This course aims to illustrate the fundamentals of differential and integral calculus for functions of several variables, of the theory of numerical and functions series, of ordinary differential equations, as well as to introduce students to modeling and solving simple problems of practical interest which exploit these mathematical tools. D1 - Knowledge and understanding skills At the end of the course, the student should know the fundamentals of differential and integral calculus for functions of several variables, of numerical and function series, of ordinary differential equations. D2 - Applying knowledge and understanding The student should be able to deal with and solve simple exercises, questions, problems, of theoretical and computational nature, related to the topics of the course. D3 - Making judgements The student should be able to describe, model and solve simple problems of interest for applications, by using the mathematical tools developed during the course. D4 - Communication skills The student should be able to describe mathematical topics with an adequate command of language, as well as to translate practical problems in mathematical terms. D5 - Learning skills The student should be able to read and understand books and articles, by using the mathematical tools learned in the course, and be able to learn more advanced ones.

Differential and integral calculus for functions of one real variable. Linear algebra and analitical geometry.

Euclidean spaces: linear, metric, topological structures.
Differential calculus in R^N: directional and partial derivatives, differentiation, differentiation rules, Taylor formula and applications, implicit function theorem and applications.
Numerical series: partial sums and series, relationships with integrals, convergence tests.
Sequences and series of functions: pointwise and uniform convergence, term by term differentiation and integration, power series, Taylor series expansion, elementary functions of complex variable.
Integral calculus in R^N: integral on N-rectangles and their properties, measure in R^N, integral on bounded sets, reduction formulas, change of variables, generalized integrals.
Curves and surfaces: curves and surfaces in parametric or implicit form, length and area, line and surface integrals of a scalar field.
Vector calculus: vector fields, line and surface integrals of a vector field, curl and divergence, conservative fields, curl theorem, divergence theorem.
Differential equations: differential equations and mathematical modeling, Cauchy problem for ordinary differential equations, resolution by integration versus qualitative study, equations and systems, linear differential equations.

Lecture notes from the teacher.
C. D. Pagani, S. Salsa, Analisi matematica, vol. 2, Masson, Milano, 1991.

Euclidean spaces: linear, metric, topological structures.
Differential calculus in R^N: directional and partial derivatives, differentiation, differentiation rules, Taylor formula and applications, implicit function theorem and applications.
Numerical series: partial sums and series, relationships with integrals, convergence tests.
Sequences and series of functions: pointwise and uniform convergence, term by term differentiation and integration, power series, Taylor series expansion, elementary functions of complex variable.
Integral calculus in R^N: integral on N-rectangles and their properties, measure in R^N, integral on bounded sets, reduction formulas, change of variables, generalized integrals.
Curves and surfaces: curves and surfaces in parametric or implicit form, length and area, line and surface integrals of a scalar field.
Vector calculus: vector fields, line and surface integrals of a vector field, curl and divergence, conservative fields, curl theorem, divergence theorem.
Differential equations: differential equations and mathematical modeling, Cauchy problem for ordinary differential equations, resolution by integration versus qualitative study, equations and systems, linear differential equations.

Lectures and classroom exercises. Tutorials.
Possible changes due to the COVID19 emergency will be communicated via the institutional websites and via Moodle.

Further information and classroom notes will be available in Moodle http://moodle2.units.it.

Written or oral examination both on the practical part (exercises) and on the theoretical part (definitions, statements and proofs of theorems). Further details in Moodle http://moodle2.units.it
Possible changes due to the COVID19 emergency will be communicated via the institutional websites and via Moodle.