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: First Semester
Teaching: With teaching
Enrolment: Enrollable
To understand, know and manage the main concepts, methods and results related to vector calculus and Lebesgue integral of several real variables:
• To manage the concepts of flow divergence and rotational of a vector field and its dynamic significance.
• To understand the concepts and properties of the line integral of scalar and vectorial fields, and its practical application to concrete examples.
• To understand the concepts and properties of the surface integral of scalar and vectorial fields, and its practical application to concrete examples.
• To check on verification examples of the theorems of Green, Stokes and Gauss.
• To know the construction of the measure and the Lebesgue integral for functions of several real variables.
• To have the ability to determine the nature of some examples of Lebesgue measurable sets and functions, and the integrability of functions on measurable sets.
• Mastering the convergence theorems of Lebesgue integral and have the ability to apply them in specific cases.
• To understand the relationship between the Riemann and Lebesgue integrals, and the importance of the extension process that involves consideration of the latter
• To know and use the theorems of Fubini and the changing variable in the Lebesgue integral.
These concepts are of fundamental importance in mathematical analysis as well as in other matters of the degree in mathematics, such as those relating to differential geometry, differential equations and applied mathematics.
Lebesgue integration (14 hours CLE*):
2.1 (3 hours CLE) Outer measure of subsets of Rn. Lebesgue-measurable sets and Lebesgue measure. Null-measure sets. The sigma-algebra of Lebesgue measurable sets. Properties of the Lebesgue measure.
2.2 (4 hours CLE) Measurable functions. Properties. Simple measurable functions. Approximation of measurable functions by simple measurable functions. Egorov’s Theorem. Luzin’s Theorem.
2.3 (4 hours CLE) Integral of nonnegative simple measurable functions. Integral of nonnegative measurable functions. Properties. Monotone convergence theorem. Fatou’s Lemma. Lebesgue-integrable functions and Lebesgue integral. Properties of the Lebesgue integral. Dominated convergence theorem. The space L1.
2.4 (3 hours CLE) Relationship between Riemann and Lebesgue integrals. Theorems of Tonelli and Fubini. Theorem of Change of variables.
Vector calculus (12 hours CLE):
1.1 (3 hours CLE) Scalar and vector fields. Gradient, divergence, and rotational. Basic identities in vector analysis. Flow associated to a vector field. Gradient fields and potential functions.
1.2 (3 hours CLE) Parametrized curves in Rn. Piecewise regular curves. Tangent vector. Line integral of a scalar field. Arc lenght. Oriented curves. Line integral of a vector field. Equivalence of curves and oriented curves. Fundamental theorems for line integrals. Characterization of conservative fields
1.3 (3 hours CLE) Parametrized surfaces in R3. Regular surfaces. Normal vector. Orientable surfaces. Surface integral of a scalar field. Area of a regular surface. Surface integral of a vector field. Equivalent surfaces.
1.4 (3 hours CLE) Theorems of Green, Stokes and Gauss. Consequences and aplications.
*CLE=Blackboard classes in big group
Basic Bibliography
Del Castillo, F.: “Análisis Matemático II”. Ed. Alhambra. 1987.
Mardsen, J.E.; Tromba, A. J.: “Cálculo Vectorial”. 5ª edición. Ed. Addison Wesley. 1987.
Course notes
Complementary Bibliography
Apostol, T. M.: “Calculus, volumen 2”. Ed. Reverté. 1973.
Bombal, F.; Marín, R.; Vera, G.: “Problemas de Análisis Matemático, 3. Cálculo Integral”. Ed. AC. 1987.
Chae, S. B.: "Lebesgue Integration." Second edition, Springer-Verlag, 1995.
Fernández Viña, J. A.: “Análisis Matemático III. Integración y cálculo exterior”. Ed. Tecnos. 1992.
Franks, J.: "A (Terse) Introduction to Lebesgue Integration". AMS, 2009.
Kurtz, D. S., Swartz, Ch. W.: “Theories of integration. The integrals of Riemann, Lebesgue, Henstock-Kurzweil, and Mcshane”. Series in Real Analysis, 9. World Scientific Publishing Co., Inc., River Edge, NJ, 2004.
Spiegel, M. R.: “Análisis Vectorial". McGraw Hill, 1991.
Weaver, N.: "Measure Theory and Functional Analysis". World Scientific, 2013.
In addition to contribute to achieve the general and transverse competences taken up in the memory of the degree,
http://www.usc.es/export/sites/default/gl/servizos/sxopra/memorias_grao…,
this subject will allow the student to get the following specific competences:
CE1 - To understand and use mathematical language;
CE2 - TO know rigorous proofs of some classical theorems in different areas of mathematics;
CE3 - To devise demonstrations of mathematical results, formulate conjectures and imagine strategies to confirm or refute them;
CE4 - To Identify errors in faulty reasoning, proposing demonstrations or counterexamples;
CE5 - To assimilate the definition of a new mathematical object, and to be able to use it in different contexts;
CE6 - To identify the abstrac properties and material facts of a problem, distinguishing them from those purely occasional or incidental;
CE9 - To use statistical analysis applications, numerical and symbolic computation, graphical visualization, optimization and scientific software, to experience and solve problems in mathematics.
The general methodological indications established in the USC Mathematics Degree Title Memory will be followed.
Teaching is programmed in theoretical classes, small group practices, small group computer practices and tutorials. In the theoretical classes the essential contents of the discipline will be presented, and will allow the work of the basic, general and transversal competences, in addition to the specific competences CE1, CE2, CE5 and CE6. On the other hand, in the interactive sessions problems or exercises of more autonomous realization will be proposed, which will allow to emphasize the acquisition of specific competences CE3 and CE4. Finally, the tutorials will be devoted to discussion and debate with the students, and to the resolution of the proposed tasks with which it is intended that the students practice and strengthen knowledge. In computer classes appropriate computer tools will be used to work, in this way, the specific competence CE9.
In a generic way to the three scenarios that will be detailed below, the virtual course will be used as a mechanism to transfer the necessary resources for the development of the subject to the students (explanatory videos, notes, exercise bulletins, etc.). The fundamental difference between the three scenarios will be, on the one hand, the way in which the tutorials will be taken and, on the other hand, the type of teaching that will be taught: totally face-to-face in scenario 1, semi-presential in scenario 2 and totally virtual on scenario 3.
SCENARIO 1 (adapted normality):
The expository and interactive teaching will be face-to-face and will be completed with the virtual course of the subject, in which the students will have bibliographic materials, problem bulletins, explanatory videos, etc. Through the virtual course the student will also carry out tests for continuous assessment, as described in the corresponding section.
The tutorials will be in person or through email.
SCENARIO 2 (distancing)
Partially virtual teaching, according to the distribution organized by the Faculty of Mathematics. For this, the virtual classroom of the course will be used, in which explanatory videos, bibliographic materials, problem bulletins, etc., provided by the teachers and, if so established, synchronous virtual classes using MS Teams will be uploaded. Students can also take tests for continuous assessment through the virtual campus, as described in the corresponding section.
The tutorials will be attended by email or by MS Teams.
SCENARIO 3 (closure of the facilities):
Totally remote teaching through the virtual course of the subject and the MS teams. For this, the virtual classroom of the course will be used, in which explanatory videos, bibliographic materials, problem bulletins, etc., provided by the teachers and synchronous virtual classes using MS Teams will be uploaded. Students can also take tests and tests for continuous assessment through the virtual campus, as described in the corresponding section.
The tutorials will be attended by email or by MS Teams.
In a generic way, there will be a continuous assessment in which a continuous formative evaluation is combined with a final test.
The continuous formative assessment will be robust against possible changes in scenario. In this sense, the virtual campus tools will be used as the only mechanism to carry out a line. In addition, it will allow verifying the degree of achievement of the specific competences previously determined, with an emphasis on the transversal competences CT1, CT2, CT3 and CT5.
Regarding the final and second chance test, the difference between the three scenarios consists of the way it is carried out: face-to-face in the case of scenario 1 and, in scenarios 2 and 3, by telematic means. Specific tests will measure the knowledge obtained by the students in relation to the concepts and results of the subject, both from a theoretical and practical point of view, also evaluating the clarity and logical rigor shown in their exposition. The achievement of the basic, general and specific competences referred to in the Memory of the Degree in Mathematics of the USC and which were previously indicated will be evaluated.
SCENARIO 1 (adapted normality):
As previously mentioned, the evaluation will be carried out by combining a continuous formative assessment with a final test.
The continuous formative assessment will consist, on the one hand, of carrying out tests that the students must carry out at the end of each of the subjects of the subject and, on the other hand, of carrying out tests that will be carried out in the virtual course . These tests will be based on standard exercises, demonstrations, etc. Since the subject is made up of two large blocks, there will be one test per block. To calculate the mark of the continuous evaluation (C) the following formula will be used:
C = 0.4 * TAE + 0.3 * P1 + 0.3 * P2.
Where TAE is the global grade of the tests (out of 10 points), P1 is the grade of the test associated with the first block (out of 10 points) and P2 is the grade associated with the test of the second block.
With the mark of the continuous formative assessment (C) and the mark of the final face-to-face test (F) the final mark of the subject (NF) will be calculated using the following formula:
NF = max {F, 0.5 * C + 0.5 * F}
NOT PRESENTED is understood as a student who at the end of the teaching period is not in a position to pass the subject without taking the final test and does not appear for said test.
In the second opportunity, the same evaluation system will be used, but with the test corresponding to the second opportunity, which will be an exam of the same type as that of the first.
SCENARIO 2 (distancing)
Same procedure as that described for SCENARIO 1, with the only difference that the final test or the second chance test will be telematic.
SCENARIO 3 (closure of the facilities)
Same procedure as that described for SCENARIO 1, with the only difference that the final test or the second chance test will be telematic.
Warning. In cases where the tests are carried out fraudulently (plagiarism or improper use of technologies), the provisions of the Regulations for evaluating the academic performance of students and reviewing grades will apply.
PRESENCE WORK IN THE CLASS (54 hours)
(CLE) Blackboard classes in big group (26 hours)
(CLIS) Interactive classes in reduced group (13 hours)
(CLIL) Interactive classes of laboratory/tutorials in reduced group (13 hours)
(TGMR) Small group tutorials or individualized (2 hours)
PERSONAL WORK OF THE STUDENT (96 hours)
Personal work will depend on the students. On average, 96 hours per student are estimated.
To study this subject is important to master the contents of the following: Introduction to Mathematical Analysis, Continuity and differentiability of functions of one real variable, Integration of functions of one real variable, topology of Euclidean spaces. Differentiation of functions of several real variables. Functional series and Riemann integral in several variables.
Moreover, it is recommended to study regularly, taking the matter up, and perform all the activities proposed in the classroom. It is also very important to consult with the teacher all the doubts that may arise along the way.
Contingency plan:
Adaptation of the methodology to Scenarios 2 and 3:
SCENARIO 2 (distancing):
Partially virtual teaching, according to the distribution organized by the Faculty of Mathematics. For this, the MS Teams will be used for synchronous telematic teaching and the virtual classroom of the course, with explanatory videos and bibliographic materials provided by the teaching staff.
The tutorials will be attended by email or through MS Teams.
SCENARIO 3 (closure of the facilities):
Same procedure as described for SCENARIO 1, with the only difference that the final test will be telematic.
In the second opportunity, the same evaluation system will be used, but with the test corresponding to the second opportunity, which will be an exam of the same type as that of the first.
Alberto Cabada Fernandez
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813206
- alberto.cabada [at] usc.gal
- Category
- Professor: University Professor
Francisco Javier Fernandez Fernandez
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813231
- fjavier.fernandez [at] usc.es
- Category
- Professor: LOU (Organic Law for Universities) PhD Assistant Professor
Daniel Cao Labora
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813174
- daniel.cao [at] usc.es
- Category
- Professor: Temporary supply professor to reduce teaching hours
| Monday | |||
|---|---|---|---|
| 09:00-10:00 | Grupo /CLE_01 | Spanish | Classroom 07 |
| 13:00-14:00 | Grupo /CLE_02 | Galician | Classroom 07 |
| Tuesday | |||
| 13:00-14:00 | Grupo /CLE_01 | Spanish | Classroom 09 |
| Wednesday | |||
| 10:00-11:00 | Grupo /CLIL_03 | Spanish | Classroom 06 |
| 11:00-12:00 | Grupo /CLE_02 | Galician | Classroom 10 |
| 12:00-13:00 | Grupo /CLIL_01 | Spanish | Computer room 3 |
| 13:00-14:00 | Grupo /CLIL_02 | Spanish | Ramón María Aller Ulloa Main Hall |
| Thursday | |||
| 10:00-11:00 | Grupo /CLIL_06 | Galician | Classroom 06 |
| 12:00-13:00 | Grupo /CLIL_05 | Galician | Computer room 3 |
| 13:00-14:00 | Grupo /CLIL_04 | Spanish, Galician | Graduation Hall |
| Friday | |||
| 09:00-10:00 | Grupo /CLIS_04 | Galician | Classroom 06 |
| 10:00-11:00 | Grupo /CLIS_03 | Galician | Classroom 06 |
| 11:00-12:00 | Grupo /CLIS_02 | Galician, Spanish | Classroom 02 |
| 12:00-13:00 | Grupo /CLIS_01 | Galician, Spanish | Classroom 06 |
| 01.11.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 02 |
| 01.11.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 03 |
| 01.11.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |
| 01.11.2021 10:00-14:00 | Grupo /CLE_01 | Ramón María Aller Ulloa Main Hall |
| 06.25.2021 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |