Qualifying Exam - Analysis (M517-M518) - Description

The analysis qualifier is based (approximately) on the content of the courses M517-M518. For the convenience of those preparing the exam and those who are taking the exam, a list of topics has been compiled. The analysis qualifier may include any of the topics included in the following list (whether or not the topics were actually covered in M517-M518 during recent semesters). The analysis qualifier will not include any topics that are not included in the list and any given exam probably will not include every topic on the list.

M 517

1. basic properties of real numbers and Rn
2. Finite, countable and uncountable sets
3. Metric space topology: open, closed compact, connected sets; Heine-Borel thm
4. Sequences: limit defn; limit thms in R and Rn; subsequences; Cauchy sequences; Monotone convergence theorem for sequences.
5. Limits and continuity: limits of functions; function limit theorems; continuity and uniform continuity; continuity and monotone functions;
6. continuity on connected, compact sets.
7. Series of numbers in R or C; tests for convergence; power series; absolute convergence and rearrangements.
8. differentiation of functions of one variable: derivative; differentiation rules; Mean value theorem and consequences; Taylor's
9. theorem in 1 variable
10. integration of functions of one variable: Riemann integral;  conditions for integrability;  properties of the integral; fundamental theorem of calculus; rectifiable curves

M 518

1. sequences and series of functions: pointwise, uniform convergence; uniform convergence and continuity, integrability, differentiability; Weierstrass M-test; the B-space C(K); Weierstrass approx theorem; Stone-Weierstrass theorem
2. differentiation in Rn: derivative as a linear tranformation;partial derivatives; Jacobian and Hessian matrices; Taylor's theorem; conditions for differentiability and consequences of differentiability
3. Inverse and implicit function theorems
4. Riemann integration in Rn: proerties of the integral;sets of measure and content zero;Lebesgue's necessary and sufficient condtions for integrability; multiple and iterated integrals; Fubini's theorem; change of variable
5. vector analysis:  vector differential calculus; divergence, gradient, curl; vector integral calculus; integral identities and integral theorems of Green, Gauss and Stokes.

November 1998