Mathematics II 1

The learning objectives of the course are as follows:

Knowledge and understanding:
The student will learn some fundamental mathematical concepts and will develop computational skills and the ability to manipulate the most common objects in Mathematical Analysis. These include integrals of functions of one and several real variables, sequences and series of functions, differential equations, differential calculus for real functions of two real variables, and linear differential forms.

Applying knowledge and understanding:
Through examples related to the applied sciences, the student will appreciate the importance of Mathematical Analysis in scientific contexts, and not just as a self-contained discipline, thereby broadening their cultural horizons.

Making judgements:
The student will be able to approach, with sufficient rigor, some simple yet significant proof techniques in Mathematical Analysis, in order to refine their logical reasoning skills. Many proofs will be presented in a schematic and intuitive manner to engage students and encourage them to reach the result independently.

Communication skills:
By studying Mathematical Analysis and practicing through guided exercises, the student will learn to communicate clearly and rigorously, both orally and in writing. They will understand that using precise language is one of the most important tools for effectively conveying scientific topics, not only in mathematics.

Learning skills:
Students, especially the most motivated ones, will be encouraged to explore some topics in greater depth, including through group work.

Integral Calculus for Functions of a Single Variable
Review of integration methods. Introduction to quadrature formulas for the approximation of one-dimensional integrals and MATLAB exercises.

Sequences and Series of Functions
Pointwise convergence; uniform convergence; Cauchy criterion for convergence.
Examples and MATLAB exercises on numerical series and convergence: geometric series, harmonic series, Mengoli’s series.
Series of functions: definition, pointwise convergence, uniform convergence, Cauchy criterion for pointwise convergence, Taylor series, trigonometric series, Fourier series.

Differential and Integral Calculus for Functions of Two or More Variables
Definition of vector space, function of two variables, scalar function, vector-valued function, limits, and continuous functions.
Definitions and examples of: domain and image of a function; injective function, inverse function, composite function.
Exercise on injective functions.
Introduction to and exercises on MATLAB function.m files.
Functions of several variables – Differential calculus for multivariable functions: definition of partial derivatives in ℝ².
Fermat’s theorem on relative extrema – Higher-order derivatives and Schwarz’s lemma.
Definition of quadratic forms and the theorem on critical points, with exercises.
Double integrals – Introduction to composite quadrature formulas.

Ordinary Differential Equations
General overview of differential equations – The Cauchy problem – First-order differential equations – Cauchy’s existence and uniqueness theorem – Systems of ordinary differential equations.
Examples and MATLAB exercises using the commands ode45, ode15, ode15s.

Student learning will be assessed periodically through guided exercises in class. The final exam consists of a written test and an oral examination

Double integrals and quadrature formulas; Critical point analysis; Series of functions; Solution of ordinary differential equations.