Università Roma Tre

Summary of real functions of a real variable. Main properties and definitions, elementary functions, derivatives and derivation rules. Second order derivatives, chain rule, monotonicity and first derivative, second order derivative and convexity NSC for the existence of local maxima and minima. Global max and min. Weierstrass theorem.
Topology and metrics of the n-dimensional real space. Sequences. Open sets. Closed sets. Compact sets.
Functions of several variables. Definition and examples. Level curves. Linear functions and quadratic forms. Continuous functions and Weierstrass theorem.
Differential calculation for several variables functions. Partial derivatives. Differential and tangent plane. Gradient and Jacobian matrix. Approximations by differentials. Darivative along a curve. Directional derivative. Chain rule. Second order derivatives and Hessian matrix.
Implicit function theorem: Case of a function in two variables. Geometric interpretation. Case of a function in several variables. Linear case: m equations, n+m variables. General case of systems of m equations and n+m variables. Inverse function theorem.
Static optimization. Free optimization. NC for the existence of local maxima and minima. Stationary points. Definite and semidefinite quadratic forms. Criterion for 2 or 3 variables. Second order SC for the existence of local free maxima and minima. Constrained optimization. Equality constraints. NC for the existence of local maxima and minima. Inequality constraints: NC for the existence of local maxima and minima. Kuhn-Tucker conditions for non-negative variables. Interpretation of the Lagrange multiplier.
Homogeneous functions. Geometric definition and properties. Euler theorem. Homothetical functions
Concave and convex functions. Definitions and geometrical properties. Properties and characterization. Concavity and sign of the Hessian matrix. Quasiconcave and quasiconvex functions. Cobb-Douglas functions. Pseudoconcave functions Free and constrained optimization in hypothesis of quasiconvexity and quasiconcavity.
Dynamical models: Malthus model in discrete and continuous time. Dynamical models in continuous and discrete time. Definition of differential equation and difference equations. Existence and uniqueness of solutions for first order difference equations.
Calculus of variations. Definition of a problem of calculation of variations. Euler equation in integral form (w.p.). Euler-Lagrange equation. Transversality conditions. Sufficient conditions in hypothesis of concavity or convexity.
Optimal control theory: Optimal control problem (in discrete and continuous time). Pontryagin maximum principle. Sufficient conditions in hypothesis of concavity.
Dynamic programming. Finite horizon and discrete time: Bellman’s principle. Dynamic programming principle. Bellman's equation and optimization conditions.
(w.p.)=”with proof”.

Textbooks
• Simon & Blume “Matematica per le scienze economiche” ed Egea.
• Salsa & Squellati “Modelli dinamici e controllo ottimo” ed. Egea: chapter 1 (pp.1-6, 17-26), chapter 3 (pp.69-73, 78-79) , chapter 9 (pp. 211-232), chapter 10 (pp. 245-249, 277-288); chapter 11 (pp. 293-296, 304-310).