74790 - Mathematics - a Course with Exercises

Course Unit Page

Academic Year 2022/2023

Learning outcomes

On successful completion of the course, students will have acquired the basic knowledge of one-variable calculus, vector calculus and linear algebra, the first elements of multivariable calculus, complex numbers and the most elementary methods for solving ordinary differential equations. In particular, students will be able to represent data or functions graphically, to apply one-variable and multivariable calculus, to perform operations on vectors and matrices and to solve systems of linear equations.

Course contents


Elementary set theory.
Algebra of real numbers.
Algebraic equations and inequalities.
Elementary functions: powers, roots, exponentials, logarithms, circular functions.
Analytic geometry in the Euclidean plane.


The course is divided into three modules lectured by Prof. I. Rivalta (Module A), Prof. M. Garavelli (Module B-1) and by Dr. L. Muccioli (Module B-2). The lectures of Module A will be held in parallel with those of modules B-1 and B-2, which will instead be lectured in sequence. Each module includes classroom exercises.

Module A (6.5 CFU):

Vector calculus: vectors, vector spaces, operations between vectors, bases, Gram-Schmidt orthogonalization. Applications to analytical geometry.
Linear transformations: condition of linearity, linear combinations, affine transformations.
Matrix calculus: properties and types of matrices, operations between matrices, matrices and linear transformations, rotation matrices and polar coordinates, determinant (calculus and properties), geometric interpretation of the determinant, rank.
Base changes and matrix diagonalization.
Linear systems and Rouché-Capelli Theorem.
Complex numbers

Module B-1 (2 CFU):

Elements of set theory. Elementary functions (polynomials, logarithms, exponentials, trigonometric functions) and their inverses. Equations andinequalities. Domain, range, image, limits and continuity of functions in a variable. Derivatives of functions in one variable. De l'Hôpital's theorem.

Module B-2 (4.5 CFU):

Differential calculus and study of functions of one variable. Taylor series of functions of one variable.
Multivariable functions: partial derivatives, Hessian matrix, maxima
and minima, Taylor expansions, gradient operators, divergence, rotor
and Laplacian.
Definite and indefinite integration. Line integrals and multiple integrals.
Ordinary differential equations of first and second order.



Presentations and notes of lectures (saved with the digital board) will be published on the IOL system.


Claudio Canuto, Anita Tabacco, ``Analisi Matematica I'', 4a edizione (Springer--Verlag, Milano 2014) --- ISBN13:978-88-470-5722-7.

Tom M. Apostol, ``Calcolo'', volume II ``Geometria'', (Bollati Boringhieri, Torino 1979) --- ISBN13:978-88-339-5034-1.

Teaching methods

Modules A and B-1

Frontal lectrues using electronic blackboard with video projector, digital platforms for learning assessment. Classroom exercises.

Module B-2

Frontal lessons using slides with video projector. Classroom exercises.

Assessment methods

For the exams (including the intermediate tests) registration through AlmaEsami is required, in compliance with the established deadlines.

All the tests are written, and there are not oral tests. The test consists of a collection of exercises (from 4 to 6) on the topics of the modules (for partial tests, first test: topics modules A and B-1; second and third test: modules A and B-2; as indicated in the "Program / Contents" section) with evaluation in thirtieths and weight of each exercise explicitly indicated. The tests are aimed at assessing the understanding of concepts and fundamental calculus skills. Support material for the solution of the exercises is provided during the examination and the use of calculators is permitted. For the intermediate tests: the overall grade is obtained through the average of the three partial tests; the evaluations of each partial test must not be less than 16/30 to be taken into consideration. Alternatively, after the end of the course students can take a complete written exam on the topics of the three modules.

Teaching tools

Digital board with video projector and slides projection. Digital platforms for testing exercises and learning.

Students with learning or other types of disabilities are invited to contact the Servizio Studenti con Disabilità e DSA of the University of Bologna, the reference teacher of the Department, or the course teacher to establish appropriate processes concerning learning material and classroom adjustments.


Considering the types of activities and teaching methods adopted, the attendance of this training activity requires all students to carry out e-learning modules 1 and 2 [https://www.unibo.it/it/servizi-e -opportunities / health-and-assistance / health-and-safety / safety-and-health-in-places-of-study-and-training] and participation in module 3 for specific training on safety and health in places of study. Information on dates and methods of attendance of module 3 can be found in the specific section of the course program website.

Office hours

See the website of Ivan Rivalta

See the website of Luca Muccioli

See the website of Marco Garavelli