B2289 - Principles of Mathematics T-2

Learning outcomes

At the end of the course, students will be familiar with the main tools of Mathematical Analysis and will have seen some of their applications, with particular emphasis on functions of several variables and differential equations.

Course contents

• Elements of Linear Algebra — Definition of vector spaces, linear dependence and independence, basis. Matrices: operations with matrices. Square matrices: trace, inverse, determinant. Systems of linear equations and the Gauss-Jordan method; rank and the Rouché-Capelli theorem; Cramer's rule. Linear maps and endomorphisms: diagonalizability and characteristic polynomial. Affine spaces, geometry of the plane and space.
• Multivariable Calculus — Real-valued and vector-valued functions of several real variables. Continuity, differentiability, partial derivatives. Gradient, Jacobian matrix, Hessian matrix. Taylor's formula for real functions of several real variables. Maxima and minima of differentiable functions. Multiple integrals and Fubini’s theorem. Change of variables. Introduction to curves in ℝⁿ, line integrals and their applications.

Readings/Bibliography

The topics covered in the course are classical subjects in linear algebra and multivariable calculus. However, there are not many textbooks that include both areas. Among those that do, the following are worth mentioning:

• M. Bramanti, F. Confortola, S. Salsa: Matematica per le scienze con fondamenti di probabilità e statistica. Editore: Zanichelli
• Marco Bramanti, Carlo Domenico Pagani, Sandro Salsa
  Matematica. Calcolo infinitesimale e algebra lineare. Editore: Zanichelli.
  
• Susanna Terracini, Davide L. Ferrario, Monica Conti, Vivina Barutello, Gianmaria Verzini, Analisi matematica. Con elementi di geometria e calcolo vettoriale: 2. Editore: Maggioli.

 

Teaching methods

Frontal lesson, dialogue and exercises.

Assessment methods

The exam consists of a written test and an oral test. Both are mandatory and cover the entire syllabus presented during the lectures.

The written test lasts a total of two and a half hours and aims to assess the student’s ability to solve exercises, provide examples and counterexamples, or give short arguments on theoretical questions. During the written test, consultation of printed materials such as books, notes, etc. is allowed and indeed recommended. The use of a scientific calculator is also permitted, but the use of any other electronic device is prohibited. The use of AI tools during the exam is prohibited in any form, including chatbots, text generators, advanced translators, content summarizers, and tools integrated into devices. Any use of AI is considered a violation of the principles of honesty and integrity set out in the Policy and may result in the cancellation of the exam by the instructor, who remains responsible for determining the validity of the assessment.

Students are admitted to the oral test if they obtain at least 16 points out of 33 on the written test. The written and oral tests must be taken in the same exam session. The schedule for the oral tests will be published together with the results of the written test.

The oral test aims to assess the student’s knowledge and understanding of the topics covered in the course, as well as their ability to present them coherently, using appropriate subject-specific language and correct mathematical formalism.

The final grade takes into account the results obtained in both tests: to pass, the grade for each test must be at least 16, and in that case the final grade is approximately the average of the two marks; however, the instructor reserves the right to adjust the evaluation in exceptional cases. If the final result is insufficient, both tests must be retaken. Students may withdraw from either the written or the oral test and may refuse the proposed final grade once.

Registration for exam sessions is carried out via AlmaEsami. Registration for the written test is mandatory; please note that registration closes 5 days before the scheduled date of the written exam.

Students must bring their university ID card to the exams.

Students with specific learning disorders (SLD) or temporary or permanent disabilities are advised to contact the relevant University office in due time. The office will propose any necessary accommodations to the students concerned; such accommodations must in any case be submitted to the instructor for approval at least 15 days in advance. The instructor will assess their appropriateness also in relation to the learning objectives of the course.

Teaching tools

All teaching materials will be published on Virtuale.  

Office hours

See the website of Alessia Cattabriga