In this review, the author's words will be cited rather often. All citations will be quoted. The first edition of the book appeared in 1969. It was reprinted many times. This fact is very significant. "This book has evolved from an introductory course in mathematics given to engineering students at the University of Newcastle-upon-Tyne, It represents the author's attempt to offer the engineering students, and the science student who is not majoring in a mathematical aspect of this subject, a broad and modern account of those parts of mathematics that are finding increasingly important application in the everyday development of his subject".
It is impossible for the reviewer to judge the necessity of the collection of topics contained in the book. It depends on the 'direction' of education in the concrete university. Maybe for other universities such topics as linear programming, queueing theory, partial differential equations, etc., are more important. So, it is better to consider the method of presentation of the suggested material.
I agree with the author that "every effort has been made to integrate the various chapters into a description of mathematics as a single subject". Let us list the titles of the chapters: 1. Introduction to Sets and Numbers; 2. Variables, Functions and Mappings; 3. Sequences, Limits, and Continuity; 4. Complex Numbers and Vectors; 5 Differentiation of Functions of One or More Real Variables; 6. Exponential, t~yperbolic and Logarithmic Functions; 7. Fundamentals of Integration; 8. Systematic Integration; 9. Matrices and Linear Transformation; 10. Functions of a Complex Variable; 11. Scalars, Vectors, and Fields; 12. Series, Taylor's Theorem and its Uses; 13. Differential Equations and Geometry; 14. First Order D~fferential Equations; 15. Higher Order Differential Equations; 16. Fourier Series; 17. Numerical Analysis; 18. Probability and Statistics.
"Of necessity, much of the material in this book is standard, though the emphasis and manner of introduction and presentation frequently differs from that found elsewhere...", namely the author rejects the idea (which is common for similar books) to omit proofs. He considers that "knowledge of the proof of a result is often as essential as its subsequent application, and the modern student needs and merits both". I do agree with this belief because the knowledge of the logical structure of mathematics often gives rise to new facts in applied fields. This thesis can be confirmed by a proverb: "The formula is more clever than me". It is impossible to supply a lot of facts (collected in the book) with proofs. Of course, the author is forced not to do so everywhere. I am not sure that the decision to give or not to give the proof is wellgrounded always. For example there are no proofs in Chapter 18., Besides, there are some doubts about the logic in the sequence of Chapters. So, I believe that Chapters 4, 9, 10, 18 are not connected with their environment~ Maybe they demand another place in the book or another book.
Nevertheless, the book is well-organized. It contains a lot of problems in each Acta Applicandae M athematicae 24: (1991).