Corso di laurea in matematica

The course aims at introducing the basic notions about the differential and integral calculus for functions of several variables, completing what has already been seen in the first year course of Analysis. We introduce metric spaces, sequences and series of functions and treat qualitative theory of first order ordinary differential equations, illustrating some  solving methods. We define rigorously and analyze  geometric entities such as scalar and vector fields, areas, surfaces and volumes. 

KNOWLEDGE AND UNDERSTANDING

Students must know:

• the notions of continuity, directional derivatives and differentiability of functions in several variables and their relationships;

• the theory of curves in space;

• the main notions on differential 1-forms and understand their parallelism with vector field theory and its physical applications;
• the Gauss-Green theorem in the plane;

• the Banach-Caccioppoli contraction theorem and recognize its role in the arguments subsequently presented;

• the various types of convergence for sequences and series of functions;

• the implicit function theorem, the local invertibility theorem and the Lagrange multiplier theorem;

• the fundamental theorems on the Cauchy problem for differential equations;

• the theory of Riemann integration in several variables and the main physical interpretations of double and triple integrals. 

APPLICATION OF KNOWLEDGE AND UNDERSTANDING

Students must be able to:

• integrate 1-forms along curves and knowing how to establish parallelism with the theory of vector fields;
• apply the Gauss-Green Theorem in the plane;

• study the various types of convergence for a sequence and for a series of functions;

• to apply the implicit function theorem, the local invertibility theorem and the Lagrange multiplier theorem; 

• discuss the qualitative properties of the solutions of a differential equation;

• solve some types of differential equations;

• calculate double and triple integrals.

COMMUNICATION SKILLS

Students must be able to explain the main concepts and theorems of the course and the relative proofs with clarity and rigor, showing that he has understood the main steps and the possibilities of application.

• Limits and continuity for functions of several variables; differential calculus for vector fields. Chain rule and Taylor Formula;
• curves; integration along curve;
• 1-forms and their integration; Gauss-Green's Theorem in the plane;
• Metric spaces, sequences and series of functions, powers series, analytic functions, completeness of continuous functions on a compact. Banach Fixed Point Theorem. Normed and Banach spaces;
• Implicit function Theorem; Local Inversion Theorem; Lagrange multipliers Theorem;
• Differential equations: Cauchy problem, local and global existence, qualitative studies. Solving methods;
• Multiple integrals:  definition and computation, reduction formula; change of variables.

The course includes 104 hours of attendance divided between theoretical lectures and exercises and will be accompanied by a weekly tutoring. 

The exam consists of a written test consisting in solving some exercises and an oral test that focuses mainly on theoretical topics. The two tests must be taken in the same session. To access the oral exam it is necessary to obtain a score of at least 18/30 in the written test.

A weekly tutoring is planned to support students in carrying out the exercises.