A Course in Mathematics for Students of Physics: 2

This textbook has been developed from a course taught at Harvard over the last decade. The course covers principally the theory and physical applications of linear algebra, and of the calculus of several variables, particularly the exterior calculus. The authors adopt the 'spiral method' of teaching covering the same topic several times at increasing levels of sophistication and range of application. Thus the student develops a deep intuitive understanding of the subject as a whole and an appreciation of the natural progression of ideas. This the second volume, opens with an introduction to algebraic topology, introduced by the analysis of electrical networks or mathematically speaking, the topology of one-dimensional complexes. Chapters 15-18 develop the exterior differential calculus as a continuous version of the discrete theory of complexes. Facts of the exterior calculus are presented: exterior algebra, k-forms, pullback, exterior derivative and Stokes' theorem. Chapter 16 presents another physical theory, electrostatics. The authors argue that the dielectric properties of the vacuum determine Euclidean geometry in three-dimensional space. The basic facts of potential theory are presented. Chapters 17 and 18 continue and conclude the study of the exterior differential calculus, developing the notions of vector fields and flows, interior products and Lie derivatives, and applying them to magnetostatics. The star operator is discussed in a general context. Chapter 19 can be thought of as the culmination of the course. It applies the results of the preceding chapters to the study of Maxwell's equations and the associated wave equations. The last two chapters covercomplex analysis and elementary asymptotics, and the book ends with a sophisticated treatment of thermodynamics.