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Economics Graduate Program :
Math Review Course : Syllabus, Summer 2005

• Sundaram, Rangarajan, A First Course in Optimization Theory, Cambridge University Press, 1996.

• Rudin, Walter, Principles of Mathematical Analysis, Third Edition, McGraw-Hill, 1976.
• Bartle, Robert, Elements of Mathematical Analysis, John Wiley & Sons, 1976.
• Suppes, Patrick, Axiomatic Set Theory, Dover, 1972.
• Simon, Carl, and Lawrence Blume, Mathematics for Economists, W. W. Norton, 1994.

1. Set Theory and Logic (Sundaram, Appendix A)
   1. Sets, Unions, Intersections
   2. Propositions: Contrapositives and Converses
   3. Quantifiers and Negation
   4. Necessary and Sufficient Conditions


2. Sequence (Sundaram, 1.1, 1.2.1-1.2.6)
   1. Notation and Definition
   2. Inner Product, Norm and Metric
   3. Sequences and Limits
   4. Subsequences and Limit Points
   5. Cauchy Sequences and Completeness
   6. Suprema, Infima, Maxima, Minima
   7. Monotone Sequences in R
   8. The Lim Sup and Lim Inf


3. Basic Topology (Sundaram, 1.2.7-1.2.10)
   1. Open Balls, Open Sets, Closed Sets
   2. Compact Sets
   3. Connected Sets
   4. Convex Combinations and Convex Sets
   5. Union, Intersection, and Other Binary Operations


4. Continuity (Sundaram 1.4.1 & Rudin Ch. 4)
   1. Continuous Functions
   2. Continuity and Compactness
   3. Continuity and Connectedness


5. Differentiation (Sundaram 1.4.2-1.4.5 & Rudin Ch. 5)
   1. Differentiation of Functions from R to R
   2. Mean Value Theorems
   3. Higher Order Derivatives and Taylor’s Theorem
   4. Partial Derivatives and Directional Derivatives
   5. Differentiation of Functions from R^m to Rⁿ


6. Integral (Rudin Ch. 6)
   1. Definition and Existence of the Integral
   2. Properties of the Integral
   3. Integration and Differentiation


7. Matrix Algebra (Sundaram 1.3)
   1. Sum, Product, Transpose
   2. Some Important Classes of Matrices
   3. Rank of a Matrix
   4. The Determinant
   5. The Inverse
   6. Calculating the Determinant


8. Eigenvalue and Eigenvector (Simon and Blume Ch. 23)
   1. Definitions and Examples
   2. Linear Difference Equation
   3. Properties


9. Quadratic Forms (Sundaram 1.5)
   1. Quadratic Forms and Definiteness
   2. Identifying Definiteness and Semidefiniteness

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