Syllabus for Analysis 1 Qualifying Exam

The Analysis 1 exam covers essentially the material taught in MTH G101 (Analysis 1).

Topics

• Metric spaces: topology of metric spaces, continuous maps, sequences and limits, compactness, connectedness, completeness, and the contraction lemma.
• Calculus of one variable: basic properties of derivatives and integration.
• Sequences and series of functions: uniform convergence, equicontinuous families of functions.
• Functions of several variables: the differential of a map, the chain rule, inverse and implicit function theorems, integration of differential forms, Stokes’ Theorem.
• Existence and uniqueness of ODE, applications.

References

• Introduction to Analysis, by Maxwell Rosenlicht, Dover, 1986.
• Principles of Mathematical Analysis, 3rd Edition, by Walter Rudin, McGraw-Hill, 1976.