Mathematics II A - L

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, sequences and series of functions, the differential equations and the differential calculus for real functions of two real variables, dlinear differential forms.

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.

Solid knowledge of the contents acquired in the course of Mathematics I

 

Elements of linear algebra

Vector spaces, matrices and determinants (Laplace's theorem) – Linear systems

(Kramer and Rouché – Capelli theorem) – Linear applications – Eigenvectors and eigenvalues ​​of a matrix – Secular equation - Diagonalization of a matrix.

Numerical series, sequences and series of functions

Numerical series - General theorems on numerical series - Various examples of series - Convergence criteria of series with constant sign - Alternate series and Leibnitz criterion - Absolutely convergent series - Sequences of functions: pointwise and uniform convergence - Continuity theorems , on integration and derivation - Series of functions: absolute and total convergence - Power series: radius of convergence - Taylor and Mac Laurin series - Mac Laurin development of some elementary functions.

Differential and integral calculus for functions of two or more variables

Recalls of topology in the plane: internal, external, boundary, accumulation points, isolates, open and closed sets, bounded sets, compact sets, convex sets, connected sets, domains - Functions of several variables: limits and continuity - Differential calculus for functions of several variables: partial and directional derivative - Total differential - Differentiable functions and n-dimensional tangent hyperplane - Higher order derivatives and Schwarz's lemma - Differential operators: gradient, divergence, rotor, Laplacian - Theorem of derivation of compound functions - Theorem of Lagrange in R2 and characterization of functions with zero gradient - Taylor's formula for functions of two variables - Critical points, maximums and minimums relative to a function of two variables - 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 - Integrals dependent on a parameter: Leibinz's rule.

Ordinary differential equations

Generalities on differential equations - The Cauchy problem - Linear differential equations of the first order - Differential equations of the first order with separable variables - Cauchy's theorem on the existence and uniqueness of the solution - Linear differential equations of the second order with constant coefficients - Lagrange method of constant variations - Applications to electrical circuits and mechanical oscillations.

Notes on the geometry of curves and on linear differential forms

Regular and generally regular curves - Rectifiable curves and their length - Curvilinear abscissa - Curvilinear integral of a function - Linear differential forms - Curvilinear integral of a linear differential form - Exact differential forms: potential of a differential form - Closed differential forms - Gauss formulas -Green and area calculation - Integrating factor - Legendre transformation - Applications to thermodynamics.

Notes on the geometry of surfaces and on quadratic differential forms

Regular surfaces - Tangent plane and normal versor - Area of ​​a surface - Surface integrals of functions - Surface integrals of quadratic differential forms - Divergence theorem and Stokes' theorem.

 

Some recommended texts:

1) G. Di Fazio, P. Zamboni, Analisi Matematica due, ed. Monduzzi

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

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

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

5) S. Giuffrida, A. Ragusa – Corso di algebra lineare con esercizi svolti, ed. Il Cigno

6) L. Moschini – Esercizi svolti di Analisi Matematica II –ed Esculapio

7) L. Moschini – Lezioni di Analisi Matematica II – ed. Esculapio