Topics in Mathematical Analysis
Server-Angel Popescu

Topics:

Continuity and Uniform Continuity

1. The topology of a metric space. R^n as a R-normed space. Two basic metrics on C[a,b]. The interior, the closure and the boundary of a set.

2. Compact sets. The characterization of a compact subset in R^n.

3. Continuity and uniform continuity for metric spaces. Continuous functions on a compact metric space.

4. Connected open subsets of R^n. Continuous functions on a connected metric space.

5. Complete metric spaces. The fixed point Banach theorem.

Sequences and Series of Functions. Power Series

1. Simple and uniform convergence for a sequence of functions. Uniform convergent sequences of convergent functions (continuity, derivability and integrability).

2. Power series. Abel lemma. The computation of the convergence set.

3. Power series in a normed algebra.

Local Inversion Theorem and Implicit Functions

1. The local inversion theorem.

2. Implicit functions and applications.

3. Local extrema. Extrema with restrictions. Lagrange's method

Basic Topics in Integral Calculus

1. Some convergence criteria for generalized integrals.

2. Integrals which depend on a parameter. Uniform convergence.

3. Path integrals.

4. Double integrals. Change of variables.

5. Triple integrals. Change of variables.

6. Surface integrals.

7. Green-Riemann, Stokes and Gauss's formulas.