Advanced Mathematical Methods

Citation

Ostaszewski, A. J. (1991). Advanced Mathematical Methods. Cambridge University Press.

Chapter Summary

The book has 23 chapters in two parts, linear algebra (Chapters 1–16) and advanced calculus (Chapters 17–23), followed by a third part of solutions to selected exercises. Cambridge describes it as a self-contained second course on mathematical methods, covering topics in linear algebra and multivariate calculus that can be applied to statistics, operations research, computer science, econometrics and mathematical economics. Its stated approach is a balance between a rigorous theoretical and a cookbook approach, with concrete and geometric explanations.

Chapters 1–4: Vector spaces (revision); Geometry in ℝⁿ; Matrices; Projections

The first part opens with a revision of vector spaces, then develops the geometry of n-dimensional real space, matrices and projections.

Chapters 5–8: Spectral theory; Reduction to upper triangular form; Reduction to tridiagonal form; Inverses

The central block of matrix theory treats eigenvalue problems (spectral theory), two reductions of a matrix to special forms, and matrix inverses.

Chapters 9–13: Convexity; The separating hyperplane theorem; Linear inequalities; Linear programming and game theory; The simplex method

These chapters move from convex sets to the separating hyperplane theorem and linear inequalities, and then to linear programming, game theory and the simplex method.

Chapters 14–16: Partial derivatives (revision); Convex functions; Non-linear programming

The first part ends by revising partial derivatives and passing from convex sets to convex functions and non-linear programming.

Chapters 17–19: The integration process; Manipulation of integrals; Multiple integrals

The advanced calculus part begins with integration: the integration process, manipulation of integrals and multiple integrals.

Chapters 20–22: Differential and difference equations (revision); Laplace transforms; Series solution of linear ordinary differential equations

The next chapters revise differential and difference equations and then present Laplace transforms and series solutions of linear ordinary differential equations.

Chapter 23: Calculus of variations

The last chapter introduces the calculus of variations.

Part III: Solutions to selected exercises

The book closes with worked solutions to selected exercises.

Key Concepts

Part I: Linear algebra (Chapters 1–16)

• Projections and spectral theory: Chapters 4–5.
• Reduction to triangular and tridiagonal form: Chapters 6–7.
• Convexity and the separating hyperplane theorem: Chapters 9–10.
• Linear programming, game theory and the simplex method: Chapters 12–13.
• Convex functions and non-linear programming: Chapters 15–16.

Part II: Advanced calculus (Chapters 17–23)

• Manipulation of integrals and multiple integrals: Chapters 18–19.
• Laplace transforms: Chapter 21.
• Series solutions of ordinary differential equations: Chapter 22.
• Calculus of variations: Chapter 23.

Critical Analysis

Strengths

• A deliberate balance of rigor and technique. Cambridge says the author keeps a balance between a rigorous theoretical and a cookbook approach, using concrete and geometric explanations so that students without deep mathematical training can follow.
• Optimization and linear algebra in one volume. The contents combine matrix theory with convexity, linear programming, game theory and non-linear programming, so a reader who needs both does not have to move between texts.
• Worked material. The publisher notes that the book is amply supplied with examples and exercises, with complete solutions to a large proportion of them, and a third part of the book collects solutions to selected exercises.
• Informal style. The publisher describes the text as written in an appealing and informal style; the American Mathematical Monthly called it "a wonderful reference book."

Limitations

• Application-oriented rather than proof-driven. Three chapters (1, 14 and 20) are revisions of earlier material, and Cambridge says the book will appeal to students whose interest is in application rather than theory; readers who want the full proof-based treatment of a pure analysis or algebra course should look elsewhere.
• Limited topic range. The second part covers integrals, differential equations, Laplace transforms and the calculus of variations; it is not a general course on partial differential equations or numerical methods, and the contents list does not name them.
• Older text. The book appeared in 1990 (paperback January 1991), so readers who want an algorithmic treatment of optimization with current software will need a more recent book alongside it.

Where It Fits in the Course

The book belongs to the applied and optimization part of the mathematics list, and its distinctive feature is that it supplies the supporting linear algebra and calculus together with the optimization. This is a methods book for students in applied fields who have finished elementary linear algebra and calculus. Ian Jacques's Mathematics for Economics and Business is the book written for economics and business students, and Guenin, Könemann and Tunçel's A Gentle Introduction to Optimization builds linear, integer and nonlinear programming from a first linear algebra course. Chong, Lu and Żak's An Introduction to Optimization and Boyd and Vandenberghe's Convex Optimization go deeper into optimization theory and algorithms, so Ostaszewski's Chapters 9–16 are a compact preparation for them. Readers who already know linear programming from the Guenin, Könemann and Tunçel book will find Chapters 12–13 a second treatment of related material, with the geometric emphasis Cambridge describes. Alain Goriely's Applied Mathematics: A Very Short Introduction is a non-technical overview of what applied mathematicians do.

How to Use This Book

The book suits second-year undergraduates and early graduate students in statistics, operations research, computer science, econometrics and mathematical economics who have taken elementary linear algebra and calculus.

• Linear algebra and optimization path: Chapters 1–16 in order; Chapters 9–13 can be read as a short course on convexity and linear programming.
• Reading order: the book is divided into linear algebra (Chapters 1–16) and advanced calculus (Chapters 17–23); choose the part that matches your field, and use the revision chapters (1, 14 and 20) to refresh background before the new material, for example Chapter 20 before Laplace transforms and series solutions.
• Practice: work the exercises and use the solutions in Part III to check method; because many solutions are complete, the book can be used for self-study.