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.

Teaching methods

Lecture-based instruction, potentially delivered remotely, with exercises completed in class. Support from a teaching tutor based on availability.

Assessment methods

The exam consists of a written test, which may be supplemented by an additional oral test.
The written test aims to assess the student's ability to explain the key concepts of the course and solve some simple exercises. During the written test, students may use books and notes, and the use of a scientific calculator is permitted, but the use of other electronic devices is prohibited. The duration of the calculus (analysis) section is at least 80 minutes, and that of the linear algebra section is at least 40 minutes.
The written test is considered passed if the grade obtained is at least 6/11 for the algebra section and 12/22 for the calculus section. The linear algebra and calculus sections may be taken separately.
Each partial grade is valid for three exam sessions, including the one in which it was obtained (six sessions for working students).
To take the written test, it is necessary to register on Almaesami and to arrive in the classroom on time, with university badge and identification document.
Students are admitted to the oral test if the grade obtained in the written test is at least 5/11 in linear algebra or 10/22 in calculus. The oral test will be held only at the student's request and at the instructor's discretion, and aims to verify understanding of the topics covered and the theoretical connections between them, the ability to state definitions and theorems, to produce examples and counterexamples, and to solve simple exercises.
A negative evaluation of the oral test also invalidates the written test.
The final grade is recorded only once both modules have been passed within the time frame indicated above.
Students with DSA/SLD (specific learning disabilities) or temporary or permanent disabilities: it is recommended to contact the relevant University office in good time : this office will propose any accommodations to interested students, which must in any case be submitted, at least 15 days in advance, for the instructor's approval, who will assess their appropriateness also in relation to the course's learning objectives.

Teaching tools

Course materials will be published on the course's Virtuale.

Office hours

See the website of Enrico Smargiassi

See the website of Maurizio Casali