These notes developed from a one semester course at Wayne State University, taught several times in the last three decades of the 1900s. The subject matter is analysis on manifolds, consisting of the theory of smooth manifolds, differential forms, integration of forms, the generalized Stokes' Theorem, de Rham cohomology, and some related topics. The course is intended for first or second year graduate students in Mathematics with a background in Advanced Calculus, General Topology, linear algebra (including quotient spaces), and a little elementary group theory (including some familiarity with the symmetric groups). Given the above background, the notes are self-contained. In particular, the notes develop from scratch the vector bundle theory and exterior algebra that they make use of. The notes include a fair number of examples, as well as exercises. The notes are divided into 19 sections.
Analysis | Geometry and Topology
Handel, David, "Notes for MAT 7500 – Winter '93, revised Winter '06" (2006). Mathematics Faculty Research Publications. 21.