MATHEMATICS 2 M - Z

The training objectives of the course are as follows:

Knowledge and understanding:

the student will learn some basic mathematical concepts and will develop the skills of calculation and manipulation of the most common objects of Mathematical Analysis: among these, the integrals for functions of one and more real variables, the numerical series and functions, the differential equations and the differential calculus for real functions of two real variables.

Applying knowledge and understanding:

through examples related to applied sciences, the student will be able to appreciate the importance of Mathematical Analysis in the scientific field and not only as a discipline for its own sake, thus broadening their cultural horizons.

Making judgments:

the student will be able to deal with some simple but significant methods of mathematical analysis with sufficient rigor to refine their logical skills. Many demonstrations will be presented in a schematic and intuitive way to engage students and encourage them to achieve the goal on their own.

Communication skills:

by studying Mathematical Analysis, and putting himself to the test through guided exercises, the student will learn to communicate with rigor and clarity both orally and in writing. He will learn that using correct language is one of the most important means of clearly communicating scientific topics, not only in mathematics.

Learning skills:

students, especially the most willing, will be stimulated to deepen some topics, also through group work.

Lectures accompanied by exercises

If the teaching is given in a mixed or remote way, the necessary changes may be introduced with respect to what was previously stated, in order to respect the program envisaged and reported in the syllabus.

Exams may take place online, depending on circumstances.

Integral calculus for functions of one variable

Indefinite integral - Integration methods: integration by decomposition and sum, integration of functions rationals, integration by parts, integration by substitution - Definition of integral according to Riemann and its properties - Some classes of integrable functions - Definite integrals - Hints of theory of the Peano-Jordan measure - Geometric meaning of the Riemann integral - Fundamental theorem of integral calculus - Hints on generalized and improper integrals and their properties.

Numerical series and series of functions

Numerical series - General theorems on numerical series - Various examples of series - Convergence criteria of series with constant sign terms - Absolutely convergent series - Alternate series and Leibnitz criterion - Series of functions - Pointwise, uniform convergence and total - Taylor and Mac Laurin series - Mac Laurin development of some elementary functions.

Differential and integral calculus for functions of two or more variables

Recall of topology in the plane: internal points, external points and boundary points, open and closed sets, accumulation points and isolated points, bounded sets, compact sets, convex sets, connected sets by arcs, domain - Functions of several variables: limits and continuity - Differential calculus for functions of several variables: partial and directional derivative - Differential and differentiable functions - Higher order derivatives and Schwartz lemma - Differential operators: gradient, divergence, rotor, Laplacian - Differentiation theorem of compound functions - Lagrange's theorem in R² and characterization of functions with zero gradient in a region - Free extrema and a function of two variables and relative theorems - Search for absolute extremes on a compact set - Integral calculus for functions of several variables: double and triple integrals according to Riemann - Change of variables - Reduction formulas: Fubini's theorem - Integrals dependent on a parameter: Leibinz rule.

Ordinary differential equations

General information on differential equations - The Cauchy problem - First order differential equations - First order differential equations with separable variables - Cauchy's theorem on the existence and uniqueness of the solution - Second order linear differential equations with constant coefficients - Applications to the study of free, damped and forced oscillations.

Some recommended texts:

- G. Di Fazio, P. Zamboni, Analisi Matematica uno e due, ed. Monduzzi

- M. Bramanti, C. D. Pagani, S. Salsa: Matematica - Analisi Matematica 1 e 2, ed. Zanichelli

- S. Salsa, A. Squellati: Esercizi di Analisi Matematica 1 e 2, ed. Zanichelli