Calculus IV: Vector Analysis, Differential Forms, Manifolds, Distributions, Variational Methods, and the Analytic Foundations of Modern Mathematical Physics ... Course in Modern Mathematics Book 4)

This fourth volume in the Scottish Science Society's calculus sequence carries the reader from the classical calculus of several variables to the modern apparatus of differential geometry and analysis on which contemporary mathematics rests. The first arc — fourteen chapters — develops vectors, partial derivatives, multiple integrals, and the integral theorems of Green, Gauss, and Stokes in their classical formulations. The second arc — eight further chapters — extends the apparatus to smooth manifolds, differential forms, de Rham cohomology, Riemannian geometry, vector bundles, Lie derivatives, and the calculus of variations, before closing with the analytic theory of distributions and Sobolev spaces.

Each chapter combines mathematical rigour with discursive exposition, in a continuous prose register inspired by Machado de Assis, Montaigne, and Dr Johnson, and is supplemented by carefully chosen exercises with full resolutions and a selective list of further references. Programmatically generated figures throughout the volume exhibit the geometric content of the text in the Society's characteristic typographic palette.

Part of the Treatises in Mathematics series from the Scottish Science Society Press, Edinburgh.