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 sequences and series of functions, differential calculus for real functions of several variables, double integrals, and ordinary differential equations.

• 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.

The course is divided into two parts:

A) Mathematical Analysis

B) Numerical Analysis

Regarding Part A, the syllabus covers the following topics:

• Sequences and series of functions: pointwise and uniform convergence for sequences of functions, various types of convergence for series of functions, power series, Taylor series, Fourier series.

• Differential calculus for functions of several variables: limits and continuity, directional derivatives and gradient, C1 functions, Fermat's theorem, higher-order derivatives, Schwarz's theorem, Hessian matrix, review of quadratic forms, Taylor's formula, sufficient conditions for local extrema.

• Double integrals: area of a planar region, reduction formulas, change of variables formula. Applications: center of mass, moment of inertia.

• Ordinary differential equations: general concepts, Cauchy problem, local existence and uniqueness theorem, separable equations, first- and second-order linear differential equations. Applications: Malthus model and harmonic oscillator.

Regarding Part B, the syllabus covers the following topics:

• Introduction to MATLAB: basic commands, for loops, while loops, and applications. Examples of user-defined functions.

• Sequences and series of functions: application of the Cauchy criterion for convergence; examples of geometric series, harmonic series, and Mengoli series; series of functions: definition, pointwise convergence, uniform convergence, examples of trigonometric series, Taylor series, and applications.

• Integral calculus: introduction to simple and composite quadrature rules for approximating integrals in one and two dimensions.

• Ordinary differential equations: first-order differential equations, exercises on the Cauchy problem, systems of ordinary differential equations. Applied examples on chemical reactions using ode45, ode15, and ode15s commands.

M. Bramanti, C.D. Pagani, S. Salsa, Analisi matematica 1, Zanichelli Ed., 2008.