*Mathematical Analysis*, the first volume of *Foundations of Applied Mathematics* by Jeffrey Humpherys, Tyler J. Jarvis, and Emily J. Evans, is a rigorous presentation of mathematical analysis for students of applied mathematics. As per the preface, its audience is upper-level undergraduates or first-year graduate students, who have already taken classes in linear algebra, vector calculus, and real analysis. It would also make an excellent reference book and a great addition to one’s mathematical library.

As a physical object, the book itself is impressive. It is large; counting the front matter, it’s just over 700 pages. The pages are heavyweight paper, and the type setting is excellent. The authors use colored text boxes to add elements to the book: red for *notate bene*, blue for examples, yellow for unexamples, green for applications, and grey for vistas. The combination results in an aesthetically pleasing read.

The *notate bene* are well placed and give students a warning about common mistakes or clarify subtle points. I found the unexamples to be as useful as the examples. For instance, after defining ring homomorphisms, we are given four examples of maps that fit the definition and two unexamples of maps that do not, such as \( \phi(x)=x^{2} \), as well as explanations of why they fail. The applications are plentiful, interesting, and relevant.

Each section starts with an introductory paragraph or two that gives the motivation for the topic and might also relate the new topic to previous material. This makes the subject matter more accessible and also gives transparency to why the material is presented in the way it is. As a reader, I was never left wondering why the topic was included nor about the utility of the subject. Each chapter ends with a good number of problems, which range in difficulty and cover both computation and theory. There are no answers in the back of the book, and there does not appear to be a solution manual available.

The book is divided into four parts of approximately four chapters each. The first part, Linear Analysis I, covers the basics of linear algebra and spectral theory, including singular value decomposition (SVD). Right away, on page 9, we are treated to an application of vector spaces and the superposition principle to noise-cancelling headphones. When introducing the SVD, the authors list many

names from other disciplines by which it is also known.

The second part, Nonlinear Analysis I, covers metric space topology, differentiation, and the contraction mapping. Instead of approaching integration in the more common way of starting with Riemann integration, the integral is constructed using step functions, which results in the regulated integral. It makes for an easier transition to Lebesgue integration. This part includes a sub-section about the application of the implicit function theorem to the design of GPS, which I found to be quite interesting.

The third part, Nonlinear Analysis II, covers Lebesgue integration, calculus on manifolds, and complex analysis. Lebesgue integration feels like the natural extension of the regulated integral, and the text does not get bogged down in measure theory. The main idea and sketches of proofs are presented in chapter 8, while the full proofs and more details are delayed until chapter 9, which is labelled as optional. This has the benefit of keeping the narrative moving without excluding the details. The chapters about calculus on manifolds and complex analysis are nicely done with many examples and useful diagrams.

The fourth part, Linear Analysis II, covers spectral calculus, iterative methods for solving systems of equations and finding eigenvalues, spectra and pseudospectra, and ring theory. The resolvent of a matrix is introduced and then spectral theory is approached through this lens. This leads to several very nice proof and applications: the spectral mapping theorem, the Cayley-Hamilton theorem, and Google’s PageRank algorithm, among others. In the final chapter, the authors draw a useful analogy between vector space and rings.

There are four appendices: foundations of abstract mathematics, complex numbers and other fields, topics in matrix analysis, and the Greek alphabet. The appendices serve as useful references and also clarify the notation used in the book. As with the main body of the book, they contain plenty of examples, unexamples, and a few notate bene. This book is the first volume in a series of four volumes. It is the result of a redesign of the undergraduate curriculum for the applied and computational mathematics major at Brigham Young University. Among the objectives for the redesign were to modernized the curriculum for the 21st century and to show the interconnectivity of mathematics with other STEM fields. These goals can clearly be seen in the text. In addition to the text, there is an accompanying lab manual available for free on github (

https://github.com/Foundations-of-Applied-Mathematics/Labs). This includes algorithms and Python code that demonstrate the theory and complement the text book material. I am looking forward to the release of Volume 2.

Rebecca Conley is an assistant professor in the mathematics department of Saint Peter's University in Jersey City, NJ. She earned her PhD in computational applied mathematics at Stony Brook University in 2016.