Ateneo - UPO

The course covers key topics in linear algebra, basic concepts of analytic geometry in the plane and in space, and mathematical analysis for functions of several real variables.
More specifically, the main topics are: vector spaces and bases, matrices and matrix operations, matrix rank, the determinant of a square matrix, invertibility of square matrices and calculation of the inverse matrix, linear systems and the Rouché-Capelli theorem, the scalar product in n-dimensional Euclidean space, the Hermitian product for complex vector spaces, the vector product in three-dimensional Euclidean space, linear maps between abstract vector spaces, eigenvalues ​​and eigenvectors, diagonalization of symmetric operators, elements of general topology in the n-dimensional Euclidean space, limits and continuity for functions of several real variables, partial differentiation, multiple integration, line and surface integrals, differential forms and vector fields, Gauss-Green, divergence, and curl theorems, and linear ordinary differential equations.

Recommended textbooks:

(Geometry Module)
- Bramanti, Pagani, Salsa, “Analisi Matematica 1 con elementi di geometria e algebra lineare”, Zanichelli, 2014.

(Calculus II Module)
- F. Gazzola, "Analisi Matematica 2", Esculapio, 2022.
- M. Bramanti, C. D. Pagani, S. Salsa, "Analisi Matematica 2", Zanichelli, 2009.
For further study, the recent edition of the following text is also recommended:
- C. D. Pagani, S. Salsa, "Analisi Matematica 2", Zanichelli, 2016.
Exercise book:
- S. Salsa, A. Squellati, ”Esercizi di Analisi Matematica 2”, Zanichelli, 2011.

The course aims to provide students with the fundamental knowledge of linear algebra, analytic geometry, and mathematical analysis required in fields that utilize advanced mathematical tools. It covers the basics of vector and matrix calculus—with applications to analytic geometry in the plane and in space—as well as differential and integral calculus for functions of several real variables.

Knowledge of the concepts and content covered in the Calculus I course—specifically differential and integral calculus for real-valued functions of a single real variable—is required. In particular, students intending to attend this course and pass the corresponding exam must have achieved the learning outcomes of the Calculus I course, outlined below:
- knowledge of the fundamentals of mathematical analysis in a single real variable (single-variable functions, limits, derivatives, integrals);
- ability to apply knowledge and understanding: students must have developed a mastery of the concepts learned, enabling them to independently connect these concepts and use them in combination to solve simple theoretical problems;
- ability to communicate what has been learned: students must be able to clearly explain both the concepts themselves and their application, whether in general terms or within the specific context of a problem.

The course is structured around lectures comprising both theoretical content and exercises. Each topic is introduced via a general discussion designed to make the material as accessible as possible. Next, the fundamental concepts of the topic are presented, followed by examples to clarify their meaning. The third stage covers the statements of key theorems and their proofs. The final part is dedicated to exercises. Active participation is encouraged through questions that also aim to gauge the level of difficulty students experience in following the lectures. The university platform (www.dir.uniupo.it) is used to provide supplementary teaching materials, such as lecture notes, various exercises, and past written exam papers.

In addition to the recommended textbooks for theory, further exam preparation material will be provided during the course. This consists primarily of the lecture slides and past written exam papers.

Students with disabilities, Specific Learning Disorders (SLD), or Special Educational Needs (SEN) may request specific services and tools by contacting the Student Career Development and Services Staff and consulting the dedicated page on the University website: https://uniupo.it/it/servizi/servizi-studentesse-e-studenti-condizione-di-disabilit%C3%A0-e-dsa
Once they have contacted the University staff, students with disabilities, SLD, or SEN may contact the course instructor to discuss the specific arrangements for the exam regarding academic aspects.

The exam consists of a written test and an oral test. The written test usually comprises three exercises for each of the two modules, covering various course topics. Students are permitted to take the written test for only one module at a time. Below is a summary list of the main topics that may appear in the exercises:
a) vectors, matrices, and related operations;
b) analysis of matrix invertibility and calculation of the inverse matrix;
c) determination of matrix rank;
d) analysis of the existence and uniqueness of solutions to linear systems using the Rouché-Capelli theorem;
e) calculation of solutions to linear systems using various methods, including Cramer's rule;
f) exercises on lines and planes in three-dimensional Euclidean space;
g) determination of eigenvalues ​​and eigenvectors for linear operators and/or their associated matrices;
h) analysis of the diagonalizability of linear operators;
i) determination and classification of stationary points;
l) analysis of differential forms and vector fields, and calculation of their line integrals;
m) calculation of double and triple integrals;
n) calculation of surface area, surface integrals, and the flux of a vector field through a surface;
o) application of Gauss-Green, divergence, and curl formulas to the calculation of line and surface integrals;
p) solving Cauchy problems for linear ordinary differential equations with constant coefficients.

The score assigned to each exercise varies depending on the type of exercise and its level of difficulty.
This score is indicated next to the text of each exercise.
Admission to the oral exam for each module is contingent upon achieving a minimum score on the written exam for that module.
The oral exam for each of the two modules covers the following points:
a) understanding of any errors made in the written exam;
b) discussion of a theoretical topic of the student's choice;
c) an additional question on a topic chosen by the instructor.
If the answers and the language used are satisfactory, it is possible to achieve a final grade higher than the one obtained in the written exam. However, the final score may in no case exceed the written exam score by more than 7 points.
It is possible for the final grade to be lower than the written exam grade; please note that if the oral exam performance is unsatisfactory, the overall outcome for the module may be considered a failure.
Achieving an excellent grade requires a strong grasp of theoretical topics combined with excellent proficiency in calculations.

The Calculus II exam as a whole is considered passed only when the exams for both modules have been successfully completed. The final grade for the Calculus II exam will be determined based on an overall assessment of the results achieved in the individual modules.

The course aims to provide the fundamentals of vector and matrix calculus, as well as differential and integral calculus for functions of several real variables. Below is a summary of the main topics to be covered (please refer to the detailed syllabi for each module for further information):
1. geometric vectors in the plane and in space; vectors in abstract real and complex vector spaces;
2. matrices and matrix operations;
3. matrix rank and the determinant of a square matrix;
4. scalar product, Hermitian product, and vector product;
5. invertibility of square matrices and calculation of the inverse matrix;
6. existence and uniqueness of solutions to linear systems, Rouché-Capelli theorem, and Cramer's rule;
7. lines and planes in three-dimensional Euclidean space;
8. linear operators, eigenvalues ​​and eigenvectors, and diagonalization of linear operators;
9. limits and continuity for functions of several real variables;
10. partial differentiation, differentiability, stationary points, and relative maxima and minima;
11. arc length, line integrals with respect to arc length, and line integrals of vector fields;
12. differential forms, vector fields, and their line integrals; closed and exact differential forms; irrotational and conservative vector fields;
13. multiple integration with a focus on double and triple integrals, reduction formulas for multiple integrals, and the change-of-variables formula;
14. surface area calculation, surface integrals, and the flux of a vector field through a surface;
15. Gauss-Green, divergence, and curl theorems, and their applications to calculating line and surface integrals;
16. linear ordinary differential equations, with a focus on those with constant coefficients.

To achieve the knowledge and skills corresponding to the minimum passing level, students must demonstrate:

(Knowledge)

- knowledge of the main concepts regarding vectors, matrices, and related operations;
- knowledge of the main concepts regarding the study of linear systems;
- knowledge of the main concepts regarding the analytic geometry of lines and planes in three-dimensional Euclidean space;
- knowledge of the main concepts regarding linear maps between vector spaces;
- knowledge of the main concepts regarding eigenvalues, eigenvectors, and the diagonalizability of linear operators;
- knowledge of the main concepts regarding limits and continuity for functions of several real variables;
- knowledge of the main concepts of differential calculus;
- knowledge of the main concepts regarding differential forms and vector fields;
- knowledge of the main concepts regarding multiple integration, as well as line and surface integrals;
- knowledge of the main concepts regarding linear differential equations with constant coefficients.

(Skills)
- ability to apply the main concepts of vector and matrix calculus;
- ability to apply the main concepts of differential and integral calculus;
- ability to identify, within a specific application area, the appropriate concepts from vector/matrix calculus and mathematical analysis required for the context at hand.

(Transversal skills)
- ability to use correct mathematical terminology within an applied project;
- ability to grasp the underlying meaning of material from sources such as textbooks and articles, even when advanced mathematical language is employed.
Achievement of an advanced level: students must demonstrate a high level of mastery of the subject matter, both in terms of problem-solving ability and knowledge of the key theoretical results in vector and matrix calculus and in the mathematical analysis of functions of several real variables.