Applied Exterior Calculus

Exterior calculus — the calculus of differential forms — is now used widely in applications. However, many introductions to the subject use a global approach that requires the heavy machinery of topology and differential geometry. The author of Applied Exterior Calculus attempts to get around this formidable introductory barrier by focusing on the local aspects of the theory and confining himself mostly to n-dimensional Euclidean space. This approach is at least partly successful. There is still quite a bit of prerequisite material (Lie algebras and subalgebras, tangent spaces and their dual spaces, derivations, operator representations of vector fields, etc.).

The real advantage of the author's approach is that it gets the reader quickly past the preliminaries. A notable strength is the early introduction of partial differential equations in the context of the exterior calculus. The method of characteristics is introduced via vector fields; this leads directly to general solutions of linear and quasilinear PDEs. Later on, PDEs are revisited during a systematic exploration of the consequences of the Darboux, Cartan and Frobenius theorems for differential systems.

The meat of the text is in the applications. Primary mathematical applications are second-order PDEs and the calculus of variations. Applications to physics include classical and irreversible thermodynamics (including Carathéodory's notion of inaccessibility for adiabatic processes), electrodynamics (Maxwell's equations) and gauge theory. It was probably with gauge theory that physicists first became convinced of the utility of the exterior calculus. The author manages to develop the field equations and geometric structure of gauge theory without any direct use of the theory of fiber bundles.

The book is aimed at advanced undergraduates or beginning graduate students who have done course work in upper-division algebra and analysis. There are lots of exercises, including many that emphasize developing the ability to compute.

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Bill Satzer ([email protected]) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.