The aim of the course is to provide basic knowledge concerning the theory of functions of several real variables (continuity, differentiability, integrability), of ordinary differential equations and their applications to concrete problems. Eventually, the student will have to prove to have understood both theoretical and applicative parts of the course and to be able to use those methods for modeling and solving economic, financial and corporate problems.
Expected learning outcomes
Knowledge and understanding: The student should be able to prove knowledge of mathematical tools and ability to identify those suitable for modeling and solving economic, financial and corporate issues.
Applying knowledge and understanding: the student should be able to apply the acquired knowledge in solving the main problems regarding the study of functions of several variables. This will involve the ability to identify appropriate theoretical tools suitable to the particular problem under study by applying correctly the tools of the infinitesimal calculus.
Making judgements: the student should be able to use the acquired knowledge also in an autonomous way, by also applying them to specific issues related to economic problems.
Communication: the student should be able to answer in a clear and thorough way to the questions of the written examination and to those of the oral examination.
Lifelong learning skills: the student should be able to show a good learning ability, by widening, for example, his/her knowledge with use of relevant bibliographic references.
Contents of the course of Calculus I
Functions of several variables: Limits and Continuity. Partial Derivatives. Higher Order Derivatives: Schwarz Theorem. Gradient vector and Differentiability. Directional Derivatives. Quadratic Forms. Convex Functions. Local and Global Maxima and Minima. Constrained Maximization Problems. (24 hours)
Integration: definite and indefinite integrals; table of integrals ; integration by parts, integrations by substitution (4 hours)
Ordinary Differential Equations: Existence and Uniqueness: the Cauchy problem, Linear First Order Equations, Linear Second Order Equations, Separable Equations, Bernoulli Equations. Systems of Differential Equations. Economic applications (24+8 hours)
Double integrals: normal domains ; change of variable (8 hours)
Curves: introduction. Tangent vector to a Curve. Orientation of a curve. Rectifiable curve: length of a curve. Integration on curves. (4 hours)
The course includes frontal lessons, during which the themes of the program are discussed, and exercises in attendance.
• N.Fusco, P.Marcellini, C.Sbordone: Lezioni di Analisi Matematica due, Zanichelli Editore, 2020.
• Paolo Marcellini Carlo Sbordone; ESERCITAZIONI DI ANALISI MATEMATICA DUE – prima parte e seconda parte – Zanichelli editore
• C.P. Simon - L.E. Blume, Mathematics for Economists
The assessment is based on written examination (the use of notes, books and informatics devices -smartphone, tablet, pc, ecc.- is not allowed) 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. The laude can be assigned if the student shows, in his/her answers, a particular ability to deepen the topics mentioned in the examination’s questions.