Subsections
- Advanced Calculus/Ordinary Differential Equations (``ODE'')
- Review of the real number system
- Metric spaces, continuity, uniform convergence
- Contraction mapping principle
- Existence and uniqueness theorems for ODE
- Global existence theorem for linear ODE
- Linear transformations, orthogonal projections and matrix exponential
- Linear systems of ODE with constant coefficients
- Derivatives in
Rn and in Banach spaces
- Newton's method and the inverse function theorem
- The implicit function theorem
- Measure theory
- Riemann integral in
Rn
- Cantor-type sets, dyadic decompositions in
Rn
- Measures arising from volume functions on open sets
- Basic properties of the Lebesgue measure
- Measurable and integrable functions
- Convergence theorems for Lebesgue integrals: monotone and
dominated convergence theorems and Fatou's lemma
- Criterion for Riemann integrability
- Daryl Geller, A first graduate course in real
analysis. Part I,
Solutions Custom Publishing (can be ordered from the campus bookstore);
- Walter Rudin, Principles of mathematical analysis,
3rd ed., McGraw-Hill, New York 1976;
- Walter Rudin, Real and complex analysis,
3rd ed., McGraw-Hill, New York 1987;
Download Graduate Handbook: PDF version or compressed PostScript version
Updated 2003-12-01, Graduate Committee.
Please email errata, comments, and suggestions to Graduate Committee <gpd@math.sunysb.edu>