This venerable textbook, now in its tenth edition, gives a thorough introduction to all areas of applied mathematics. It is designed for a two-year course, but can also be used for several specialized one-semester courses, and is organized to be a useful reference. The author, Erwin Kreyszig, died in 2008, and this revision was prepared by his son Herbert Kreyszig and by Edward J. Norminton.
The terms “advanced” and “engineering” in the title should be taken with a grain of salt. The stated prerequisite is calculus, and the material is advanced in comparison with calculus, but the book gives you only the most important and commonly-used parts of the many subjects it covers, and in mathematical terms is not very advanced. These topics cover all aspects of applied mathematics, not just those of interest to engineers. Coverage includes not only the expected differential equations, complex analysis, vector calculus, and linear algebra, but also combinatorics, probability, statistics, optimization, and numerical methods.
The book is organized as a collection of mathematical techniques for solving applied problems. It is not a cookbook (it proves nearly everything), but it is also not a systematic exposition (related techniques are grouped together, but there is no real narrative thread). Its nature is much like the typical calculus book, giving you all the most useful bits of the subject without trying to be exhaustive. Compared to the average calculus book, though, this book is very concise, and usually gives only one worked example per technique. There is little hand-holding.
The book spends very little time on modeling; for the most part the examples postulate the mathematical model and then show the techniques to handle it. The exercises are heavily slanted toward drill, but include some problems that introduce more advanced topics. There is no computer work in the body of the text, but there is some discussion of software packages and some of the exercises use computers.
Bottom line: a valuable reference for the target engineering audience, and possibly a good text for some curricula, but in most courses you would want to work in more depth and less breadth than is given here.
Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.
CHAPTER 1 First-Order ODEs.
CHAPTER 2 Second-Order Linear ODEs.
CHAPTER 3 Higher Order Linear ODEs.
CHAPTER 4 Systems of ODEs. Phase Plane. Qualitative Methods.
CHAPTER 5 Series Solutions of ODEs. Special Functions.
CHAPTER 6 Laplace Transforms.
PART B Linear Algebra. Vector Calculus.
CHAPTER 7 Linear Algebra: Matrices, Vectors, Determinants. Linear Systems.
CHAPTER 8 Linear Algebra: Matrix Eigenvalue Problems.
CHAPTER 9 Vector Differential Calculus. Grad, Div, Curl.
CHAPTER 10 Vector Integral Calculus. Integral Theorems.
PART C Fourier Analysis. Partial Differential Equations (PDEs).
CHAPTER 11 Fourier Series, Integrals, and Transforms.
CHAPTER 12 Partial Differential Equations (PDEs).
PART D Complex Analysis.
CHAPTER 13 Complex Numbers and Functions.
CHAPTER 14 Complex Integration.
CHAPTER 15 Power Series, Taylor Series.
CHAPTER 16 Laurent Series. Residue Integration.
CHAPTER 17 Conformal Mapping.
CHAPTER 18 Complex Analysis and Potential Theory.
PART E Numeric Analysis.
CHAPTER 19 Numerics in General.
CHAPTER 20 Numeric Linear Algebra.
CHAPTER 21 Numerics for ODEs and PDEs.
PART F Optimization, Graphs.
CHAPTER 22 Unconstrained Optimization. Linear Programming.
CHAPTER 23 Graphs. Combinatorial Optimization.
PART G Probability, Statistics.
CHAPTER 24 Data Analysis. Probability Theory.
CHAPTER 25 Mathematical Statistics.
APPENDIX 1 References A1.
APPENDIX 2 Answers to Odd-Numbered Problems A4.
APPENDIX 3 Auxiliary Material A63.
A3.1 Formulas for Special Functions A63.
A3.2 Partial Derivatives A69.
A3.3 Sequences and Series A72.
A3.4 Grad, Div, Curl, 2 in Curvilinear Coordinates A74.
APPENDIX 4 Additional Proofs A77.
APPENDIX 5 Tables A97.
PHOTO CREDITS P1.