Advanced Modern Engineering Mathematics

Citation

James, G., Burley, D., Dyke, P., Clements, D., Steele, N., & Wright, J. (2018). Advanced Modern Engineering Mathematics (5th ed.). Pearson Education.

Chapter Summary

Pearson's catalogue lists twelve entries, in this order: an introduction, followed by eleven topic chapters. The publisher describes the book as building on the foundations of the companion text Modern Engineering Mathematics and giving "an extensive treatment" of advanced areas of mathematics with applications in engineering, particularly as tools for computer-based system modelling, analysis and design. The print edition runs to about 1,000 pages. The catalogue gives titles without section lists, so the entries below describe what each title names.

Introduction

The opening entry precedes the technical chapters. The catalogue gives no further description of it.

Matrix Analysis

The first topic chapter is on matrices, placed ahead of the chapters on differential equations, vector calculus and transforms.

Numerical Solution of Ordinary Differential Equations

The only chapter on ordinary differential equations in the list takes the numerical route. There is no separate chapter on analytic solution methods, which fits the publisher's statement that the book builds on a first-level companion volume.

Vector Calculus

A chapter on the calculus of vector fields, placed before the complex-variable material.

Functions of a Complex Variable

A single chapter on complex functions.

Laplace Transforms and The z Transform

Two consecutive chapters on transforms: Laplace transforms, then the z transform. Taken together they cover the continuous-time and discrete-time versions of the same idea.

Fourier Series and The Fourier Transform

Two further consecutive chapters: Fourier series, then the Fourier transform.

Partial Differential Equations

This chapter follows the Fourier material in the list.

Optimization

A chapter on optimization.

Applied Probability and Statistics

The final chapter, on probability and statistics in applied form.

Key Concepts

The catalogue names the following topics; the explanations are general definitions of the named ideas, not claims about how the book treats them.

Matrix Analysis; Numerical Solution of Ordinary Differential Equations

• Matrix analysis: the study of matrices and their properties, named as the first topic chapter.
• Numerical solution of ordinary differential equations: approximating solutions step by step on a computer when no closed form is practical.

Vector Calculus; Functions of a Complex Variable

• Vector calculus: differentiation and integration of vector-valued quantities and fields.
• Functions of a complex variable: functions whose argument and value are complex numbers.

Transforms and Fourier Methods

• Laplace transform: a transform of functions of time, widely used for continuous systems.
• z transform: the corresponding tool for discrete sequences and discrete-time systems.
• Fourier series and Fourier transform: expansions in sines and cosines for periodic functions and the transform for non-periodic ones.

Partial Differential Equations; Optimization; Probability and Statistics

• Partial differential equations: equations in several independent variables, the subject of its own chapter.
• Optimization: finding best values of a quantity under stated conditions.
• Applied probability and statistics: the final topic chapter.

Critical Analysis

Strengths

• A stated learning method. The publisher describes a "philosophy of learning by doing" aimed at helping students use mathematics with understanding to solve engineering problems, supported by "a wealth of engineering examples".
• Computing is part of the book. Pearson says the text integrates MATLAB, MAPLE and R, which fits the publisher's emphasis on tools for computer-based system modelling, analysis and design.
• Continuous-time and discrete-time tools side by side. Laplace transforms and the z transform are separate, adjacent chapters, and the list also includes optimization and applied probability and statistics, so the book reaches beyond classical analysis.
• A mature title. This is the fifth edition. The University of Bristol research record lists a second edition from 1999 at 989 pages, which shows that the book is a long-running teaching text; the 2018 edition is recorded at 1,016 pages in Google Books.

Limitations

• First-year material is assumed. No chapter title covers calculus, analytic methods for ordinary differential equations or basic complex numbers; the publisher says the book builds on Modern Engineering Mathematics. A student without that background needs it or an equivalent.
• One chapter per large subject. Complex variables, partial differential equations, optimization and probability and statistics each occupy a single chapter in the list, so each is likely a compact treatment and a dedicated text will be needed for a full course in any of them.
• Length. At roughly 1,000 pages, a typical module will use only some of the twelve topics.

Where It Fits in the Course

Within this course list, the book is an alternative to Kreyszig's Advanced Engineering Mathematics for the second-level mathematics that follows a first-year course. Kreyszig's single volume divides into seven parts and 25 chapters, gives complex analysis six chapters and analytic ODE methods a part of their own. James and colleagues use twelve entries, with one chapter on complex variables and none on analytic ODE methods, but they add the z transform and put matrix analysis first.

The books to read before it are the first-year texts: Stroud's Engineering Mathematics, a self-paced programmed text from foundation algebra through calculus, matrices, complex numbers, differential equations and Laplace transforms, and Croft and Davison's Mathematics for Engineers, a first-year text from arithmetic and algebra to calculus, matrices, complex numbers and differential equations. The publisher's own companion volume, Modern Engineering Mathematics, is the most direct preparation. After this book, Riley, Hobson and Bence's Mathematical Methods for Physics and Engineering goes on to Green's functions, variational calculus, tensors and group theory, and Bender and Orszag's volume on asymptotic methods and perturbation theory is a graduate text on approximate solutions.

How to Use This Book

The book suits engineering students who have finished a first-year mathematics course and now need the advanced methods used for computer-based modelling and analysis. Because Pearson describes it as following Modern Engineering Mathematics, check that you are comfortable with that book's material before starting.

A practical plan, based on the order of the chapters:

• Start with Matrix Analysis, the first topic chapter in the list.
• Read the transform chapters in sequence: Laplace transforms, then the z transform, then Fourier series and the Fourier transform, then Partial Differential Equations, which follows them in the list.
• Treat the remaining chapters as modular: Numerical Solution of Ordinary Differential Equations, Vector Calculus, Functions of a Complex Variable, Optimization and Applied Probability and Statistics can be matched to the modules you are taking.

Use the MATLAB, MAPLE or R work the publisher mentions to check hand calculations, rather than to replace them. Before planning a semester, check the section lists in the contents pages of your own copy.