Some Engineering Mathematics

C. Pozrikidis

Forthcoming in 2021


Why this book?

A good way of learning, teaching, and refamiliarizing oneself with methods of applied mathematics and scientific computing is to study a selected set of topics related to practical science and engineering applications that are a blend of the old, the new, and the emerging. This book discusses a pertinent collection of carefully selected topics, problems, algorithms, and complete computer codes, including mainstream and original material. The discussion of the individual topics provides opportunities for recalling, clarifying, and learning methods and developing skills.

One main goal of this book is to provide insights into a fair question often asked: \emph{what are the applications of a particular method or topic in applied science and engineering?} The particular method of topic can be tensor, separation of variables, curvilinear coordinates, algorithms, numerical methods, complex variables, fractional calculus, and others. This book helps answer this question by providing motivation and by making cases by example. Regarded in hindsight, the discussion illustrates the usefulness of applied mathematics, scientific computing, and computer coding to problems encountered in a broad range of applications. A junior or even lower level mathematical background on calculus, and elementary familiarity with computer programming are assumed for most of the topics covered.

This book is suitable as a text for instruction in a mainstream, special topics, or special projects advanced undergraduate course or introductory graduate course in applied mathematics, engineering mathematics, computer programming, and scientific computing in the applied sciences and engineering. More broadly, the book offers a point of departure for exploring, performing online research from different points of view, and pursuing individual study. The intended audience includes students, course instructors, researchers, and anyone interested in science, technology, and mathematics.

In line with the intended character of this book, the notion of a physical or conceptual vector or tensor is developed from the ground up in a way that can be understood by the non-expert, the concept of volume and surface curvilinear coordinates is discussed in detail from a theoretical and computational standpoint with applications in physical transport and mechanics, the solution of differential equations by the method of separation of variables is exemplified with emphasis on engineering applications, and the Moore--Penrose inverse of a matrix is explained in the context of data regression and based on the solution of the Poisson equation with Neumann boundary conditions around the entire boundary of a solution domain.

Elementary and advanced Matlab/Octave computer codes performing miscellaneous computations, simulations, and graphics visualizations are listed and discussed in the text. The codes illustrate explicitly how mathematical equations translate into computer code instructions. The computer programs are arranged in a software library named SEMLIB, which is available freely on the Internet under the terms of the current GNU license agreement.

The book complements mainstream textbooks, online resources, and lecture notes provided by others in a way that circumvents the need for referencing background material and invoking the work of others. Consistent with this goal, references and citations are not provided with an understanding that, if needed, they can be obtained readily by searching the Internet.

Errata and supplements