88149 - Calculus P

Learning outcomes

The student learns the fundamental concepts and the main properties of the real or vector functions of one or more real variables (limits, differential calculus, integral calculus) and arguments complementary to these (complex numbers, linear differential equations). The student acquires the methods of solving simple exercises on these topics, with particular reference to their use in Physics and in professional courses. The student acquires the basic tools of linear algebra (matrices, vector structure of R^n, linear systems, eigenvalues), guaranteeing the ability to operate on simple examples, with particular reference to their use in Informatics and in the professionalizing teachings. The student has knowledge of the fundamentals of a symbolic and / or numerical calculation program and uses it to solve simple problems related to the topics listed above. At the end of the course the student is able to perform, by hand and with computer tools, the calculation of limits, derivatives and integrals of functions in one or (in simple cases) several real variables, the study of a function in a variable real, the integration of linear differential equations, the calculation of the determinant and eigenvalues of a square matrix and of the rank of a matrix, the discussion and possible resolution of a system of linear equations. He is also able to have an overall idea of the theoretical links between the topics studied.

Course contents

Linear Algebra

The geometric vectors: algebraic structure, scalar and vector product.

The space  R^n: vector structure, standard scalar product, norm, orthogonality; linear combinations and linear dependence; vector and affine subspaces; generator systems, bases and dimension.

Matrices: vector structure and product of matrices; echelon form; definition of rank and calculation techniques; linear transformation associated with a matrix.

Square matrices: invertible matrices; definition of determinant and calculation techniques; eigenvalues and eigenvectors; diagonalization of a matrix.

Symmetric matrices: spectral theorem, signature and Sylvester's theorem.

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



Analysis

Numbers: real numbers and complex numbers.

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

Differential calculus for real functions of real variable: derivative, growth and decrease, local extremes, study of the graph of a
function, Taylor's formula.

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

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

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

Linear differential equations.


Laboratory

Use of a symbolic and / or numerical calculation program (MATLAB) to solve simple problems related to the theoretical topics of the course.


The detailed and complete teaching program will be published on Virtuale at the end of the lessons.

Readings/Bibliography

Recommended text:

• Bramanti, Pagani, Salsa, Matematica. Calcolo infinitesimale e algebra lineare, Zanichelli.

 

Other texts:

• Barozzi, Dore, Obrecht, Elementi di Analisi Matematica 1, Zanichelli.
• Plazzi, Ritelli, Elementi di calcolo in piu' variabili, Pitagora Editrice, Bologna.
• Ritelli, Bergamini, Trifone, Fondamenti di Matematica, Zanichelli.
• Barnabei, Bonetti, Sistemi lineari e matrici, Pitagora Editrice.
• Guerraggio, Matematica, Pearson-prentice-Hall.
• Naldi, Pareschi, Aletti, Calcolo differenziale e algebra lineare, McGraw-Hill.

Text devoted to exercises:

• Salsa, Squellati. Esercizi di Analisi matematica 1,
  Zanichelli Editore.
• Salsa, Squellati. Esercizi di Analisi matematica 2,
  Zanichelli Editore.
• Parigi, Palestini, Manuale di Geometria, Esercizi, Pitagora Editrice.
• Mulazzani, Di Fabio, Prove d'esame risolte di Matematica Generale per il corso di Laurea in Economia Aziendale, Esculapio.
  

Teaching methods

Traditional lesson with classroom and laboratory exercises.

Assessment methods

The exam consists of a written test and an oral exam. The written test embrace the program of the linear  algebra and analysis part, while the oral exam regards also the laboratory part. Both the exams take place remotely (EOL and Zoom for the written part, Teams for the oral). Please subscribe to the platforms EOL, Zoom  and Teams  and get acquainted with them. Detailed information on the exams will be sent to the registered students.

The written test aims to test the student's ability to solve exercises. During the written test it is allowed and even recommended to use books and notes. It is possible to use the calculator, but the use of other electronic devices is prohibited. The part of analysis lasts 80 minutes, that of linear algebra 40 minutes.

To take the written test you must register on Almaesami.

In order to enter the oral exam  the mark obtained in the written test must be at least 15/33 (5/11 linear algebra+10/22 analysis). Each final written test is valid for three calls, the one of the written test included. 

The oral exam, which begins with the discussion of the written test, aims to verify the student's understanding of the topics covered and of the theoretical links between them, his ability to state definitions and theorems and to produce examples or counterexamples. For the laboratory part, a test will be carried out at the terminal.

It will be possible to take separately the part of linear algebra, the part of analysis and that of laboratory. The mark obtained in just one or two of the three part of the exam lasts only for the A.Y. in which it has been obtained. 

 

Teaching tools

All the materials will be published on Virtuale.

 

The tutor is Dott. Giorgio Tortone.

Office hours

See the website of Alessia Cattabriga

See the website of Enrico Smargiassi

See the website of Enrico Smargiassi