B2289 - Principles of Mathematics T-2

Learning outcomes

At the end of the course the student acquires the tools of Linear Algebra and Mathematical Analysis for functions of several variables.

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:

• 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 compulsory and encompass the entire lesson plan.

The written test has a total duration of three hours and aims to assess the student's ability to solve exercises, give examples and counterexamples, enunciate and provide brief arguments on theoretical issues. The written test consists of 6 theoretical exercises or questions and the maximum score is 300. Students are admitted to the oral test if the mark in the written test is at least 180/300. Written and oral must be carried out in the same call. The oral schedule will be published with the results of the written test.


The oral exam consists of an in-depth study of the questions in the written test and has the aim of assessing the student's knowledge and understanding of the topics covered in the teaching and his ability to expose them consistently using a specific language and a correct formalism.

The final result of the exam takes into account the performance in both parts: in order to pass, the grade for each part must be at least 18, and in that case, the final grade is approximately the average of the two. However, the instructor reserves the right to adjust the final grade in exceptional cases. If the final result is not sufficient, both parts of the exam must be retaken. The student has the right to withdraw from either the written or the oral part, and may refuse the final proposed grade once.


Registrations for the exams are made on AlmaEsami. 


Registration for the written exam is mandatory, please note that registrations close 5 days before the date scheduled for the written exam. It is necessary to show up for the tests with a university card or an identity document.

Students with Specific Learning Disorders (SLD) or temporary/permanent disabilities: they are strongly advised to contact the relevant University office [https://site.unibo.it/studenti-con-disabilita-e-dsa/en] in advance. The office will be responsible for proposing any necessary accommodations to the interested students. These accommodations must be submitted for approval to the course instructor at least 15 days in advance. The instructor will assess their suitability 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