This is some kind of twin sister of the encyclopedic Princeton companion to mathematics [{\it T. Gowers} (ed.) et al., The Princeton companion to mathematics. Princeton, NJ: Princeton University Press (2008; Zbl 1242.00016)]. Again, the best way to get some idea of its contents is to resume first the editor’s description in the preface which reads as follows. This book describes what applied mathematics is about, why it is important, its connections with other disciplines, and some of the main areas of current research. It also explains what applied mathematicians do, which includes not only studying the subject itself but also writing about mathematics, teaching it, and influencing policy makers. The companion differs from an encyclopedia in that it is not an exhaustive treatment of the subject, and it differs from a handbook in that it does not cover all relevant methods and techniques. Instead the aim is to offer a broad but selective coverage that conveys the excitement of modern applied mathematics while also giving an appreciation of its history and the outstanding challenges. The companion focuses on topics felt by the editors to be of enduring interest, and so it should remain relevant for many years to come. With online sources of information about mathematics growing ever more extensive, one might ask what role a printed volume such as this has. Certainly, one can use Google to search for almost any topic in the book and find relevant material, perhaps on Wikipedia. What distinguishes the companion is that it is a self-contained, structured reference work giving a consistent treatment of the subject. The content has been curated by an editorial board of applied mathematicians with a wide range of interests and experience, the articles have been written by leading experts and have been rigorously edited and copyedited, and the whole volume is thoroughly cross-referenced and indexed. Within each article, the authors and editors have tried hard to convey the motivation for each topic or concept and the basic ideas behind it, while avoiding unnecessary detail. It is hoped that the companion will be seen as a friendly and inspiring reference, containing both standard material and more unusual, novel, or unexpected topics. It is difficult to give a precise definition of applied mathematics, as discussed in [the item] What is applied mathematics? and, from a historical perspective, in the item The history of applied mathematics. The companion treats applied mathematics in a broad sense, and it cannot cover all aspects in equal depth. Some parts of mathematical physics are included, though a full treatment of modern fundamental theories is not given. Statistics and probability are not explicitly included, although a number of articles make use of ideas from these subjects, and in particular the burgeoning area of uncertainty quantification brings together many ideas from applied mathematics and statistics. Applied mathematics increasingly makes use of algorithms and computation, and a number of aspects at the interface with computer cience are included. Some parts of discrete and combinatorial mathematics are also covered. The target audience for the companion is mathematicians at undergraduate level or above; students, researchers, and professionals in other subjects who use mathematics; and mathematically interested lay readers. Some articles will also be accessible to students studying mathematics at pre-university level. Prospective research students might use the book to obtain some idea of the different areas of applied mathematics that they could work in. Researchers who regularly attend seminars in areas outside their own specialities should find that the articles provide a gentle introduction to some of these areas, making good pre- or post-seminar reading. In soliciting and editing the articles the editors aimed to maximize accessibility by keeping discussions at the lowest practical level. A good question is how much of the book a reader should expect to understand. Of course, “understanding” is an imprecisely defined concept. It is one thing to read along with an argument and find it plausible, or even convincing, but another to reproduce it on a blank piece of paper, as every undergraduate discovers at exam time. The very wide range of topics covered means that it would take a reader with an unusually broad knowledge to understand everything, but every reader from undergraduate level upward should find a substantial portion of the book accessible. The companion is organized in eight parts, which are designed to cut across applied mathematics in different ways. Part I, Introduction to applied mathematics, begins by discussing what applied mathematics is and giving examples of the use of applied mathematics in everyday life. The language of applied mathematics then presents basic definitions, notation, and concepts that are needed frequently in other parts of the book, essentially giving a brief overview of some key parts of undergraduate mathematics. This article is not meant to be a complete survey, and many later articles provide other introductory material themselves. Methods of solutions describes some general solution techniques used in applied mathematics. Algorithms explains the concept of an algorithm, giving some important examples and discussing complexity issues. The presence of this article in Part I reflects the increasing importance of algorithms in all areas of applied mathematics. Goals of applied mathematical research describes the kinds of questions and issues that research in applied mathematics addresses and discusses some strategic aspects of carrying out research. Finally, the history of applied mathematics describes the history of the subject from ancient times up until the late twentieth century. Part II, Concepts, comprises short articles that explain specific concepts and their significance. These are mainly concepts that cut across different models and areas and provide connections to other parts of the book. This part is not meant to be comprehensive, and many other concepts are well-described in later articles (and discoverable via the index). Part III, Equations, laws, and functions of applied mathematics, treats important examples of what its title describes. The choice of what to include was based on a mix of importance, accessibility, and interest. Many equations, laws, and functions not contained in this part are included in other articles. Part IV, Areas of applied mathematics, contains longer articles giving an overview of the whole subject and how it is organized, arranged by research area. The aim of this part is to convey the breadth, depth, and diversity of applied mathematics research. The coverage is not comprehensive, but areas that do not appear as or in article titles may nevertheless be present in other articles. For example, there is no article on geoscience, yet earth system dynamics, inverse problems, and imaging the earth using green’s theorem all cover specific aspects of this area. Nor is there a part IV article on numerical analysis, but this area is represented by approximation theory, numerical linear algebra and matrix analysis, continuous optimization (nonlinear and linear programming), numerical solution of ordinary differential equations, and numerical solution of partial differential equations. Part V, Modeling, gives a selection of mathematical models, explaining how the models are derived and how they are solved. Part VI, Example problems, contains short articles covering a variety of interesting applied mathematics problems. Part VII, Application areas, comprises article on connections between applied mathematics and other disciplines, including such diverse topics as integrated circuit (chip) design, medical imaging, and the screening of luggage in airports. Part VIII, Final perspectives, contains essays on broader aspects, including reading, writing, and typesetting mathematics; teaching applied mathematics; and how to influence government as a mathematician. As the companion in mathematics, published 7 years ago, this is again a real masterpiece in both contents and style. It should be available in every math library all over the world, and perhaps in some private bookshelf, too. Referring to the third paragraph of the above summary, the reviewer does not agree that one may find the same, or even large part of the material in Wikipedia, or through a Google search: these sources are by no means reliable and contain quite often lots of errors. In contrast, only renowned and leading specialists are able to write a book as this companion, with such a large variety of topics, with so many illuminating examples, and in such a clear and convincing language.

Reviewer:

Jürgen Appell (Würzburg)