This program aims to provide an in-depth review and to fill in gaps in some background material in Advanced Calculus expected in standard first year graduate courses. The material to be covered is also part of the syllabus of the qualifying exams on the subjects of Real Variables and Elementary Point-Set Topology and Complex Variables and Advanced Calculus.
Below is a tentative list of topics to be covered; the actual coverage varies depending on instructor.
- Basic properties of the reals: Limits (including upper and lower limits), Cauchy sequences, completeness, sequential compactness (Bolzano–Weierstrass theorem) and compactness (Heine-Borel Theorem).
- Basic tools: Cauchy-Schwarz inequality. Summation (integration) by parts.
- Sequences and series of numbers and functions, including absolute and uniform convergence, and equicontinuity. Applications involving power series, integration and differentiation.
- Basic topological notions such as connectivity, Hausdorff spaces, compactness, product spaces and quotient spaces. Emphasis on examples in Euclidean and metric spaces.
- Compactness criteria in metric spaces. Arzelà–Ascoli Theorem and applications.
- Review of multiple, line and surface integrals, theorems of Green and Stokes and the divergence theorem.
- Jacobians, implicit and inverse function theorems, and applications.
Change of variables formula. Role of exterior calculus.