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Le persone

Mathematical Analysis

Aim of the course

The course presents the classical tools of infinitesimal and differential calculus, and integral calculus for functions in one and two variables, in addition to geometry and linear algebra, illustrating wherever possible their applications in other disciplines.

The topics will be covered with a view to:

- Acquiring analysis skills: i.e. introducing and becoming accustomed to rigorous discussion and analytical reasoning.

- Attaining, through practice, command of an exact and apposite language.

- Applying the tools of calculus to prepare for better managing the use of quantitative methods.

- Attaining, through practice, command of an exact and apposite language.

- Applying the tools of calculus to prepare for better managing the use of quantitative methods.

Students must be conversant with the following topics:

- Arithmetic and algebra.

- Analytical geometry and trigonometry on a plane.

- Functions: in particular, a knowledge of the definition, graphs and principal properties of the fundamental functions (power, exponential, logarithmic, sine, cosine, tangent). They must also be able to solve algebraic, exponential, logarithmic and trigonometric equations and inequalities.

Syllabus

1. *Numerical sets:*

- Sets and logic symbols.

- Numerical sets and their algebraic and topological structure (natural numbers, integers, rational numbers, real numbers).

- Subsets of the real numbers: definition of upper bound and lower bound, supremum and infimum, maximum and minimum.

- Sets and logic symbols.

- Numerical sets and their algebraic and topological structure (natural numbers, integers, rational numbers, real numbers).

- Subsets of the real numbers: definition of upper bound and lower bound, supremum and infimum, maximum and minimum.

- Complex numbers

- Correspondences and functions between sets.

2.

-Definition of sequence and calculation of the limits.

- Numerical series and their characteristics.

- Geometric series.

- Harmonic and generalised harmonic series.

- Series with terms of alternating sign. Absolute convergence and Leibniz criterion.

- Series of powers.

3.

- Definition of univocal function of a real variable.

- Symmetric, increasing, decreasing, periodic, limited functions.

- Maximum and minimum of a function.

- Definition of compound and inverse functions.

- Limits of functions.

- Continuity.

- Derivative and differential.

- Maximum and minimum of a function.

- Definition of compound and inverse functions.

- Limits of functions.

- Continuity.

- Derivative and differential.

- Elasticity

- Rules of differential calculus.

- Searching for local or global maxima and minima.

- Higher order derivatives.

- Convex or concave functions.

- Study of a function.

- Taylor - Mac Laurin series.

- Rules of differential calculus.

- Searching for local or global maxima and minima.

- Higher order derivatives.

- Convex or concave functions.

- Study of a function.

- Taylor - Mac Laurin series.

4.

- Definition of primitive and the indefinite integral.

- Rules of integration: by decomposition, by parts, by substitution.

- Definite integral.

- Geometric applications.

- Integral function.

- Generalised integrals

- Integral function.

- Generalised integrals

5.* Elements of geometry and linear algebra.*

- Vectors on a plane and in 3-space. Scalar and vector product.

- *n* –dimensional vectors: the **R**^{n} space. Vector space.

- Linear dependence and independence, basis and dimension.

- Scalar product in **R**^{n}.

- Lines and planes in three-dimensional space.

- Matrices and matrix calculus.

- Determinant.

- Inverse matrix.

- Matrices and matrix calculus.

- Determinant.

- Inverse matrix.

- Rank of a matrix.

- Linear systems.

- Cramer's theorem.

- Rouchè-Capelli theorem.

- Linear systems.

- Cramer's theorem.

- Rouchè-Capelli theorem.

- Linear transformation and representation theorem.

- Eigenvalues and eigenvectors. Matrix diagonalization.

6. *Real functions of one or more real variables.*

- Functions in R^{2}: domain, level curves.

- Continuity. Partial derivatives, gradient vector, tangent plane and differentiability.

- Functions in R

- Continuity. Partial derivatives, gradient vector, tangent plane and differentiability.

- Directional derivatives.

- Implicitly defined functions.

- Free or constrained optimisation of functions in two variables.

- Free or constrained optimisation of functions in two variables.

7.

- Equations with separable variables.

- First and second order linear equations.

- Homogeneous and non-homogeneous second order equations with constant coefficients.

- Euler equations.

8.* **Line integral and double integral.*

- Smooth curves and line integrals in **R**^{2} and **R**^{3}.

- Definition of double integral. Geometric interpretation.

- Calculation of double integral using Cartesian and polar coordinates.

- Calculation of double integral using Cartesian and polar coordinates.

- Gauss-Green formula in the plane.

Examinations

There will be a written exam, held during the sessions scheduled in the academic calendar, and possibly completed with an additional oral exam.

There will also be four intermediate written tests during the course.

Students who earn a grade of eighteen or higher on each of the four intermediate tests can be exempted from sitting the final written exam.

Reading list

Bramanti M., Pagani C.D., Salsa S., *Matematica,* Zanichelli.

Salsa S., Squellati A., *Esercizi di Matematica*, volumi: 1^{o} 2^{o}, Zanichelli.

Lecture notes and exercises prepared by the lecturer.