LEARNING OUTCOMES

The knowledge of the mathematical tools introduced and the skill of using them to solve problems eventually in a physical setting. The skill of  setting up a problem correctly and the development of intuitive and deductive skills in solving problems.

AIMS AND LEARNING OUTCOMES

The knowledge of mathematical basic tools useful in physical problems modelling.  The skill of setting up and solving problems by using intuitive and deductive reasoning as well as recognizing and using the suitable mathematical tools in solving problems in a physical setting. At the end of the course the student will be able

1. to state the concepts ( theorems and definitions ) introduced during the course ( f.i. level set, partial derivatives, optimization, line integral, integral in R^2 and R^3 );

2. to give physical and geometric interpretation of the basic concepts of Mathematical Analysis;

3. to select the suitable mathematical tools in problem solving;

4. to solve problems with deductive reasoning.

TEACHING METHODS

The course consists of 36 hours of lectures and 24 hours of practices. In the lectures the topics of the syllabus are explained with definitions and theorems and some proofs which can be useful for the comprehension of the topics and to develop the logical and deductive skills. Every theoretical topic is explained with easy examples and some exercises. In the practices, many exercises are solved with the aim of going into the knowledge of theoretical topics treated in the lectures and preparing the student for the exam. Some guided practices will be held to help the student to valuate one's preparation. Several intermediate tests are provided.

Students  have several exercises at their disposal on Aulaweb.

SYLLABUS/CONTENT

Euclidean spaces. Toplology in R^n. Functions of several variables. Level sets. Continuity and differentiability. Directional and partial derivatives. Derivatives of higher order. Schwartz Theorem. Taylor approximation with Peano's and Lagrange's reminder. Quadratic forms. Unconstrained optimization. Necessary first order condition and sufficient second order condition. Implicit function theorem. Change of coordinates. Constrained optimization.

Systems of nonlinear differential equations. Cauchy problem. Existence and uniqueness of the solution.

Double and triple integrals. Normal domains in R^2. Integration formulas and theorem of change of variables. 

Lines in R^n and line integral.  

Vector fields. Irrotational and conservative vector fields. Gauss - Green formulas and Divergence Theorem in R^2.

RECOMMENDED READING/BIBLIOGRAPHY

C. Canuto, A. Tabacco, "Analisi Matematica II", Springer, 2014.

M. Bramanti, C. Pagani, S. Salsa. “Analisi matematica 2”, Zanichelli, 2009.

S. Salsa, A. Squellati. “Esercizi di Analisi matematica 2”, Zanichelli 2011.