1. Book Review
Recent changes in mathematics education in Japan have significantly influenced the preparation of students entering university chemistry programs. Topics such as matrices and linear transformations, which were previously included in high school curricula, are no longer taught in the same way, while greater emphasis has been placed on statistics and the complex plane. At the same time, the growing importance of data science has increased the need for mathematical literacy among students in the natural sciences. Under these circumstances [1], many students encounter a gap between introductory university mathematics courses and the mathematical tools required in chemistry and related disciplines.
The Japanese translation of Mathematical Methods for Molecular Science by John E. Straub [2], translated by Tetsuro Nagai and co-workers and published by Kagaku-Dojin, provides a timely introduction to mathematical concepts from the perspective of molecular science. The book is intended for students and researchers who wish to understand not only the mathematical foundations (high school or university freshman level) of chemical phenomena but also the practical applications of mathematical methods in modern molecular science. For example, Chapter 14 of the Japanese edition is translated as “Matrices and Matrix Algebra” rather than “Matrices and Linear Algebra”, employing more natural, less abstract terminology in Japanese language.
The book consists of sixteen chapters that cover a broad range of topics, including calculus, differential equations, linear algebra, probability, statistics, and symmetry. Rather than treating these subjects independently, the author presents them within a common framework centered on molecular science. Throughout the text, mathematical concepts are illustrated with examples drawn from chemistry, physics, and materials science, making the content accessible to readers with scientific interests. Interestingly, Figure 1.8 indicates a ln[p(T)] vs 1/T ‘straight’ line on a semi-logarithmic plot, which is frequently used in chemical experiments.
From the perspective of crystallography, Chapter 13, which deals with Fourier series, Fourier transforms, and harmonic analysis, is particularly noteworthy. Fourier transforms as well as phase problem form the theoretical basis for determining electron density distributions from diffraction data, yet many students encounter these methods only through software packages and automated procedures [3]. The chapter provides a clear explanation of the underlying mathematical principles and their physical significance, helping readers understand how diffraction experiments are connected to structural models.
The final chapters, covering matrices, eigenvalues, geometric transformations, and molecular symmetry are especially relevant to students of inorganic and coordination chemistry [4]. Symmetry operations, point groups, and group theory are fundamental concepts in these fields, but they are often introduced after only limited exposure to matrix methods. For students who have not previously studied matrices in depth, the relationship between matrix operations and molecular symmetry may appear abstract.
This book addresses that difficulty by introducing matrices in a practical context and demonstrating how they can be used to describe rotations, reflections, and other geometric transformations. The discussion of eigenvalues and diagonalization further supports the understanding of symmetry-based approaches used in spectroscopy, crystallography, and quantum chemistry [5]. Readers are able to see how mathematical concepts are applied in the analysis of molecular structures and properties.
Another strength of the book is its extensive use of visual representations and worked examples. Abstract mathematical ideas are frequently accompanied by graphical illustrations, making it easier for readers to connect equations with physical and chemical phenomena. This approach is particularly valuable for students whose primary interests lie in chemistry rather than mathematics.
The publication of the Japanese edition is likely to be welcomed by students and educators alike. As interdisciplinary research continues to expand, the ability to communicate across the boundaries of mathematics, chemistry, physics, and data science has become increasingly important. In this context, Mathematical Methods for Molecular Science provides a useful resource for understanding how mathematical methods contribute to contemporary molecular science.
Overall, this book offers a well-organized introduction to the mathematical foundations of molecular science. It will be valuable not only for undergraduate and graduate students in chemistry and related disciplines but also for researchers seeking a broader understanding of the mathematical tools commonly used in modern scientific research.
Appendix: The Cover of Mathematical Methods for Molecular Science: Theory and Applications, Visualizations and Narrative (Japanese Edition)