The editor behind *The Princeton Companion to Applied Mathematics* (PCAM) is Nicholas J. Higham, a numerical analyst who has been recognized with multiple awards for his contributions to the accuracy and stability of numerical algorithms. He is assisted by more than 150 experts in applied mathematics. The book is an excellent reference that successfully compiles into a readable and engaging form the broad range of topics that an applied mathematician might encounter in their career.

**Organization**

While clicking on the “Table of Contents” tab above will give access to a detailed table of contents, here is a summary:

**Part** |
**Pages** |
**Articles** |
**Average**
Length |
**Organization** |

I: Introduction |
81 |
6 |
14.3 |
Logical |

II: Concepts |
55 |
36 |
2.5 |
Alphabetical |

III: Equations, Laws, and Functions
of Applied Mathematics |
39 |
31 |
2.2 |
Alphabetical |

IV: Areas of Applied Mathematics |
419 |
40 |
11.5 |
Thematic |

V: Modeling |
143 |
21 |
7.8 |
Thematic |

VI: Example Problems |
51 |
18 |
3.8 |
Thematic |

VII: Application Areas |
115 |
25 |
5.6 |
Thematic |

VIII: Final Perspectives |
65 |
9 |
8.1 |
Random |

The nature of the book is clearly described in the preface. The goal of the book is to treat applied mathematics in a broad sense, rather than cover all subjects exhaustively. While ideas from mathematical physics, probability, and statistics are discussed in the book, these subjects are not explicitly included. A number of ideas from computer science and combinatorial mathematics are also discussed. The book aims to be approachable by “keeping discussions at the lowest practical level” and assumes an audience at the undergraduate mathematics level and above. Many of the sections start with a conversation that would be accessible to an advanced calculus students.

This book builds on the *Princeton Companion to Mathematics* (PCM) and the editors have made an effort not to overlap with that text. There are relevant articles about basic mathematical concepts in the PCM that are not repeated in the PCAM and some material from the PCM chapter on “The Influence of Mathematics,” while very applied, is not made redundant in the PCAM.

**Part I:**

The introduction begins with the difficult question “What is Applied Mathematics,” followed by a brief, but very well written, compilation of the Language of Mathematics. This section works as a review of the topics typical of an undergraduate mathematics degree that form the basis of the language of applied mathematics, such as complex numbers, differential equations, special functions, power series, matrices and vectors. Next, an overview of solution methods is presented with topics ranging from function approximations to computer algorithms. As a capstone to the introduction we are given an engaging section on the history of applied mathematics. The introduction would be a fabulous read for a student who is thinking about pursuing a career in applied mathematics. It serves as a strong foundation upon which all the following chapters may be built.

**Parts II and III:**

The next two chapters of the PCAM serve as an encyclopedia of important concepts and equations in applied math. The sections are brief but informative, with articles that define and put into context important concepts in applied mathematics. Both parts are arranged alphabetically and many of the sections have references to later chapters or suggestions for further reading. These sections summarize many of the mathematical concepts, or well known equations, that are often assumed known in specific application areas. They are referenced extensively in the following chapters of the book. The range of topics is comprehensive: it includes Asymptotics, Boundary Layers, Chaos, Numerical Methods, Integral Transforms, Tensors and Manifolds, along with and introduction to many important equations such as Laplace’s Equation, Maxwell’s Equations, the Gamma Function, and the Navier-Stokes Equations. See the table of contents for a full listing of topics. These chapters act as a concise reference for the working mathematician or as an overview of important topics for a student venturing into higher level topics in applied mathematics.

**Part IV**

The next section is the largest and most in-depth section of the book. It bridges the gap between the classical foundations of applied math and the current areas of interest and research. Each section emphasizes important mathematical concepts, theorems, or equations. Most sections give brief example problems along with references for extended study or more mathematical formalism. They emphasize mathematical intuition and focus on giving a readable and complete overview of the topic. The writing style is conversational, in most instances building from very simple ideas, definitions, or historical context.

One example of this chapter’s approach can be found in the section on “Fluid Dynamics”, which gives the reader a brief explanation of the continuum approximation, fluid properties, incompressible flows, streamlines and voracities, without overwhelming the reader with details such as the full derivation of the Navier-Stokes equations, which can be built up later should the reader need more information. The mathematics and physics is described in a natural way, giving enough information for the reader to easily fill in the gaps; for example, when deriving the vorticity equation the needed vector identities are supplied. The following fluid cases are considered: simple flows where exact solutions are possible, Stokes flows and the motion of a particle, inviscid flows and vortex rings, and aerodynamics with boundary layers. Finally, the important issue of flow instability is summarized. This chapter gives an overview of topics from a graduate level fluid mechanics class, has a nice balance of detail and readability, and gives the reader many jumping off points for further inquiry. It does a good job of taking a very complex area of study and compressing the big ideas into just a few pages.

**Part V**

Part V gives examples of a variety of techniques that can be employed to model physical systems. These sections are helpful in that they demonstrate the process that goes into building a model for a new system, and give some insight into the complications that arise in the modeling process. Some sections do assume an understanding of the system being modeled; for example, the section “Financial Mathematics” expects some familiarity with derivatives pricing and portfolios. Most sections, however, are completely approachable, even to a student reader, and give references for further inquiry.

As an example, the section on “The Spread of Infectious Disease” does an outstanding job of building a strong background into the formulation of the problem. It then follows up with a variety of model types including the simple SIR model with or without demographics, SIS models, and models for HIV/AIDS. It leaves the reader interested in the topic and wanting to learn more.

**Part VI and VII**

These two sections, taken together, tackle the impossible task of cataloging the broad range of problems being solved by applied mathematicians today. While no list could be complete, the PCAM does a commendable job of introducing the reader to many of the current research areas and important problems in applied mathematics.

Part VI illustrates the wide variety of application problems, with examples ranging from bubbles and foams to insect flight and macromolecules, the chapter gives an assortment of topics that can be investigated using applied mathematics. The example problems are all similar in both their brevity and approachability. Each section represents a quick glimpse into the problem area, often with an easy to follow calculation. For example, the section “Ranking Web Pages” gives a simple example network and shows the calculations for how the pages would be ranked. Most of the example problems are easy enough to follow without extensive background.

Part VII is organized similarly, but rather than looking at specific problems, it takes a wider view of applied mathematics and it's interdisciplinary nature. The sections include topics from computer science, aerospace engineering, social networks, image processing, solid state physics, biology, neuroscience, and many more. Each chapter gives an introduction to the area, describes some of the fundamental problems being solved, and suggests further reading and future directions. These sections give the reader a glimpse into the more recent work being done in applied mathematics and most would be perfect reading for senior level undergraduates who are looking for ideas and motivation for thesis projects. The accessibility ranges quite a bit from section to section. For example, the chapter “Evolving Social Networks, Attitudes, and Beliefs - and Counterterrorism” grabs the readers attention but very quickly jumps into notation and definitions that require at least some background specific to graph or network theory. This makes the chapter less approachable, though much of background can found earlier in the book. In contrast, the chapters “Chip Design”, “Programming Languages, An Applied Mathematics View”, and “Systems Biology” are more conversational and thus more approachable.

**Part VIII**

Part VIII is a compilation of final perspectives. It includes articles about how to do reproducible research, teaching applied mathematics, and how mathematics interacts with popular culture and policy.

PCAM serves as a valuable reference for the applied mathematician. For students, undergraduate and above, it illustrates the diversity of problems that can be approached mathematically, providing references and support for how to learn more. It puts into context the way in which the beauty of pure mathematics can be applied to real world problems and serves as a jumping off point for further independent study of applied mathematics. For the professional mathematician, PCAM can serve a variety of needs. It would be a wonderful teaching reference when trying to answer the question “when will this ever be used?” or trying to motivate students in their study of applied mathematics. It is a concise and readable reference book for the mathematical concepts used in every day research and it is a way to expand mathematical breadth or think about research approaches from another mathematical viewpoint. As a reader, I find myself flipping through the pages and becoming engaged in new and interesting ideas from the world of applied math. PCAM does a good job of extending the breadth and accessibility of PCM to the world of applied mathematics.

Joanna Bieri is an Associate Professor of Mathematics at The University of Redlands in Redlands, California.