Ateneo - UPO

The course covers the main topics of mathematical analysis for functions of several real variables.
Specifically, the key topics include: elements of general topology in n-dimensional Euclidean space, limits and continuity of 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:

F. Gazzola, "Mathematical Analysis 2", Ed. Esculapio, 2022.

M. Bramanti, C. D. Pagani, S. Salsa, "Mathematical Analysis 2", Ed. Zanichelli, 2009.

For a deeper analysis of the subject we also suggest the more recent edition of the following textbook:

C. D. Pagani, S. Salsa, "Mathematical Analysis 2", Ed. Zanichelli, 2016.

Workbook:

S. Salsa, A. Squellati, ”Esercizi di Analisi Matematica 2”, Ed. Zanichelli, 2011.

The course aims to enable students to acquire the fundamental knowledge of mathematical analysis required in application areas that utilize advanced mathematical tools. The course will cover the basics of differential and integral calculus for functions of multiple 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 associated exam must have achieved the learning outcomes of the Calculus I course, outlined below:
- knowledge of the fundamental concepts 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 suggested books, further material for the preparation of the exam will be provided during the development of the course. It is made up of the slides used during the course of the last year and of the written exams of the previous years.

Students with physical disabilities, Learning Disabilities or Special Education Needs can request specific services and tools via the Staff Sviluppo e Coordinamento Carriere e Servizi alle Studentesse e agli Studenti, consulting the University webpage:
https://uniupo.it/it/servizi/servizi-studentesse-e-studenti-condizione-di-disabilit%C3%A0-e-dsa
Students with disabilities, learning disabilities or special education needs, once they have contacted the University Staff, can refer to the tutor in charge of the course to define the examination modalities, concerning academic aspects.

The exam consists of a written test and an oral exam. The written test usually comprises three exercises covering various topics from the course. The main topics that may appear in these exercises are as follows:
a) differentiability and calculating the equation of the tangent space to a graph;
b) determining stationary points and local extrema;
c) analysis of differential forms: closed and exact forms, calculation of line integrals;
d) analysis of vector fields: irrotational and conservative fields, calculation of line integrals;
e) calculating curve length and line integrals with respect to arc length;
f) calculating double and triple integrals;
g) calculating surface area, surface integrals, and the flux of a vector field through a surface;
h) applying Gauss-Green, divergence, and curl theorems to calculate line and surface integrals;
i) 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 difficulty.
This score is indicated next to the text of each exercise.

Admission to the oral exam is contingent upon achieving a minimum score on the written test.
The oral exam involves an assessment of the following points:
a) understanding of any errors made in the written test;
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 test. In no case may the final score exceed the written test score by more than 7 points.
It is also possible for the final grade to be lower than the written test grade; please note that if the oral examination proves unsatisfactory, the overall exam result may be deemed negative.
Achieving an excellent grade requires a thorough understanding of the theoretical topics combined with excellent proficiency in calculations.

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 differential and integral calculus for functions of several real variables. The main topics covered in the course are as follows:
1. limits and continuity for functions of several real variables;
2. partial differentiation and differentiability;
3. stationary points, relative maxima and minima, Fermat's Theorem, Taylor's formula for functions of several real variables, analysis of the Hessian matrix for classifying stationary points;
4. arc length, line integrals with respect to arc length, line integrals of vector fields;
5. differential forms and vector fields and their associated line integrals, closed and exact differential forms, irrotational and conservative vector fields;
6. multiple integration with a focus on double and triple integrals, reduction formulas for multiple integrals, change-of-variables formula;
7. calculation of surface area, surface integrals, flux of a vector field through a surface;
8. Gauss-Green, Divergence, and Stokes' (curl) theorems and their applications to the calculation of line and surface integrals;
9. linear ordinary differential equations, with particular emphasis on those with constant coefficients.

To achieve the minimum level of proficiency, students are required to demonstrate:

(Knowledge)
- knowledge of the key concepts regarding limits and continuity for functions of several real variables;
- knowledge of the key concepts of differential calculus;
- knowledge of the key concepts regarding differential forms and vector fields;
- knowledge of the key concepts regarding multiple integration, as well as line and surface integrals;
- knowledge of the key concepts regarding linear differential equations with constant coefficients.

(Competencies)
- the ability to apply the key concepts of differential and integral calculus;
- the ability to identify, within a specific application area, the correct mathematical analysis concept required for the context at hand.

(Transversal skills)
- the ability to use appropriate mathematical language within an applied project;
- the ability to grasp the deeper 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 degree of mastery of the subject matter, both in terms of problem-solving skills and knowledge of the key theoretical results of mathematical analysis for functions of several real variables.