ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 99 Hours of tutorials: 3 Expository Class: 24 Interactive Classroom: 24 Total: 150
Use languages Spanish, Galician
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Departments: Statistics, Mathematical Analysis and Optimisation
Areas: Mathematical Analysis
Center Faculty of Mathematics
Call: Second Semester
Teaching: With teaching
Enrolment: Enrollable
To understand, know and handle the main concepts, results and methods related to sequences and function's series, which have a basic importance in the mathematical analysis. More immediately for the students, it is equally relevant in other subjects of undergraduate mathematics such as subjects on complex, functional analysis, Lebesgue's integration and Fourier series.
The mentioned contents can be divided in two contents blocks. In the first one, sequences and series of functions are introduced and different ways of convergence for them are studied. In the second one, the knowledge on integrals of funcions of one real variable are enlarged, presenting the basics of improper integrals of functions of one real variable and Riemann multiple integrals.
The achievement of the objectives will be obtained if the theoretical and practical contents are known and can be applied in concrete problems of different nature, occasionally, maybe, by means of the computer. If some software is needed, then Maple will be used.
I) IMPROPER INTEGRALS OF FUNCTIONS OF ONE REAL VARIABLE (6 h. expositivas apprx.)
1.1 Improper integrals.
Integration in non compact intervals. Convergent and divergent integrals. Properties of improper integrals . Cauchy's condition for the convergence of an integral.
1.2 Criteria of convergence.
Characterization of convergence of integrals of non negative functions. Comparison test, ratio and comparison limit test. Study of some integrals of relevance. Convergence and absolute convergence of integrals. Dirichlet's criteria.
1.3 Improper integrals and numerical series.
II) SEQUENCES AND SERIES OF FUNCTIONS (12 h. expositivas apprx.)
2.1 Sequences of functions.
Point-wise and uniform convergence. Cauchy's condition for uniform convergence. Results on continuity, derivability and integrability of a limit.
2.2 Series of functions.
Point-wise, absolute and uniform convergence of a series of functions. Cauchy's condition for uniform convergence of a series. Weierstrass's test for convergence. Results on continuity, derivability and integrability of a sum.
2.3 Power series.
Radius of convergence. Cauchy-Hadamard's theorem. Uniform convergence. Properties on continuity, derivability and integrability of a sum. Taylor series. Analytic functions.
III) RIEMANN MULTIPLE INTEGRALS (10 h. expositivas apprx.)
3.1 Riemann integral in R^n.
Partitions of an interval. Darboux's sums. Lower and higher integrals. Riemann integrable functions and their integrals. Properties of Riemann integral. Riemann sums for integrals. Riemann theorem.
3.2 Riemann integrable functions on an interval.
Zero content and zero measure sets. Oscillation of a function arround a point. Lebesgue's characterization of integrability in Riemann sense.
3.3 Integration in Jordan measurable sets.
Jordan measurable sets. Integration in Jordan measurable sets. Riemann integrable functions in measurable sets. Properties of Riemann integral. Fubini's theorem.
3.4 Integral calculus.
The Theorem for change of variable. Polar, spherical and cylindrical coordinates.
COMPLEMENTARY BIBLIOGRAPHY
T. M. Apostol: "Análisis Matemático". Ed. Reverté. 1977.
T. M. Apostol: "Calculus, volumen 2". Ed. Reverté. 1973.
R. G. Bartle, D. R. Sherbert: "Introducción al Análisis Matemático de una variable". Ed. Limusa.
R. G. Bartle: "Introducción al Análisis Matemático". Ed. Limusa.
F. Bombal, L .R. Marín, G.Vera: "Problemas de Análisis Matemático. Vol. 3: Cálculo integral". Ed. AC.
J. de Burgos: "Cálculo infinitesimal de una variable". Ed. McGraw-Hill.
J. de Burgos: "Cálculo infinitesimal de varias variables". Ed. McGraw-Hill.
F. del Castillo: "Análisis Matemático II". Ed. Alhambra. 1980.
J.A. Fernández Viña: "Análisis Matemático III, Integración y Cálculo exterior". Ed. Tecnos 1992.
J.A. Fernández Viña, Eva Sánchez Mañes: "Ejercicios y Complementos de Análisis Matematico III". Ed. Tecnos, 1994.
M. Spivak: "Cálculo en Variedades". Ed. Reverté. 1988.
The competences mentioned in the Guidelines for the Bachelor of Mathematics at USC for this subject: CG1, CG2, CG3, CG4 y CG5; CT1, CT2, CT3 y CT5; CE1, CE2, CE3, CE4, CE5, CE6 y CE9.
Such competences will be achieved by doing the following activities:
• Analysis of sequences and function's series, distinguishing the notions of point-wise and uniform convergence, and showing conditions that allow the functional limit (the sum) to inherit the properties of regularity of corresponding series (sequences) of functions.
• Study of the convergence of improper integrals, calculating its value when it is posible.
• Construction of Riemann integral of functions of several variables in Jordan measurable sets.
• Study of the integrability in Riemann sense for functions of several variables in measurable sets in the sense of Jordan.
• Calculation of multiple integrals in measurable sets in the sense of Jordan, use of Fubini's theorem and the theorem for change of variable with the most frequent transformations (polar, cylindrical and spherical).
• Use certain softwares to support the visualizations and the calculations.
In general, the resources needed for the developing of the subject (notes, explanatory videos, folders...) will be provided to the students by means of the virtual classroom, as established for the three scenes in the document entitled "Directrices para o desenvolvemento dunha docencia presencial segura".
Scene 1 (adapted normality, face-to-face lessons)
The contents of the subject can be presented in different orders. The order given above may be modified if the circumstances advise so, during the face-to-face lessons destined by the Faculty of Mathematics to this end.
The developing of the subject will promote the students' learning as well as the continuous assessment, by means of different (voluntary) proposals of exercises. Moreover, the participation during the lessons will be also encouraged.
MAPLE program will be used for computer classes.
To make easier the learning of the subject, instructional materials (in galician) will be prepared. This includes the following: notes about the contents of the subject, explanatory videos, Maple practices for students use and others. All this material will be available for students in the virtual classroom of the subject.
For Scenes 2 and 3, see the modifications presented in the item Comments below.
Scene 1
In the developing of the subject, we will support the continuous assessment, which will be face-to-face, for the students that wish it. In this way, being participatory and hard-working students, they will have the opportunity of obtaining a percentage of their final grade by means of different (voluntary) activities, which could be per individual or in group and need be presented in written or oral form before the due date as recommended by the teacher.
In this kind of assessment (which will be called Modality 1 and where the students are expected to participate during the lessons and to do at least the 80% of the proposed activities) the final examination (which will be face-to-face) will be just another activity, but mandatory for the assessment of the students. These activities will be useful for assessing the understanding as well as the general, specific and transversal competences achieved by the students.
The final grade will be obtained following the general criteria on assessment established in the Guidelines for the Bachelor of Mathematics at USC. In the right conditions, the percentage of the final grade assigned to the work of students during the course (without the final examination) could reach the 25% of the final grade (CF), as in the following formula: CF = E + min{T/4, 10 - E}, where E means the final examination mark and T represents the mark obtained in the continuous assessment. Both E and T will be between 0 and 10.
A second alternative for the assessment, which will be called Modality 2, will be available. It will consist of an intermediate examination with a previously fixed schedule. In this case, the following formula will be used: CF=máx{E, 0’7*E+0’3*PI}, where PI stands for the intermediate examination, whose mark will be between 0 and 10. As in Modality 1, it will be mandatory to do the final examination in order to be assessed.
At the beginning of the course, the students will have the opportunity to choose between both alternatives: Modality 1 or 2. This will be possible thanks to a choice activity in Moodle. If the choice is not made in the established period, then Modality 2 will be assigned automatically.
Despite this, any student may be able to achieve the maximum grade just with the final examination, independently of the marks obtained in the rest of activities or in the intermediate examination. The students who does not make the final examination will be assessed as "Non presentado".
The assessment system will be the same in both opportunities.
For Scenes 2 and 3, see the modifications presented in the item Comments below.
150 hours: 58 face-to-face hours and 92 hours of independent study.
It is important to understant the contents of the following topics to study this subject: Introduction to mathematical analysis, Continuity and derivability of functions of one real variable; Integration of functions of one real variable; Differentiation of functions of several real variables.
On the other hand, it is recommended to study regularly, and to make all the activities that are proposed during the (face-to-face or virtual) lessons. It is also very important to clarify all the doubts a student may face during the course.
Contingency Plan
In general, during Scenes 2 (blended learning course) and 3 (distance learning), we will follow as much as possible the program elaborated for Scene 1, except with respect to the face-to-face activities (lessons, tutorials, oral expositions on the subject by the students...), which will be developed virtually when face-to-face were not possible. This will be made by means of Moodle platform (virtual classroom), Teams or others provided that the suitable technical equipment be supplied.
Teaching methodology in Scenes 2 and 3
*Scene 2 (blended learning course)
Teaching will be partially virtual, following the guidelines provided by the Faculty of Mathematics. The technical platforms for the virtual teaching as well as Moodle platform will be used.
*Scene 3 (distance learning)
Virtual teaching by means of the technical platform provided to this end.
In both scenes, tutorials will be telematic. The participation of the students in the class will be (partially or totally) replaced by "virtual participation", for instance, in thematic forums specially created for some activity or discussions by means of a virtual platform.
Assessment system in Scenes 2 and 3
*Scene 2 (blended learning course)
As described for Scene 1, with the exception that the activities will be virtual instead of face-to-face activities. Expositions on the subject will be made by means of virtual platforms, if face-to-face is not possible. If possible, the final examination will be face-to-face.
*Scene 3 (distance learning)
As described for Scene 1, with the exception that the activities and expositions on the subject will be virtual instead of face-to-face activities. If possible, the final examination will be face-to-face.
Rosa Mª Trinchet Soria
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813205
- rosam.trinchet [at] usc.es
- Category
- Professor: University Lecturer
Jorge Rodríguez López
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- jorgerodriguez.lopez [at] usc.es
- Category
- Professor: LOU (Organic Law for Universities) PhD Assistant Professor
Jorge Rodríguez López
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- jorgerodriguez.lopez [at] usc.es
- Category
- Professor: Temporary supply professor to reduce teaching hours
| Monday | |||
|---|---|---|---|
| 15:00-16:00 | Grupo /CLE_02 | Galician | Classroom 08 |
| 17:00-18:00 | Grupo /CLE_01 | Galician | Classroom 07 |
| Tuesday | |||
| 15:00-16:00 | Grupo /CLE_02 | Galician | Classroom 08 |
| 17:00-18:00 | Grupo /CLE_01 | Galician | Classroom 07 |
| Wednesday | |||
| 15:00-16:00 | Grupo /CLIS_03 | Galician | Ramón María Aller Ulloa Main Hall |
| Thursday | |||
| 15:00-16:00 | Grupo /CLIS_04 | Galician | Ramón María Aller Ulloa Main Hall |
| 16:00-17:00 | Grupo /CLIS_01 | Galician | Classroom 06 |
| 16:00-17:00 | Grupo /CLIL_05 | Galician | Computer room 3 |
| 17:00-18:00 | Grupo /CLIS_02 | Galician | Classroom 06 |
| 17:00-18:00 | Grupo /CLIL_06 | Galician | Computer room 3 |
| 18:00-19:00 | Grupo /CLIL_03 | Galician | Computer room 2 |
| 19:00-20:00 | Grupo /CLIL_01 | Galician | Computer room 2 |
| Friday | |||
| 15:00-16:00 | Grupo /CLIL_04 | Galician | Computer room 3 |
| 16:00-17:00 | Grupo /CLIL_02 | Galician | Computer room 4 |
| 05.24.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 02 |
| 05.24.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 03 |
| 05.24.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |
| 05.24.2021 10:00-14:00 | Grupo /CLE_01 | Ramón María Aller Ulloa Main Hall |
| 07.05.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |