The course aims at providing the rigorous mathematical framework needed to understand and use quantitative economic analysis models and computational methods. Specifically, after reviewing the necessary topics of linear algebra and multivariable calculus, the course deals with optimization, measure theory, dynamical systems, and linear partial differential equations.
Prerequisiti
In order to fruitfully follow this course, the students should be familiar with the basic notions of linear algebra (vector spaces, matrix algebra, linear applications) and single-variable calculus (e.g., limits, derivatives, and integrals).
Metodi didattici
The course is composed by lectures (44 hours/year) and exercise sessions (22 hours/year). Both sections are taught at the blackboard. Lecture notes, in pdf format, are available to students.
Verifica Apprendimento
The final examination is a written test, where students are requested to solve exercises and to answer short theory questions. There is no oral examination. Books and calculators will not be allowed in the exam.
Testi
There is no prescribed textbook, lecture notes will be provided by the lecturer during the course. The following textbooks may be useful: _Gilbert Strang. Introduction to Linear Algebra. Wellesley-Cambridge Press, 1993. _G. F. Simmons, Calculus with analytic geometry. McGraw-Hill, 1996. _T. Mikosch, Elementary Stochastic Calculus with Finance in View. World Scientific, 1998. _M.W. Hirsch and S. Smale, Differential equations, Dynamical systems, and Linear algebra. Academic Press, 1974. _L. C. Evans, Partial Differential Equations. American Mathematical Society, 2002.
Contenuti
1. Linear algebra. (a) Eigenvalues, eigenvectors, diagonalization. (b) Spectral Theorem, bilinear forms. 2. Multivariable calculus. (a) Partial derivatives, gradient, differential, directional derivatives. (b) Hessian, Taylor formulas, chain rule. (c) Multiple integrals. 3. Optimization. (a) Unconstrained optimization. (b) Parametric curves and surfaces. (c) Level sets and Implicit function Theorem. (d) Constrained minimization: Lagrange multipliers. 4. Integration theory. (a) Riemann integral and Riemann-Stieltjes integral. (b) Multiple integrals. Reduction formula on rectangles. Change of variables. 5. Dynamical systems. (a) Ordinary differential equations. Existence and uniqueness theorem. Linear equations. (b) Systems of linear equations with constant coefficients. Equilibrium and stability. 6. Partial Differential Equations. (a) Introduction to PDEs. Classification and examples. (b) The transport equation. (c) The heat equation in R. Fundamental solution and Cauchy Problem.
