The Structures of Mathematical Physics

This text approaches the reader with shocking breadth and niggardly depth. As the author notes in the Preface, "This textbook serves as an introduction to groups, rings, fields, vector and tensor spaces, algebras, topological spaces, differentiable manifolds and Lie groups..." 

 

Each chapter of this book is comprised of short sections that focus on the definition of one or two mathematical objects. Around each definition, there is short---and pleasant---narrative and then a number of examples are described. Chapters end with a list of straightforward exercises which mainly focus on the verification of further examples (and the occasional non-example). 

 

There are essentially no results demonstrating the relationship between mathematical objects within a chapter or between chapters. For instance, in Chapter 6, Fundamental Concepts of General Topology, product spaces, quotient spaces, and continuous functions are all introduced. However, there is no discussion of the universal properties of the product (resp. quotient) topology, i.e., understanding continuous mappings into (resp. out of) such spaces, which severely handicaps the reader should they ever want to actually use topology in their studies. 

 

At times, the author provides a bit more context and intuition regarding mathematical structures. For instance, Chapter 7 on differentiable manifolds is quite readable and useful. The description of multivariable differentiation, at the beginning of this chapter, is clear and well-motivated. 

 

As a stand-alone mathematical dictionary, the text under review may serve a purpose, though few become fluent in a foreign language by solo study of a dual language dictionary. 

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Ryan Grady is an Assistant Professor of Mathematics at Montana State University