Mathematics for the Environment (II MOD)
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.
Expected learning outcomes
Knowledge and understanding
The student will demonstrate that he has acquired the tools of mathematical analysis and has to be able to apply them to model and solve real problems.
Applying Knowledge and understanding
The student should be able to apply the acquired mathematical techniques to real problems, especially to optimal management and planning and, in particular, he has to be be able to solve optimization problems.
The student will demonstrate that he is able to translate into mathematical terms a problem that describes a real phenomenon.
The student will demonstrate his ability to expose, with a certain rigor, the knowledge acquired, answering in a exhaustive way to the questions of the oral examination.
Lifelong learning skills
The student should be able to show a good learning ability of the acquired methodologies and he should know how to use mathematical tools to solve application problems.
According the regulations of the Course of the Studies , it is necessary to have taken and passed the exam of Introduction to Mathematics (of the first year). It is, in fact, essential to have acquired basic knowledge such as: elementary functions (in particular linear function, power, exponential and logarithmic), analysis of a real function of a real variable, calculation of derivatives of a function of a variable and rules of derivation, definition of indefinite and definite integral, elements of linear algebra (in particular calculation of the determinant of a square matrix, product between matrices and vectors).
FUNCTIONS OF TWO OR SEVERAL VARIABLES (24 hours)
Functions of two or several variables. Graphs and domain of functions of two variables; level curves. Limits and continuity. Partial derivatives. Hessian matrix. Schwarz's theorem. Differentiability and continuity. Sufficient condition for differentiability. Regular curve. Directional derivatives.
UNCONSTRAINED AND CONSTRAINED OPTIMIZATION (24 hours)
Local and Global Maxima and Minima. Concave and convex functions. Constrained maxima and minima with Laplace multipliers. Applications.
Constrained maximums and minimums: Lagrange multiplier method. Conditions necessary for determining the optimum. Quadratic forms. Sufficient conditions.
INTEGRATION AND CONTINUOUS MODELS (24 hours)
Definite and indefinite integrals. Double integrals.
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.
The course includes frontal lessons, during which the themes of the program are discussed, and exercises in attendance.
P.Marcellini, C.Sbordone: Esercitazioni di Matematica 2, parte I e II, Liguori Editore, Napoli, 1991.
C.P. Simon, L.E. Blume, Matematica 2 per l'economia e le scienze sociali, Università Bocconi editore, 2002.
- Slides of the lessons on platform elearning of Ateneo “Parthenope”
The assessment is based on written examination (duration 90minutes) and an oral interview.
The written test is composed of exercises in order to assess the achievement by the student of the learning objectives. The oral exam focuses on the theoretical topics dealt with during the course and it is designed to evaluate the student's ability to express and formalize mathematical concepts. The vote of the examination is expressed in scale from 0 to 30, and it is results of written and oral examination.
During the examination, the use of notes, books and informatics devices (smartphone, tablet, pc, ecc.) is not allowed.