96993 - MATEMATICA E STATISTICA (9 CFU)

Learning outcomes

On successful completion of the course, the student will have acquired the basic knowledge of calculus, linear algebra and geometry, essential to describe geological processes and to deal with other courses of the degree programme, especially those related to physics. In particular, the student will be able to represent functions graphically, to apply one-variable and multivariable calculus, to compute solutions of first order differential equations, to perform operations on vectors and matrices, and to solve systems of linear equations and easy geometric problems on the plane and the three-dimensional space.

Course contents

Prerequisites

Mathematical knowledge normally acquired in upper secondary school is required: elementary algebra, equations and inequalities, analytic geometry of the plane, elements of trigonometry, exponentials, and logarithms. Students who feel they have gaps in their initial preparation are strongly encouraged to attend the tutoring activities organized during the semester.

Review of algebra and set theory

• Sets and set operations.

• Cartesian product.

• Natural, integer, rational, and real numbers.

• Powers and logarithms.

• Equations and inequalities.

• Absolute value.

• Review of trigonometry.

Functions of a real variable

• Domain, codomain, image.

• Injective, surjective, and bijective functions.

• Composition and inverse function.

• Polynomial, rational, exponential, logarithmic, and trigonometric functions.

• Limits and continuity.

• Fundamental theorems on limits.

• Qualitative study of the graph of a function.

• Applications to models of exponential growth, radioactive decay, capacitor discharge, and oscillatory phenomena.

Differential calculus

• Derivative and its geometric meaning.

• Differentiation rules.

• Taylor's formula.

• l'Hôpital's rule.

• Maxima and minima. Complete study of functions.

• Introduction to functions of several variables: gradient and Hessian matrix, local maxima, minima, and saddle points.

Integral calculus

• Antiderivatives (primitives).

• Definite integral according to Riemann.

• Fundamental properties.

• Fundamental theorem of calculus.

• Mean value theorem.

• Integration techniques.

• Geometric and physical applications of the integral.

Differential equations (time permitting)

• Separable equations.

• First-order linear equations.

• Cauchy problem.

Analytic geometry

• Vectors in the plane and in space.

• Norm, dot product, and cross product.

• Distances.

• Lines and planes.

• Parametric and Cartesian equations.

• Geometric and physical applications.

Linear algebra

• Matrices and operations.

• Determinant.

• Inverse matrix.

• Gaussian elimination method.

• Rank.

• Systems of linear equations.

• Rouché-Capelli Theorem.

Elements of descriptive statistics

• Data organization.

• Measures of position.

• Measures of dispersion.

• Normal distribution.

• Standardization (Z-test).

• Linear regression and correlation.

• Introduction to hypothesis testing.

Readings/Bibliography

Recommended readings/Bibliography

Textbook:

• P. D'Ancona, M. Manetti, Istituzioni di Matematiche. The textbook is available for free online and can be purchased in paperback format on Amazon.

In addition, the following materials will be made available on the Virtuale platform during the course:

• exercise sheets;

• any supplementary lecture notes;

• material used in class;

• course-related announcements and notices.

Teaching methods

Teaching methods

Each week, the course includes:

• 6 hours of lectures dedicated to the development of theoretical contents;

• 2 hours of practical exercises dedicated to the guided resolution of exercises.

The purpose of the practical exercises is to consolidate the topics covered in class and to develop the ability to apply mathematical tools to problem-solving.

Every week, a list of recommended exercises for self-study will also be proposed.

Subject to the availability of the tutoring service, optional support meetings will be organized in which additional quizzes and exercises will be discussed, with a particular focus on the topics that present the greatest difficulties.

Assessment methods

Assessment methods and criteria

The exam consists of a written test and an oral exam.

Registration via AlmaEsami is mandatory to participate in both tests.

Written test

The written test consists of solving exercises and problems and verifies the ability to correctly apply the mathematical tools developed during the course.

Oral exam

The oral exam is aimed at verifying:

• understanding of the main theoretical results;

• knowledge of the fundamental definitions;

• the ability to rigorously present mathematical procedures;

• the appropriate use of scientific language.

The written and oral tests must be completed in the same session. Unlike in previous years, a passing grade in the written test is not carried over to subsequent sessions.

The final grade takes into account both the practical skills demonstrated in the written test and the theoretical preparation verified during the oral interview.

Detailed organizational procedures for the exams will be illustrated at the beginning of the course and published on the Virtuale platform.

Assessment criteria

Passing the exam requires achieving adequate preparation in both theoretical aspects and exercise resolution.

The assessment will take into account:

• the mathematical correctness of the procedures;

• the ability to apply the studied techniques;

• the understanding of theoretical concepts;

• the clarity of exposition and correct use of mathematical language.

Academic integrity

During exams, the use of tools not expressly authorized by the teacher, including generative Artificial Intelligence systems, is prohibited.

Students are expected to respect the principles of fairness, transparency, and academic integrity set forth by the University regulations.

Teaching tools

Teaching tools

Students experiencing difficulties in their preparation are strongly encouraged to participate in tutoring activities.

During these sessions, additional exercises will be solved, fundamental topics will be reviewed, and methodological guidance for studying the discipline will be provided.

The teacher also receives students during the office hours published on their institutional web page.

Inclusion and student support

Students with disabilities, Specific Learning Disabilities (SLD/DSA), or specific educational needs are invited to contact the teacher in a timely manner and refer to the dedicated University services, in order to agree on the compensatory measures and any adaptations provided for by current regulations.

Office hours

Office hours are held by appointment.

Updated hours and procedures are available on the teacher's web page.

Office hours

See the website of Riccardo Biagioli