MATH 445 Analysis on Manifolds

This course rebuilds calculus on curved spaces: instead of integrating over intervals or regions in Rⁿ, you learn what it means to differentiate and integrate on smooth manifolds, with differential forms as the unifying language that makes the classical Green's, Stokes', and divergence theorems collapse into a single statement. Expect a steady diet of problem sets working through tangent spaces, wedge products, pullbacks, and orientations, with two midterms and a final that lean on computation as much as proof. Building on advanced calculus and point-set topology, it's the bridge into differential geometry and a prerequisite mindset for anything later involving Lie groups, de Rham cohomology, or general relativity.