Unit MATHEMATICAL ANALYSIS II

Course
Mathematics
Study-unit Code
55001909
Curriculum
In all curricula
Teacher
Tiziana Cardinali
Teachers
  • Tiziana Cardinali
Hours
  • 73 ore - Tiziana Cardinali
CFU
9
Course Regulation
Coorte 2021
Offered
2022/23
Learning activities
Base
Area
Formazione matematica di base
Academic discipline
MAT/05
Type of study-unit
Obbligatorio (Required)
Type of learning activities
Attività formativa monodisciplinare
Language of instruction
Italian
Contents
The goal of Mathematical Analysis II is to continue the study of calculus on the real line, which you started in Mathematical Analysis I, with a focus on functions of several variables: continuity, differentiability, optimization and Lebesgue integration.
Integrals on curves. Differential forms and their integration.
Reference texts
The main material introduced during lectures and is contained in Text-book:

1) C. CANUTO, A. TABACCO Analisi Matematica II, Springer-Verlag , 1nd Ed 2008 or 2nd Ed., 2014

or the English version:

1)' C. CANUTO, A. TABACCO Mathematical Analysis II, Springer-Verlag , 2nd Ed., 2015

On the Unistudium platform, lecture notes by the teacher on "The integration of Lebesgue in R^n" (Italian) and exercises on the program will be made available.

Other recommended books:

M. BRAMANTI, C.D. PAGANI, S. SALSA, Analisi matematica 2, Zanichelli, 2009.

M. BRAMANTI, Esercitazioni di Analisi Matematica 2, Ed. Esculapio, Bologna, 2012.

G. BUTTAZZO, V. COLLA, Temi di esame di Analisi Matematica II, Pitagora, 2001.

M. AMAR, A. M. BERSANI, Esercizi di Analisi Matematica per i Nuovi Corsi di Laurea, Progetto Leonardo Ed. Esculapio, 2002.

Working students, non-attending students, disabled and / or SLD students are invited to report this to the teacher in charge of the course in order to better interact with the student.
Educational objectives
The course represents the second course of Mathematical Analysis and examines the basic elements related to the functions of several variables with particular attention to their differentiation and Lebesgue-integration and the basic elements related to conservative fields and their integration.
The main objective of the course is to provide students with the basics to tackle the study of topics such as ordinary differential equations or partial differential equations and topics that interest differential geometry. Furthermore, the Lebesgue presentation of the theory of integration aims to lay the foundations of some arguments that will be presented in the study of probability.

On successful completion of the course, students should be able to:

- have acquired the properties of continuity, differentiability, optimization (both free and constrained) and integration (both on domains and on curves) for functions of several variables and have acquired the main elements on the study of conservative fields and their integration on curves.

- possess computational skills for solving exercises on the topics of the course.

- be able to explain in an appropriate language the properties and proofs on the differential and integral calculus for functions of several variables and on conservative fields.

- knowing how to communicate the mathematical knowledge acquired during the course and the related problems to others.

- understand the procedures that allow you to apply the contents of the course with other disciplines, in particular to Physics.

- acquire an analytical method in dealing with problems and exercises

- be able to autonomously provide a demonstration of simple propositions relating to the topics covered in the course

- to be able to recognize correct demonstrations and identify incorrect reasoning;

- to think critically, and express mathematical concepts precisely in writing

- to be prepared to take Mathematical Analysis III,

- be able to read and understand any text of Mathematical Analysis that deals with the topics described in the program

- to apply the knowledge gained in this course to other situations and disciplines,

- to apply knowledge and skills acquired in mathematical analysis to analyze and handle novel situations in a critical way.

- to be able to communicate the mathematical knowledge acquired in the course.

The skills listed are in my opinion indispensable for a mathematician who wishes to dedicate himself to teaching but also for a mathematician who wishes to carry out a professional technical and / or industrial activity.
Prerequisites
This course assumes that the student has a good working knowledge of Mathematical Analysis I topics including inequalities, limits, continuity, derivatives, basic integration and improper integrals on the real line. These prerequisites are concepts that students meet not only in the mentioned basic course of Mathematics but also in their pre-university education.

For non-attending students the prerequisites are the study of inequalities, study of limits, derivation and integration properties for functions of one variable, topics that are present in the mathematics programs of secondary schools, in pre-university education.
Teaching methods
The course is organized as follows:

- Lectures and exercises on all topics of the course (73 frontal hours: 63 hours for carrying out the course program and 10 hours in the classroom for carrying out exercises)

- student reception at the teacher's office or online on Teams: two hours a week will be dedicated to a more personalized reception to which students are invited to participate even in small groups (discussing it together also helps students who think they can do the exercise or get to know the proposed topic).
Furthermore, to support teaching, the lessons will be integrated with Assisted Study activities that will allow the teacher to carry out a training activity for learning the concepts presented in class. During the hours dedicated to Assisted Study, which will take place for the entire duration of the course, students are invited to clarify the topics covered in the previous week with the teacher and to discuss the topics covered with each other and with the teacher. These hours are aimed at a greater understanding and deepening of the properties, definitions and demonstrations presented in class).
For each topic the teacher will illustrate exercises during the hours of Assisted Study, which will be those proposed in the written exams of the previous years. (See: https://www.unistudium.unipg.it/unistudium/)

- another strategy used as a support to teaching is the tutoring activity carried out by a capable and deserving student, as established by the Intercourse Council. In additional hours to the lessons, the capable and deserving student will carry out, under the guidance of the teacher, further exercises in preparation for the exemption tests and the exams.

- finally, it is the intention of the teacher to prepare two exemption tests that invite students to study the topics in a calibrated way and to also have the possibility with a study distributed throughout the course to be able to take the exam more easily and in the exam sessions exam set in the academic year in progress.

- the textbook used meets the needs of the course which are to present a part of the reformed contents of Mathematical Analysis indicated for a second level course for the three-year degrees in Italian mathematics.

- the arguments presented are accompanied by examples and counterexamples in order to achieve a good understanding of the definitions and statements of the theorems.

- at the end of one or more related topics, the teacher will carry out exercises taken from the exams that allow them to deepen the topics covered and to relate them.

- If the course will also be accompanied by an stage-activity, then the course can also be supported by the presence of a student, who has already successfully passed the course, who will carry out further support activities on the topics covered in lesson.

- Numerous exercises and didactic material are available on the web page edited by the teacher (or at the address: https://www.unistudium.unipg.it/unistudium/) . Further material for the preparation of the written test, in addition to the one reported in the bibliography, can be found in all the books having Mathematical Analysis 2 as their topic, which are available for consultation in the library.

- If there are working students, non-attending students or students with disabilities, the teacher has prepared slides and handouts in Italian, available on the Unistudium platform. In any case, I advise non-attending students to report this to the teacher at the beginning of the lessons to decide together the most suitable strategy to arrive at a good preparation for the exam.

Some advices for to study the course, the book text and the exam tests:

1. Read the example problems carefully, completing some steps that are left out (ask someone for help if you can't follow the solution to a worked example).

2. Later use the worked examples to study by covering the solutions, and seeing if you can solve the problems on your own.

3. Keep in mind that sometimes an answer could be expressed in various ways that are analitically equivalent, so don't assume that your answer is wrong just because it doesn't have exactly the same form as the answer in the back.
Other information
The course is divided in 6+2 hours per week and its time schedule is available at the web page :
http://www.dmi.unipg.it/didattica/corsi-di-studio-in-matematica/matematica-triennale/orario-lezioni

Lecture halls : http://www.dmi.unipg.it/didattica/corsi-di-studio-in-matematica/matematica-triennale/orario-lezioni


- Attendance of the lectures is warmly recommended.

- Tutoring or Internship activities could be envisaged. This Tutoring / Internship, coordinated by the teacher, will aim to help students in the study and understanding of the course topics, with particular attention to carrying out the exercises on the topics of the course.








- During the reception hours (https://www.unipg.it/personale/tiziana.cardinali/didattica) students will also be followed in a personalized way.

- Material and news relating to the course can be found at:
https://www.unistudium.unipg.it/unistudium/

- The lectures will be companied by exercises sessions.
The teacher will distribute educational material on the argument : Lebesgue integration (in italian) useful for a better understanding of this topic.

- It is divided into 8 exam sessions available on the web page
http://www.dmi.unipg.it/didattica/corsi-di-studio-in-matematica/matematica-triennale/calendario-esami.

Commission: T. Cardinali, I. Benedetti, (A.Boccuto, R. Filippucci, P. Pucci, P. Rubbioni, A. Sambucini, E. Vitillaro).

- You can find the room where the exam takes place on the web page: http: //www.segreterie.unipg.it/self/gissweb.home

- For information on support services for students with disabilities and / or SLD, visit the page
http://www.unipg.it/disabilita-e-dsa

- Some tips:
Don't miss class. Ask questions. Go to office hours as often as necessary.

The final exam may be conducted in the English language at the request of the student.

Some suggestions for the exam:
You need to know the terminology used throughout this course.
Finally, many of the problems in this course will have multiple solution techniques and so you’ll need to be able to identify all the possible techniques and then decide which will be the easiest technique to use.

- All cell phones and electronic devices that transmit wirelessly must be turned off during the written exam. Vibrate or silence modes are not allowed. Laptops, iPods, language translators, or any devices that can receive a wireless signal are not allowed.



The final written exam is not given in the usual classroom. You will find the room assignments at
http: //www.segreterie.unipg.it/self/gissweb.home
There is a Web page which contains this course description as well as other information related to this course. Point your Web browser to:

https://www.unipg.it/personale/tiziana.cardinali/didattica

There is a webpage that contains the course description and other information related to this course::
https://www.unistudium.unipg.it/unistudium/
Learning verification modality
The final exam consist: written exam with open answer questions and oral exam.

- the written exam consists of three exercises one of which divided into several questions and takes three hours. The calculus abilities and knowledge are verified.

- the oral exam consists of a discussion on three topics one of which divided into several questions and takes about 30/40 minutes.
It is possible to do partial exams that, in case of a positive evaluation (with a grade> = 18), despense the student to do the final written-exam.

The partial exams are valid until the last call of the session of January / February 2024.

For the preparation of the written test it is important to attend actively, that is by asking questions about the program carried out and carefully reviewing the exercises presented in class or during the Assisted Study or during the Tutoring hours, then trying to carry out autonomously also those inserted by the teacher on the Unistudium page relating to the course in question.


The oral exam is designed to verify the level of knowledge attained by the student on the theoretical contents and on the methodologies of the course. Moreover, the oral examination allows the teacher to assess the performance of the student and his/her ability to organize the presentation in autonomy. The method, the language and the theoretical knowledge of the matter are verified. It is necessary that the student will need to know all definitions introduced in the course. Moreover, the student will need to understand them, how they work, and more importantly whether they can be used or not.
You can take the oral test in the same appeal of the written test of whether the latter has achieved a higher vote than or equal to 15/30. You can take the oral examination in appeals after the date on which the written exam is supported if on the latter it has achieved a higher vote than or equal to 18/30. If the oral examination is not enough, the teacher will decide if the written test is necessary to repeat (it depends on the reasons that have led to an assessment not enough of the oral test).
In the event that it is necessary according to the University regulations, the exams and partial exams could take place on the online platform.

For the preparation of the theoretical test it is important to attend in an active way, that is by asking questions about the program carried out. The right place for clarification is the hours dedicated to Assisted Study and the hours dedicated to student reception.

Students with DSA certification must submit the same at least two weeks prior to the test.

Valutation (it is advisable to take the oral exam with an evaluation> 14/30):
- Written test score < 18/30 and the oral exam is SATISFACTORY ALSO AS REGARDS THE PART OF THEORY,
the final mark of the exam can be at most rated 23/30.

- Written test score = o > 18/30 and < 20/30 and the oral exam is SATISFACTORY ALSO AS REGARDS THE PART OF THEORY, the final mark of the exam can be at most evaluated 28/30 .

- Written test score = o > of 20/30 and the oral exam is SATISFACTORY ALSO AS REGARDS THE PART OF THEORY,
the final mark of the exam can be at most 30/30 and honors.

If the oral test is not sufficient, the teacher will decide whether it is necessary to repeat the written test as well, it depends on the reasons that led to an insufficient evaluation of the oral test (for example: an insufficient oral test that takes place following a written test with a grade <18 it requires to repeat the written test as well, while an insufficient theoretical test following a written test with an assessment> 18 can lead to the decision to repeat only the theoretical test).

For the needs of didactic programming, being a teaching
without the obligation to attend, based on the requirements set out in the Syllabus (Rev. 3 of 11 March 2022) the following is specified:

"In the event that the student intends to anticipate the exam in a year prior to that
programmed in the study plan, it is recommended to attend the cycle of lessons e
to take the exam in the first useful session after the lessons themselves are
completed, thus respecting the semester of teaching planning ".

Information on support services for students with disabilities and / or DSA see: http://www.unipg.it/disabilita-e-dsa
Extended program
Vector functions and curves. Functions of several variables: continuity, partial derivability, directional derivability, differentiability, maximums and minimums with and without constraints. Lagrange Multipliers. Chain Rules. Implicit functions. Lebesgue integration in R^n. Polar coordinates in R2, cylindrical coordinates spherical coordinates. Integrals on curves. Differential forms and their integration. Gauss and Green's theorem, divergence theorem, Stokes' theorem in R^2.
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