Mathematics for the Environment (I MOD)
Expected learning outcomes
The course aims to provide basic knowledge concerning the theory of functions of several real variables (continuity, differentiability, integrability), unconstrained and constrained optimization problems, ordinary differential equations. The student will be able to use the studied methods for modeling and solving economic and environmental problems.
Contents of the course of Introduction to Mathematics of the first year.
FUNCTIONS OF TWO OR SEVERAL VARIABLES (8 hours)
Graphs and domain of functions of two variables; limits and continuity; partial derivatives, directional derivatives.
QUADRATIC FORMS (8 hours)
Definiteness of a quadratic form. Principal minors and leading principal minors. Criterion for the definiteness of a quadratic form.
UNCONSTRAINED AND CONSTRAINED OPTIMIZATION (16 hours)
Local and Global Maxima and Minima. Concave and convex functions. Constrained maxima and minima with Laplace multipliers. Applications.
INTEGRATION (16 hours)
Definite and indefinite integrals. Double integrals.
CONTINUOUS MODELS (24 hours)
First and second order linear differential equations. Applications: population growth models; predator-prey systems, competition and cooperation; epidemic model; heat loss of a body; the oxygen debt; passage of a substance across a membrane; some applications in Physic.
P.Marcellini, C.Sbordone: Esercitazioni di Matematica 2, parte I e II, Liguori Editore, Napoli, 1991.
C.P. Simon - L.E. Blume, Mathematics for Economists