2008 Math Camp Part II
Course Information
July 28 - August 8, 2008
10:00 AM - 12:00 Noon and 1:30 PM - 3:30 PM
Steinberg Hall-Dietrich Hall (SH-DH) 1206
Instructor Office: McNeil #317
Lecture Notes
0. Four Motivating Examples
4 Economic Examples: Cournot Equilibrium, Stackeberg Equilibrium, Job Search Model, First-Price Auction
1. Topology of Rn
Very Basic(Metric space, Normed linear space), Open and Closed Sets, Continuous Functions, Separability, Compactness, Connectedness, Completeness and Baire's Theorem.
2. Review of Linear Algebra
Vector Spaces, Basis and Dimension, Linear Maps, Isomorphism, Theorem of Image and Kernel and Determinant.
3. Differentiation in Rn
Differentiability, Partial and directional derivatives, Mean Value Theorem, Schwarz's Theorem, Higher Order Derivatives, Inverse Function Theorem, Implicit Function Theorem.
4. Riemann Integration
Definition, Measure Zero, Lebesque's Theorem, Fubini's Theorem, Change of Variables.
5. Sard's Theorem and Transversality Theorem
The most simplified version. For more general treatment, look at Eduardo's lecture note.
6. Some Probability (Measure) Theory
Probability Space and Random Variables, Multidimensional distributions, Independence, Change of Variable, Conditional Expectation
Appendix: Some previous lecture notes by Eduardo Faingold
Implicit Function Theorem, Inverse Function Theorem, Sard's Theorem and Transversality Theorem
Homework
Problem Set 1 Suggested Solution
Problem Set 2 Suggested Solution