MATH 1130, Fall 2007. Analysis: Functions of Several Variables

Instructor: Prof. Sergei Treil,
office: 216 Kassar,         ph.: x1122
e-mail: treil<at>math.brown.edu
Web: http://www.mth.brown.edu/treil

Office hours: MWF 2:00-3:00 pm and by appointment.

Syllabus.

Outline of the course and homework assignments.

Solutions

Solutions for the homework 1. 

Solutions for the test 1, take-home part. 

Solutions for the test 2.     

Lecture Notes taken by students (pdf format).

9/7: Relations, function
9/10: Cardinality. Countable sets.
9/12: Cardinality. Cantor-Bernstein theorem. Continuum
9/14: Real numbers.
9/17: Review of sequences. Limsup, liminf. Cauchy criterion. 
9/19: Metric spaces. Completeness of R^d. Open and closed sets.  
9/21: Interior, exterior, closure and boundary. Properties of open sets.
9/24: Topological spaces. Continuity     
9/26: Base of topology. Relative (inherited) topology.   
9/28: Connected sets. Continuous induction.   
10/1. Limits, continuity and sequences. Hausdorff spaces.   
10/3. Compactness and sequential compactness.  
10/5. Product topology and compactness.    
10/10. Review. Definition of derivatives.   
10/15. Review of linear algebra. Norms in R^n. Jacobi matrix.  
10/17. Partial derivatives an differentiability.   
10/19 Matrix norms. Chain rule.   
10/22 Mean Value Theorem.    
10/24 Taylor's formula.
10/26 Sufficient condition of extremum. Inverse function and implicit function theorems    
10/29 Proof of implicit function and inverse function theorems.    
10/31. Manifolds in R^N   
11/2. Manifold in R^N.  Tangent spaces.   
11/5. Measure Theory. Main definitions.  
11/7. Countable additivity of length. Outer measures.   
11/9. Outer measures.  
11/12. Caratheodory Theorem.  
11/14. Applications of Caratheodory theorem.  
11/16. Uniqueness of extension. Complete measures. Borel sigma algebra.     
11/19. Uniqueness of translation invariant Borel measure. Hausdorff measures.  
11/26. Hausdorff dimension. Metric outer measure.     
11/28. Metric outer measures. Example of a non-measurable set.     
11/30. Measurable functions.   
12/03. Lebesgue integral.   
12.05. Monotone convergence theorem. Integration of non-positive functions.