Syllabus for the Qualifying Examination in Basic Analysis

1. Metric Spaces
   1. Convergence of sequences in metric spaces
   2. Cauchy sequences
   3. completeness
   4. contraction principle
2. Topological spaces
   1. continuous maps
   2. Hausdorff spaces
   3. compactness
   4. connectedness
3. The real numbers
   1. The real numbers as a complete ordered field
   2. closed bounded subsets are compact
   3. intermediate value theorem
   4. maxima and minima for continuous functions on a compact set
4. Sequences and series of complex numbers
   1. standard tests for convergence and divergence of series
   2. absolute convergence and rearrangements
5. Differentiation
   1. Differentiation of a function in one real variable
   2. Mean Value Theorem
   3. L'Hopital's Rule
   4. Taylor's Theorem with error estimates
6. Riemann integration of functions in one real variable
   1. Definition
   2. Riemann integrable functions
   3. integration and anti-differentiation
7. Sequences and series of functions
   1. power series and radii of convergence
   2. uniform convergence of sequences of functions
   3. uniform convergence and integration
   4. integration and differentiation of power series
8. Differential Calculus for functions from n-space to reals and reals to n-space
   1. Parametrized curves
   2. tangent vectors
   3. velocity
   4. acceleration
   5. partial derivatives
   6. directional derivatives
   7. the gradient
   8. the chain rule
   9. Taylor's theorem
   10. local maxima and minima
   11. level surfaces of functions
   12. tangent planes to surfaces in 3-space
   13. Lagrange multipliers
9. Differential Calculus for functions from n-space to m-space
   1. notion of derivative
   2. affine function with best approximates a differentiable function at a point
   3. chain rule
   4. inverse function theorem
   5. implicit function theorem
10. Integral Calculus in several variables
    1. The integral, path and surface integrals
    2. Green's theorem in the plane
    3. the divergence theorem in 3-space
    4. the change of variables formula

    References:

    The first nine chapters of Principles of Mathematical Analysis, 3rd edition, by Walter Rudin.

    Other useful references:
    • Wendell Fleming: Functions of Several Variables
    • Kennan T. Smith: Primer of Modern Analysis
    • Harold Edwards: Advance Calculus of Several Variables
    • Marsden and Hoffman: Elementary Classical Analysis