This text was produced for the second part of a two-part sequence
on advanced calculus, whose aim is to provide a firm logical
foundation for analysis. The first part treats analysis in one
variable, and the text at hand treats analysis in several
variables. After a review of topics from one-variable analysis and
linear algebra, the text treats in succession multivariable
differential calculus, including systems of differential equations,
and multivariable integral calculus. It builds on this to develop
calculus on surfaces in Euclidean space and also on manifolds. It
introduces differential forms and establishes a general Stokes
formula. It describes various applications of Stokes formula, from
harmonic functions to degree theory. The text then studies the
differential geometry of surfaces, including geodesics and
curvature, and makes contact with degree theory, via the
Gauss-Bonnet theorem. The text also takes up Fourier analysis, and
bridges this with results on surfaces, via Fourier analysis on
spheres and on compact matrix groups.
General
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