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**A course in mathematics for students of physics: 2.**
*(English)*
Zbl 0706.00002

Cambridge etc.: Cambridge University Press. xvii, 407-850, £60.00/hbk; $ 54.50/hbk (1990).

The pedagogical approach of this volume continues that of the first part [see Zbl 0669.00030]. Clearly the material presented here is more advanced than in the previous volume. Like in the first part the mathematical theory is linked with motivations from physics and detailed descriptions of physical examples. In this volume they generally are taken from the theory of electromagnetism. Each single chapter is followed by a long list of exercises and a summary describing the main techniques which the student should have learned from the preceding text. Also some non-standard tools from mathematics are included in this volume which generally do not belong to the contents of a common course in mathematics for physicists. Thus the two textbooks by P. Bamberg and S. Sternberg must be considered as an interesting and valuable addition to the literature for students of physics.

Some description of the contents should be given: The first three chapters of this volume present a gentle introduction to some elements of algebraic topology. In Chapter 12 the interest in algebraic topology is motivated by the study of electric networks which in mathematical terms are given by 1-dimensional complexes. Several applications to physically interesting networks are described. This is continued in Chapter 13 by examining the boundary-value problems associated with capacitive networks and the solution of some classical problems in electrostatics involving conductors. In Chapter 14 the 1-dimensional considerations are generalized to higher dimensions including the discussion of complexes, homology, Euler’s theorem, dual spaces and cohomology. The next four chapters develop the exterior differential calculus as a continuous version of the discrete theory of complexes. Chapter 15 is rather technical and presents the following topics: exterior algebra, k-forms and exterior differentiation, integration of forms, Stokes’ theorem, differential forms and cohomology. The presentation is very detailed and several single cases are discussed in order to become acquainted with the general theory. Then in Chapter 16 electrostatics is developed as a continuous analogy to the theory of networks. The basic facts of potential theory are exhibited too. The general introduction is continued in Chapter 17. The main topics are vector fields, flows, Lie derivatives and interior products. On the physical side magnetostatics is developed in this context. As an important technical tool the Hodge star operator is introduced in Chapter 18. Its relations to vector algebra and geometry are exhibited and a short introduction to Clifford algebras is given. Also some vector calculus operators and identities are discussed in this context. Finally, all these preparations are applied to the study of Maxwell’s equations and the associated wave equation. The remaining three chapters include some complementary material which may be of interest for students of physics. Chapter 20 presents a short treatment of the theory of functions of a complex variable with emphasis on the geometric point of view. Chapter 21 discusses elementary aspects of asymptotics: Laplace’s method, the method of stationary phase, Gaussian integrals, the Fourier inversion formula etc. The last chapter shows how the exterior calculus can be used in classical thermodynamics. Though the presentation of the material is considerably different from that in the textbooks commonly used for a course in mathematics for students of physics, these two volumes can be highly recommended as a background for such a course. Several students will consider the pedagogical approach of this course as a very convenient method to learn mathematics as a useful method for the description of physical theories. Some of them will prefer the conventional presentations and use this course as a reference to get some idea of the mathematics which is needed to understand the modern developments in theoretical physics.

Some description of the contents should be given: The first three chapters of this volume present a gentle introduction to some elements of algebraic topology. In Chapter 12 the interest in algebraic topology is motivated by the study of electric networks which in mathematical terms are given by 1-dimensional complexes. Several applications to physically interesting networks are described. This is continued in Chapter 13 by examining the boundary-value problems associated with capacitive networks and the solution of some classical problems in electrostatics involving conductors. In Chapter 14 the 1-dimensional considerations are generalized to higher dimensions including the discussion of complexes, homology, Euler’s theorem, dual spaces and cohomology. The next four chapters develop the exterior differential calculus as a continuous version of the discrete theory of complexes. Chapter 15 is rather technical and presents the following topics: exterior algebra, k-forms and exterior differentiation, integration of forms, Stokes’ theorem, differential forms and cohomology. The presentation is very detailed and several single cases are discussed in order to become acquainted with the general theory. Then in Chapter 16 electrostatics is developed as a continuous analogy to the theory of networks. The basic facts of potential theory are exhibited too. The general introduction is continued in Chapter 17. The main topics are vector fields, flows, Lie derivatives and interior products. On the physical side magnetostatics is developed in this context. As an important technical tool the Hodge star operator is introduced in Chapter 18. Its relations to vector algebra and geometry are exhibited and a short introduction to Clifford algebras is given. Also some vector calculus operators and identities are discussed in this context. Finally, all these preparations are applied to the study of Maxwell’s equations and the associated wave equation. The remaining three chapters include some complementary material which may be of interest for students of physics. Chapter 20 presents a short treatment of the theory of functions of a complex variable with emphasis on the geometric point of view. Chapter 21 discusses elementary aspects of asymptotics: Laplace’s method, the method of stationary phase, Gaussian integrals, the Fourier inversion formula etc. The last chapter shows how the exterior calculus can be used in classical thermodynamics. Though the presentation of the material is considerably different from that in the textbooks commonly used for a course in mathematics for students of physics, these two volumes can be highly recommended as a background for such a course. Several students will consider the pedagogical approach of this course as a very convenient method to learn mathematics as a useful method for the description of physical theories. Some of them will prefer the conventional presentations and use this course as a reference to get some idea of the mathematics which is needed to understand the modern developments in theoretical physics.

Reviewer: Bernd Wegner

### MSC:

00A69 | General applied mathematics |