear second-order differential equations and their applications. Chapter 4 is devoted to the solution of differential equations and systems of differential equations by the important Laplace transform.

Part 2: Vectors, Matrices, and Vector Calculus (Chapters 7–9) Chapter 7, Vectors, and Chapter 9, Vector Calculus, include the standard topics that are usually covered in the third semester of a calculus sequence: vectors in 2- and 3-space, vector functions, directional derivatives, line integrals, double and triple integrals, surface integrals, Green’s theorem, Stokes’ theorem, and the divergence theorem. In Section 7.6 the vector concept is generalized; by defining vectors analytically we lose their geometric interpretation but keep many of their properties in n-dimensional and infinite-dimensional vector spaces. Chapter 8, Matrices, is an introduction to systems of algebraic equations, determinants, and matrix algebra, with special emphasis on those types of matrices that are useful in solving systems of linear differential equations. Optional sections on cryptography, error correcting codes, the method of least squares, and discrete compartmental models are presented as applications of matrix algebra.

Part 3: Systems of Differential Equations (Chapters 10 and 11) There are two chapters in Part 3. Chapter 10, Systems of Linear Differential Equations, and Chapter 11, Systems of Nonlinear Differential Equations, draw heavily on the matrix material presented in Chapter 8 of Part 2. In Chapter 10, systems of linear first-order equations are solved utilizing the concepts of eigenvalues and eigenvectors, diagonalization, and by means of a matrix exponential function. In Chapter 11, qualitative aspects of autonomous linear and nonlinear systems are considered in depth.

Part 4: Partial Differential Equations (Chapters 12–16) The core material on Fourier series and boundary-value problems involving second-order partial differential equations was originally drawn from the text Differential Equations with Boundary-Value Problems, Ninth Edition, by Dennis G. Zill (Cengage Learning). In Chapter 12, Orthogonal Functions and Fourier Series, the fundamental topics of sets of orthogonal functions and expansions of functions in terms of an infinite series of orthogonal functions are presented. These topics are then utilized in Chapters 13 and 14 where boundary-value problems in rectangular, polar, cylindrical, and spherical coordinates are solved using the method of separation of variables. In Chapter 15, Integral Transform Method, boundaryvalue problems are solved by means of the Laplace and Fourier integral transforms.

Part 5: Complex Analysis (Chapters 17–20) The final four chapters of the hardbound text cover topics ranging from the basic complex number system through applications of conformal mappings in the solution of Dirichlet’s problem. This material by itself could easily serve as a one quarter introductory course in complex variables. This material was taken from Complex Analysis: A First Course with Applications, Third Edition, by Dennis G. Zill and Patrick D. Shanahan (Jones & Bartlett Learning).