The author taught courses in mathematical physics at Utrecht University that were directed primarily to students in geophysics. He later used a draft at the Colorado School of Mines so the subject material covered in this book has been tested on students from two different cultures. It is intended to be at the level of the text by Boas that is appropriate to Canadian third year honours physics programs but does not attain the level of the classic by Arfken that is appropriate in the fourth year of these programs. Canadian geophysics undergraduate students do not receive a strong mathematical education and it is entirely possible that this book would provide a good means of supplementing and broadening their exposure to mathematical techniques. However, the approach used by the author is not rigorous, as the emphasis is on learning techniques and their application.

Each chapter includes examples of “real” examples chosen primarily from geophysics. Consequently, this book is full of interesting examples of geophysics problems that will encourage students to want to learn more techniques. The emphasis is on relevance and it is to be hoped that it will inspire such students to take more rigorous mathematics courses. It is the assessment of this reviewer that this text is not appropriate as a course textbook because there are not many geophysicists with the mathematical background necessary to teach a course using this book but that it has a definite role as a supplementary text to encourage students with a weak mathematical background to become more comfortable with the power of this essential subject.

This text consists of 22 chapters that are organized such that the material learned in one chapter can be used in succeeding chapters. The subject matter organisation is very innovative. Chapter 2 on Power Series includes examples on the growth of the earth by cosmic dust, and reflection and transmission by a stack of layers. Chapters 3 through 9 cover coordinate systems and the various vector operators as well as the theorems of Gauss and Stokes. Chapter 10 concerns the conservation laws starting with their general form. It then covers the various conservation equations with an emphasis on the application of the vector calculus developed in the preceding chapters. Chapter 11 introduces scale analysis that is of considerable utility in geophysics and helps to reduce complicated problems to more tractable forms. Chapter 12 covers those elements of linear analysis that have important geophysical application. Examples include Coriolis forces, normal modes of vibrating systems and eigenvalues. Chapter 13 introduces the Dirac delta function and applies it to the self-energy of the electron. Chapter 14 on Fourier analysis enables filter design to be used as an application. Chapters 15 and 16 cover the elements of analytic functions and complex integration. Chapters 17 and 18 cover Green’s functions carefully from first principles and then give a number of examples of their application to linear systems. Chapter 19 on normal modes enables Bessel and Legendre functions to be introduced and applied to vibrating systems as well as to discuss guided waves, leaky modes, and radiation damping. Chapter 20 on potential theory leads to the gravitational field of the earth and multipoles. Chapter 21 on Cartesian tensors covers the concepts necessary to understand stress and strain tensors and their applications in geophysics and develops the tensor notation used in special relativity. Finally, Chapter 22 covers perturbation theory. This outline attempts to give both the organisation and flavour of the subject material covered in this innovative text.

In summary, there is a great deal of interesting and important mathematics covered in this text. The focus is on those mathematical topics that every geophysicist should be familiar with and be comfortable using as the tools of their trade. The author, being both a geophysicist and a mathematician, has been very successful in demonstrating the utility of these topics. The reviewer recommends this text to geophysicists who teach courses in mathematical methods for their students but does not recommend it as a text for a course in mathematical physics for physics students. It is a fascinating attempt to bridge the gap between being too rigorous and too applied; it would be a useful way for anyone interested in reviewing the methodologies covered to strengthen their knowledge base as it makes interesting and thought-provoking reading.

Harvey Buckmaster
University of Victoria