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Advanced Engineering Mathematics

Peter V. O'Neil
Cengage Learning
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

This is the eighth edition of a text whose first edition was published in 1982. It is aimed at engineers and is pitched as a survey course, so it has a lot of breadth but usually not much depth. It has good coverage of the traditional engineering topics of differential equations (ordinary and partial), Fourier series, Fourier and Laplace transforms, linear algebra and eigenvalues, and some complex variables. The term “advanced” in the title means that it is more advanced than first-year calculus, and calculus is the only prerequisite.

The advertised differences from the seventh edition are the addition of Math in Context sidebars, and reorganization of the order of presentation. There are a few new advanced topics, and several new online-only modules. An unadvertised difference is the removal of all MAPLE material (although there was not a lot of this before). The new Math in Context items are valuable and cover a lot of application areas, but they are usually very lightweight and not a substitute for real applications, because there’s not enough detail to teach you how to solve these kinds of problems. They’re good as motivators.

There are a large number of worked examples, and they are explained in detail. This is a techniques book, not a proofs book, and nearly all the exercises are drill in the techniques, with only a few proofs. Final answers (not complete solutions) to the odd-numbered exercises are in the back of the book.

The book advertises that it comes with a large amount of supplementary material on the web. For instructors this includes a complete solutions manual and a set of PowerPoint slides. For students this includes supplementary chapters on additional topics, and a student solutions manual that has complete solutions to half the problems. These materials are all password protected and I was not able to examine them.

The book makes good use of graphs, particularly phase portraits. It’s weak on labeling the parts of the graph: if there are several curves in the same plot, it usually doesn’t tell you which is which. For example, Figure 6.1 on p. 178 is intended to show a plucked string when it is released and at several times afterward. It is captioned “\(y(x,t)\) at \(t=0, 1/3, 2/3, 3/4, 1, 4/3\) in Example 6.1”. The individual curves are not labeled, so we don’t know which is which. (An additional complication is that there are seven curves drawn but only six values in the caption. Experimentally I think the extra curve is for \(x=1/2\).) Another example is Figure 17.19 on p. 622, which is captioned “\(S_{10}(t)\) and \(\sigma_{10}(t)\)” and is intended to compare a partial sum of a Fourier series for a step function with the Cesàro sum, but the two curves are not labeled (and the target step function is not drawn either).

Some comparable books are Kreyszig’s Advanced Engineering Mathematics, Arfken et al.’s Mathematical Methods for Physicists, and Mary L. Boas’s Mathematical Methods in the Physical Sciences. These books cover the same subject areas as O’Neil, except that all three also cover probability and statistics (the present book has online modules for this but it is not in the print book). Compared to O’Neil, Kreyszig is much richer, covering more topics and more applications (including a good section on numerical methods), but has few worked examples and provides little hand-holding. It’s more of a reference than a textbook. Arfken is similar to Kreyszig in many ways; it is not as extensive, but also is more a reference. Boas really is designed as a textbook and I think would be the best choice for most courses. Both Arfken and Boas are slanted to physics majors rather than engineers, but the mathematics involved is the same. O’Neil has almost no discussion of numerical methods, which is a serious drawback for engineering students, although your curriculum may cover this elsewhere.

I am uneasy about how free the present text is of typographical and substantive errors, based on a little bit of spot checking. The proof of the maximum principle (for harmonic functions) on p. 693 is simply wrong. It claims that a theorem of calculus says a continuous function of two variables on a closed and bounded set achieves its maximum on the boundary, which is obviously false. The plucked-string example above has an error in the formula for the Fourier series (\(\cos(2 n t)\) should be \(\sin (2 n t)\)). On p. 245 there’s a missing division sign so that the electrostatic potential appears to be directly proportional to the distance instead of inversely proportional; the subsequent calculations are correct.

There are also many typographical errors in the index. I only spot-checked this, but found several misspelled names (Archmedes, Reimann, Legrendre). Some of the entries refer to the wrong pages; I noticed compact set referencing p. 60 (should be p. 660), curve referencing p. 520 (should be p. 529), and simple curve referencing p. 519 (should be p. 529).

Bottom line: easy to follow and with reasonable coverage and lots of worked examples, but may be loaded with typographical errors.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.

1. First-Order Differential Equations.
Terminology and Separable Equations. Singular Solutions, Linear Equations. Exact Equations. Homogeneous, Bernoulli and Riccati Equations.
2. Second-Order Differential Equations.
The Linear Second-Order Equation. The Constant Coefficient Homogeneous Equation. Particular Solutions of the Nonhomogeneous Equation. The Euler Differential Equation, Series Solutions. Frobenius Series Solutions.
3. The Laplace Transform.
Definition and Notation. Solution of Initial Value Problems. The Heaviside Function and Shifting Theorems. Convolution. Impulses and the Dirac Delta Function. Systems of Linear Differential Equations.
4. Eigenfunction Expansions.
Eigenvalues, Eigenfunctions, and Sturm-Liouville Problems. Eigenfunction Expansions, Fourier Series.
5. The Heat Equation.
Diffusion Problems in a Bounded Medium. The Heat Equation with a Forcing Term F(x,t). The Heat Equation on the Real Line. A Reformulation of the Solution on the Real Line. The Heat Equation on a Half-Line, The Two-Dimensional Heat Equation.
6. The Wave Equation.
Wave Motion on a Bounded Interval. The Effect of c on the Motion. Wave Motion with a Forcing Term F(x). Wave Motion in an Unbounded Medium. The Wave Equation on the Real Line. d’Alembert’s Solution and Characteristics. The Wave Equation with a Forcing Term K(x,t). The Wave Equation in Higher Dimensions.
7. Laplace’s Equation.
The Dirichlet Problem for a Rectangle. Dirichlet Problem for a Disk. The Poisson Integral Formula. The Dirichlet Problem for Unbounded Regions. A Dirichlet Problem in 3 Dimensions. The Neumann Problem. Poisson’s Equation.
8. Special Functions and Applications.
Legendre Polynomials. Bessel Functions. Some Applications of Bessel Functions.
9. Transform Methods of Solution.
Laplace Transform Methods. Fourier Transform Methods. Fourier Sine and Cosine Transforms.
10. Vectors and the Vector Space Rn.
Vectors in the Plane and 3 – Space. The Dot Product. The Cross Product. n-Vectors and the Algebraic Structure of Rn. Orthogonal Sets and Orthogonalization. Orthogonal Complements and Projections.
11. Matrices, Determinants and Linear Systems.
Matrices and Matrix Algebra. Row Operations and Reduced Matrices. Solution of Homogeneous Linear Systems. Solution of Nonhomogeneous Linear Systems. Matrix Inverses. Determinants, Cramer’s Rule. The Matrix Tree Theorem.
12. Eigenvalues, Diagonalization and Special Matrices.
Eigenvalues and Eigenvectors. Diagonalization. Special Matrices and Their Eigenvalues and Eigenvectors. Quadratic Forms.
13. Systems of Linear Differential Equations.
Linear Systems. Solution of X’ = AX When A Is Constant. Exponential Matrix Solutions. Solution of X’ = AX + G for Constant A.
14. Nonlinear Systems and Qualitative Analysis.
Nonlinear Systems and Phase Portraits. Critical Points and Stability. Almost Linear Systems, Linearization.
15. Vector Differential Calculus.
Vector Functions of One Variable. Velocity, Acceleration, and Curvature. The Gradient Field. Divergence and Curl. Streamlines of a Vector Field.
16. Vector Integral Calculus.
Line Integrals. Green’s Theorem. Independence of Path and Potential Theory. Surface Integrals. Applications of Surface Integrals. Gauss’s Divergence Theorem. Stokes’s Theorem.
17. Fourier Series.
Fourier Series On [-L, L]. Fourier Sine and Cosine Series. Integration and Differentiation of Fourier Series. Properties of Fourier Coefficients. Phase Angle Form. Complex Fourier Series, Filtering of Signals.
18. Fourier Transforms.
The Fourier Transform. Fourier Sine and Cosine Transforms.
19. Complex Numbers and Functions.
Geometry and Arithmetic of Complex Numbers. Complex Functions, Limits. The Exponential and Trigonometric Functions. The Complex Logarithm. Powers.
20. Integration.
The Integral of a Complex Function. Cauchy’s Theorem. Consequences of Cauchy’s Theorem.
21. Series Representations of Functions.
Power Series. The Laurent Expansion.
22. Singularities and the Residue Theorem.
Classification of Singularities. The Residue Theorem. Evaluation of Real Integrals.
23. Conformal Mappings.
The Idea of a Conformal Mapping. Construction of Conformal Mappings.