88149 - Calculus P

Learning outcomes

Working knowledge of linear algebra, differential and integral calculus. Students will be able to formulate examples and counterexamples, understand the basic notions of the theory and connections among them, solve simple exercises and problems, both by hand and by computer, particularly in connection with other subjects of the Bachelor program.

Course contents

Linear Algebra

Review of numbers and functions.

R^n as a vector space. Linear combinations and linear dependence; linear and affine subspaces; systems of generators, bases and dimension; operations on subspaces and Grassmann's formula.

Linear systems: matrix notation; Gaussian elimination; parametric and Cartesian representation of subspaces of R^n; structure theorem for linear systems; Rouché-Capelli theorem; parametric systems and resolution techniques.

Linear functions, kernel and image, rank theorem. Matrix representations, composition of functions and matrix product, invertibility, change of basis. Invariants for conjugation: rank, trace, determinant, characteristic polynomial; calculation techniques. Eigenvalues, eigenvectors, triangulability and diagonalizability.

If time allows: standard scalar product and vector product in R^3. Metric concepts: orthogonal projection, Gram-Schmidt orthogonalisation. Orthogonal matrices. Symmetric matrices, spectral theorem, quadratic forms and signature.

 

Analysis

Numbers: real and complex numbers.

Real functions of a real variable: definition, injectivity, surjectivity, monotonicity; graph of a function; elementary functions (powers, roots, exponentials, logarithms, circular functions); limits and continuity.

Differential calculus for real functions of a real variable: derivative, monotonicity, local extrema, study of the graph of a function, Taylor's formula.

Integral calculus for real functions of real variables: primitives, fundamental theorem of integral calculus, integration by substitution and by parts.

Differential calculus for vector functions of several variables: partial derivatives, gradient, local extrema.

Integral calculus for real functions of several real variables: reduction theorems, change of variables.

Linear differential equations.

Readings/Bibliography

Recommended readings (in Italian):

Testo consigliato:
■ Bramanti, Pagani, Salsa, "Matematica. Calcolo infinitesimale e algebra lineare", Zanichelli.

■ Abate, de Fabritiis, "Geometria analitica con elementi di algebra lineare", McGraw-Hill

Other readings:

■ Schilling, Nachtergaele e Lankham, "Linear Algebra", LibreTexts
■ Boyd e Vandenberghe, "Introduction to
Applied Linear Algebra", Cambridge University Press

 ■ Sernesi, "Geometria I", Bollati Boringhieri (chapters 1 and 2)

■ Plazzi, Ritelli, Elementi di calcolo in più variabili, Pitagora Editrice, Bologna.
■ Guerraggio, Matematica, Pearson-prentice-Hall.
■ Naldi, Pareschi, Aletti, Calcolo differenziale e algebra lineare, McGraw-Hill.

Exercise books:
■ Salsa, Squellati. Esercizi di Analisi matematica 1, Zanichelli Editore.
■ Salsa, Squellati. Esercizi di Analisi matematica 2, Zanichelli Editore.

■ Abate, de Fabritiis, "Esercizi di geometria", McGraw-Hill
■ Parigi, Palestini, Manuale di Geometria, Esercizi, Pitagora Editrice.
■ Mulazzani, Di Fabio, Prove di esame risolte di Matematica Generale per il corso di Laurea in Economia Aziendale, Esculapio.

Teaching methods

Lectures (possibly remote) with examples and exercises. Tutorials, subject to availability.

Assessment methods

The exam consists of a written test, possibly supplemented by an additional oral test. The written test covers the contents of analysis and linear algebra, while the oral test also includes laboratory contents.

The written test aims to assess the student's ability to illustrate the key concepts of the course and solve some simple exercises. During the written test, students are allowed to use books, personal notes, and calculators, but the use of other electronic devices (smartphone, computer) is prohibited. The analysis part lasts 80 minutes, while the linear algebra part lasts 40 minutes.

The written exam is passed if the grade obtained is at least 6/11 for the algebra part and 12/22 for the analysis part. It is possible to take the linear algebra and analysis parts separately. Each partial grade is valid for three sessions, including the one in which it was obtained (six sessions for student workers).

In order to be admitted to the written exam, you must register on Almaesami, arrive in the classroom on time, and bring your university badge and ID card with you.

Admission to the oral exam is conditional upon having obtained at least 5/11 in linear algebra or 10/22 in analysis at the written exam. The oral exam will only be held at the request of the student and at the discretion of the lecturer, and aims to assess the student's understanding of the topics covered and the theoretical links between them, ability to state definitions and theorems, produce examples and counterexamples, and solve simple exercises.

The grade will only be recorded after both modules have been passed within the above-mentioned time frame.

 

Students with learning disorders and\or temporary or permanent disabilities: please, contact the office responsible (https://site.unibo.it/studenti-con-disabilita-e-dsa/en/for-students) as soon as possible so that they can propose acceptable adjustments. The request for adaptation must be submitted in advance (15 days before the exam date) to the lecturer, who will assess the appropriateness of the adjustments, taking into account the teaching objectives.

Teaching tools

The teaching material will be posted on Virtuale.

Office hours

See the website of Luca Battistella

See the website of Enrico Smargiassi