Manifold Theory

Manifold theory is where the local tools of multivariable calculus get rebuilt to work on curved spaces that only look like Euclidean space up close, giving you a coordinate-free language for doing geometry and analysis on surfaces, Lie groups, and the configuration spaces that show up across physics and topology. You will spend the semester pushing through four problem sets and a midterm computing with tangent vectors, tensor fields, and differential forms, culminating in Stokes' theorem as the single statement that swallows the classical Green, Gauss, and Stokes results you saw in calculus. It assumes comfort with real analysis and point-set topology, and it is the gateway course for differential geometry, de Rham cohomology, and most of geometric analysis at the graduate level.