Teaching code

4S00031

Teacher

Franco Zivcovich

Credits

6

Also offered in courses:

• Mathematical analysis [Matricole dispari] - Analysis II of the course Bachelor's degree in Computer Science

Language

Italian

Scientific Disciplinary Sector (SSD)

MAT/05 - MATHEMATICAL ANALYSIS

Period

Semester 2 dal Mar 3, 2025 al Jun 13, 2025.

Courses Single

Authorized

Lessons timetable Moodle Seminars 0

Learning objectives

The aim of the course is to provide students with the fundamental notions of differential and integral calculus in many variables, generalizing and mastering the notions learned in the course “Mathematical Analysis I” and employing, if needed, the notions of the other courses attended during the first year of the Bachelor in Computer Science. At the end of the course the student must prove: - to know and to be able to understand the tools and the advanced notions of the mathematical analysis and to use such notions for the solution of problems; - to be able to use the notions learned in the course for the comprehension of the topics of further courses, not necessarily in the mathematical area, where the knowledge of mathematical analysis can be a prerequisite; - to be able to choose which mathematical tool or theoretical result can be useful for the solution of a problem; - to be able to appropriately use the language and the formalism of the mathematical analysis; - to be able to broaden the knowledge in Mathematics, Computer Science or in any scientific area using, when needed, the notions of the course.

Prerequisites and basic notions

Basic notions of the courses of Analysis I and Linear Algebra.

Program

Introduction to ordinary differential equations.
- First-order linear equations. Cauchy problem and existence and uniqueness theorem of the solution. Equations with separable variables. Direction field and graphical analysis of a differential equation in simple cases. Second-order linear equations, homogeneous and non-homogeneous: structure of the set of solutions; linear equations with constant coefficients and similarity method; Wronskian and method of variation of arbitrary constants.
Differential calculus in multiple variables.
- Limits and continuity for functions of multiple variables, level curves.
- Directional derivatives and differential of functions of multiple variables, total differential theorem, gradient of scalar functions.
- Higher-order derivatives, Hessian matrix, Schwarz theorem, Taylor expansion.
- Optimization problems for functions of multiple variables. Critical points, free optimization, study of the Hessian matrix to determine free relative maxima and minima.
- Constrained optimization, Lagrange multipliers, Dini's theorem, inverse function theorem.
Integral calculus in multiple variables.
- Multiple integrals for continuous functions defined on n-dimensional rectangles. Fubini's theorem. Formula for the change of variables in double and triple integrals.
- Curvilinear integral of the first kind. Regular surfaces and surface integrals of the first kind, Gauss-Green formula.
- Curvilinear integral of the second kind, conservative vector fields.

Didactic methods

Lectures, classroom exercises. Multimedia material available on the course e-learning pages.

Learning assessment procedures

The final exam consists of a written test including a series of exercises to be solved related to the program covered (specific instructions will be communicated during the course). The final test may be replaced by two in itinere tests, the second coinciding with the first available appeal.

Evaluation criteria

The exam aims to verify the ability to solve problems on the course program, the possession of an adequate capacity for analysis, synthesis and abstraction, starting from requests formulated in natural language or in specific language.

Exam language

Italiano