Tensor analysis provides the necessary tools for investigating complex phenomena, from the highly abstract general relativity theory to elasticity theory of materials with daily real-life applications. The wide range of applications of tensor theory has attracted scientists from different disciplines. Thereby, this multi-disciplinary interest in tensors implicates a number of challenges for authors of books of an introductory nature. It is all mathematics and yet it is not. While mathematics is the underlying systematic and consistent language of tensor analysis, physicists, engineers, and mathematicians tend to read it differently. Therefore, it is of utmost importance that a monograph on this subject is written in a manner appropriate for its intended readership.

In my opinion, the present monograph by Nguyen-Schäfer and Schmidt has achieved this goal astonishingly well for engineers. Schmidt is a mathematician and the book clearly reveals his signature. It starts with a comprehensive introduction to the necessary notations and provides a clear definition of tensors and rules of transformation, with a special focus on calculus on general curvilinear coordinates. Using the curvilinear coordinates as the continuous link, elementary concepts of differential geometry are introduced and subsequently applications are discussed. Thereby, Nguyen-Schäfer’s touch makes the book less abstract and more goal oriented. This is what I believe an engineer would look for. Instead of losing the reader in abstract mathematics, the authors provide an accurate how-to of the analytic methods. They further treat the subject with particular care in order to avoid any possible errors, which is a hard task given that tensor analysis relies on the accurate combination of indicies. The required methods to apply tensor analysis in fluid dynamics, continuum mechanics, electrodynamics, and general relativity theory are provided in detail; nevertheless, the book could have benefited from a few explicit problems and examples of how to calculate specific engineering problems.

In summary, this book is an excellent systematic realization of tensor analysis for engineers and physicists at all levels, from undergraduate students to experts.