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
The general objective of this course is to understand, know, and handle the main concepts, results, and methods related to functional sequences and series, and to the theory of Riemann integration for real functions of several real variables.
More specifically, the following objectives are set:
OB1 - Analyze the behavior of functional sequences and series, distinguishing between pointwise and uniform convergence.
OB2 - Know sufficient conditions for the limit (sum) of a functional sequence (series) to inherit the regularity properties of the terms of the corresponding functional sequence (series).
OB3 - Study the convergent or divergent nature of improper integrals and, when possible, calculate their value.
OB4 - Develop the theory of Riemann integration for functions of several real variables on Jordan-measurable sets.
OB5 - Study the Riemann-integrable nature of functions of several real variables on Jordan-measurable sets.
OB6 - Calculate multiple integrals on Jordan-measurable sets using Fubini's theorem and the change of variable theorem with some of the most common transformations in the plane and in space.
OB7 - Use MAPLE as support for completing activities with the aim of enhancing conceptual understanding, discovery, and comparison of results inherent to the course.
1 - Improper Integrals of a Real Variable (6 hours of lectures)
1.1 - Improper Integrals
Integration on non-compact intervals. Convergent and divergent integrals. Properties of improper integration. Cauchy condition for the convergence of an integral.
1.2 - Convergence Criteria
Characterization of the convergence of integrals of non-negative functions. Comparison criteria, quotient comparison, and limit comparison. Study of some reference integrals. Conditional convergence and absolute convergence of integrals. Dirichlet's criterion.
1.3 - Improper Integrals and Numerical Series.
2 - Functional Sequences and Series (12 hours of lectures)
2.1 - Functional Sequences
Pointwise convergence and uniform convergence. Cauchy condition for uniform convergence. Results on continuity, differentiability, and integrability of the limit function.
2.2 - Functional Series
Pointwise, absolute, and uniform convergence of a series of functions. Cauchy condition for uniform convergence of a series. Weierstrass Majorant criterion. Results on continuity, differentiability, and integrability of the sum function.
2.3 - Power Series
Radius of convergence. Cauchy-Hadamard formula. Uniform convergence. Properties of continuity, differentiability, and integrability of the sum. Taylor series. Analytical functions.
3 - Riemann Multiple Integral (10 hours of lectures)
3.1 - Riemann Integral in Compact Rectangles of R^n
Partitions of a rectangle. Riemann sums. R-integrable functions in compact rectangles and Riemann integral. Upper and lower sums. Lower and upper integrals. Equivalent formulations of the concept of integrable function in the sense of Riemann. Properties of the integral.
3.2 - Riemann-Integrable Functions in Compact Rectangles
Null-content and null-measure sets. Lebesgue's characterization of integrability in the sense of Riemann. Fubini's Theorem in rectangles.
3.3 - Integration on Measurable Sets in the Sense of Jordan
J-measurable sets. Integration on J-measurable sets. R-integrable functions on J-measurable sets. Properties of the Riemann integral. Fubini's theorem on J-measurable sets.
3.4 - The Change of Variable Theorem. Some Special Change of Variables.
Note: The content of the course is subject to development in an order different from the one presented.
Basic Bibliography
Apostol TM. Análisis matemático. 2a ed. Barcelona: Reverté; 1979.
Bombal F, Rodríguez Marín L, Vera Botí G. Problemas de análisis matemático. 2a ed, Madrid: AC; 1994.
Fernández Viña JA, Sánchez Mañes E. Ejercicios y complementos de análisis matemático III. Madrid: Tecnos; 1992.
Supplementary Bibliography
Apostol TM. Calculus. 2a ed. Barcelona: Editorial Reverté; 1984.
Bartle RG, Sherbert DR. Introducción al análisis matemático de una variable. 2a ed. México: Limusa Wiley; 2010.
Boss, V. Lecciones de matemática. Tomo 1. Análisis. Moscú: URSS, 2008.
de Burgos J. Cálculo infinitesimal de una variable. 2a ed. Madrid: McGraw-Hill; 2006.
de Burgos J. Cálculo infinitesimal de varias variables. 2a ed. Madrid: McGraw-Hill; 2008.
del Castillo F. Análisis matemático II. Madrid: Alhambra; 1980.
Fernández Viña JA. Análisis matemático III. Integración y Cálculo exterior. Madrid: Tecnos; 1992.
Spivak M. Cálculo en variedades. Barcelona: Reverté; 1982.
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.
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 students in group CLE01, two evaluation modalities are proposed: Modality 1 and Modality 2. At the beginning of the course, within the established period and through the Virtual Campus, students must individually choose the modality they deem appropriate. In order for the teaching staff to organize and schedule continuous assessment activities in advance, it is essential that students choosing either of the two assessment modalities make their selection in the Virtual Campus within the designated deadlines.
Modality 1 requires active participation in class and the completion of at least 80% of the proposed activities, either individual or group-based. The final exam, which is also in-person, is mandatory and is considered an additional activity in this modality. The final grade (FG) will be calculated according to the following formula: FG=E+min{T/4, 10-E}, where E is the final exam grade and T is the grade for the activities completed during the course (both between 0 and 10). The coursework can account for up to 25% of the final grade.
Modality 2, designed for those who prefer more autonomy, consists of an intermediate test (IT), with a pre-established and announced date. The final grade will be calculated according to the following formula: FG=max{E, 0.7E+0.3IT}, where E is the final exam grade and IT is the intermediate test grade.
For students in group CLE02, only Modality 2 will be available, which consists of two intermediate tests with pre-established and announced dates. The final grade will not be lower than the grade obtained using the following formula: FG=max{E, 0.7E+0.3IT}, where E is the final exam grade and IT is the arithmetic average of the two intermediate tests.
The final exam may differ for the lecture groups. In any case, coordination and equivalence in training across all groups will be guaranteed.
In the second opportunity, the same evaluation system will be used, maintaining the grades for activities and intermediate tests completed during the course and updating the final exam grade.
In cases of fraudulent completion of tasks or tests (plagiarism or improper use of technology), the provisions of the Regulation for Academic Performance Evaluation and Grade Review will be applied.
Students who do not take the final exam will receive a "Not Presented" grade.
TOTAL HOURS
150 hours: 58 in-class hours and 92 non-classroom hours.
IN-CLASS TEACHING (26 hours CLE + 14 hours CLIS + 14 hours CLIL + 2 hours TGMR + 2 hours CLE test sessions),
(CLE) Lectures (26 hours)
(CLE) Test sessions (2 hours)
(CLIS) Interactive seminar classes (14 hours)
(CLIL) Interactive laboratory classes/group tutorials (14 hours)
(TGMR) Small group tutorials (2 hours)
NON-CLASSROOM PERSONAL WORK TIME
On average, 92 hours are estimated to be necessary.
It is recommended to have completed and passed the following subjects: Introduction to Mathematical Analysis, Continuity and Differentiability of Functions of a Real Variable, Integration of Functions of a Real Variable, Topology of Euclidean Spaces, and Differentiation of Functions of Several Real Variables.
It is advisable to study the subject regularly and to try to complete the proposed exercises independently. It is recommended to consult the teaching staff with any questions that may arise throughout the course.
Rosa Mª Trinchet Soria
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813205
- rosam.trinchet [at] usc.es
- Category
- Professor: University Lecturer
Jorge Losada Rodriguez
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813215
- jorge.losada.rodriguez [at] usc.es
- Category
- PROFESOR/A PERMANENTE LABORAL
Daniel Cao Labora
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813174
- daniel.cao [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
- PROFESOR/A PERMANENTE LABORAL
| Tuesday | |||
|---|---|---|---|
| 16:00-17:00 | Grupo /CLE_02 | Galician, Spanish | Classroom 03 |
| 19:00-20:00 | Grupo /CLE_01 | Galician | Classroom 02 |
| Wednesday | |||
| 15:00-16:00 | Grupo /CLIL_02 | Galician | Computer room 2 |
| 16:00-17:00 | Grupo /CLIL_01 | Galician | Computer room 2 |
| 18:00-19:00 | Grupo /CLIL_04 | Galician | Computer room 2 |
| 19:00-20:00 | Grupo /CLIL_03 | Galician | Computer room 2 |
| Thursday | |||
| 15:00-16:00 | Grupo /CLIL_06 | Spanish, Galician | Computer room 3 |
| 16:00-17:00 | Grupo /CLIL_05 | Spanish | Computer room 3 |
| 18:00-19:00 | Grupo /CLIL_07 | Galician, Spanish | Computer room 3 |
| 19:00-20:00 | Grupo /CLIL_08 | Galician, Spanish | Computer room 3 |
| Friday | |||
| 15:00-16:00 | Grupo /CLIS_01 | Galician | Classroom 06 |
| 15:00-16:00 | Grupo /CLIS_04 | Galician, Spanish | Classroom 09 |
| 16:00-17:00 | Grupo /CLIS_02 | Galician | Classroom 07 |
| 16:00-17:00 | Grupo /CLIS_03 | Spanish, Galician | Classroom 09 |
| 05.26.2026 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |
| 07.09.2026 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |