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
The theoretical basis and the basic skills of the differential calculus in the area of real multivariate functions will be provided. Being this course a basic one, the main goal is the mathematical training of the student on the field of the differential calculus of (real) multivariate functions that is needed on the Bachelor of Mathematics as well as the resolution of some simple real life problems related with the topic.
1. Calculation of limits of real multivariate functions. Directional and iterated limits. Continuity. (2 hours CLE)
2. Partial derivative of a function at a point. Derivative according to a vector. Concept of differential and differentiable function. Properties. Necessary and sufficient conditions of differentiability. The gradient vector. Geometric interpretations of the previous concepts. (5 hours CLE)
3. Differentiability of functions into R^m. Jacobian matrix. Differentiation rules. (3 hours CLE)
4. The mean value theorem in several variables. (2 hours CLE)
5. Higher order derivation and differentiation. Study of the second order differential. The hessian matrix. Symmetry of the second order differential. (4 hours CLE)
6. Functions of class m. Taylor's formula. Relative extrema. (4 hours CLE)
7. Implicit function theorem and inverse function theorem. (4 hours CLE)
8. Applications of the implicit and inverse function theorems. Constrained extrema. Change of variable. Geometrical problems. (2 hours CLE)
APOSTOL, T. M., Análisis Matemático, Ed. Reverté, 1991.
BARTLE, R. G., Introducción al Análisis Matemático, 1st ed., Limusa, 1991.
BESADA, M.; GARCÍA, F. J.; MIRÁS, M. A.; VÁZQUEZ, C. Cálculo de Varias Variables. Cuestiones y Ejercicios Resueltos. Prentice Hall, 1, 2001.
BESADA, M.; GARCÍA, F. J.; MIRÁS, M. A.; VÁZQUEZ, C. Cálculo Diferencial en Varias Variables. Cuestiones Tipo Test y Ejercicios Resueltos. Garceta Grupo Editorial, 2011.
BOMBAL, F.; RODRÍGUEZ, L.; VERA, G. Problemas de Análisis Matemático 2º. Cálculo diferencial. Ed. AC. 1991.
BURGOS ROMAN, JUAN de. Cálculo Infinitesimal de Varias Variables. McGraw-Hill/ Interamericana de España. 2008.
FERNÁNDEZ VIÑA, J.A. Análisis Matemático II:Topologia y Cálculo diferencial. 2nd ed. Tecnos. 1993.
FERNÁNDEZ VIÑA, J.A.; SÁNCHEZ MAÑÉS, E. Ejercicios y Complementos de análisis Matemático II. 2nd ed. Tecnos. 1993.
LARSON; HOSTETLER; EDWARDS. Cálculo II. Ed. McGraw Hill. 2006
RODRÍGUEZ, G. Diferenciación de Funciones de Varias Variables Reales. Manuais Universitarios. Nº 4. Publicacións da Universidade de Santiago. 2003.
THOMAS, G. B. Cálculo de Varias Variables. Pearson. Addison Wesley. 2005.
Online references:
APOSTOL, T. M., Análisis Matemático, 2ª Ed. https://doku.pub/download/analisis-matematico-2da-edicion-tom-apostolpd…
BÚCARI, N. D., LANGONI, L., VALLEJO, D., Cálculo diferencial. https://openlibra.com/es/book/download/calculo-diferencial
TRENCH, W. F., Introduction to Real Analysis, 2013. http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF
KRANTZ, S. G.; PARKS, H. R., The Implicit Function Theorem, 2013. https://link.springer.com/book/10.1007%2F978-1-4614-5981-1
Our aim is to contribute to prepare the students in the competences mentioned for the Bachelor of Matemáticas at USC: the basic and general competences CB1, CB2, CB3, CB4, CB5, CG1, CG2, CG3, CG4, CG5, the transversal competences CT1, CT2, CT3, CT5, and the specific competences CE1, CE2, CE3, CE4, CE5, CE6, CE9.
It will be followed the general methodological indications established in the Title of Bachelor of Mathematics of the USC.
SCENARIO 1 (adapted normality)
Teaching is scheduled in theoretical and interactive classes.
The theoretical classes will be devoted to the presentation and development of the essential contents of the subject.
The interactive classes will be devoted to the presentation of examples and problem solving (combining both theory and applications).
It will be promoted the maximum participation of students on the various classes of interactive teaching laboratory, where the discussion and debate with students on aspects of the subject and the resolution of the proposed tasks will aim to practice and improve their knowledge, and to work to achieve some of the competences mentioned.
Tutorials will be in person or through email.
SCENARIO 2 (distancing)
The teaching will be partially virtual, according to the distribution organized by the Faculty of Mathematics. The virtual classroom of the course will be used, in which students will be provided complete notes of the contents of the subject, as well as explanatory videos of the same.
Tutorials will be through email or Microsoft Teams.
SCENARIO 3 (closure of the facilities)
Teaching will be totally remote. The virtual classroom of the course will be used, in which students will be provided complete notes of the contents of the subject, as well as explanatory videos of the same. Periodic interactive sessions will also be planned through Microsoft Teams, in which the contents of the subject will be worked on through problem solving.
Tutorials will be through email or Microsoft Teams.
It will follow the general criterion of assessment established in the Memory of the Title of Degree in Mathematics of the USC.
For the calculation of the final qualification (FQ) we will take into account the qualification of the continuous assessment (CA) and the qualification of the final exam (FE).
The final qualification will be computed using the following formula:
FQ=Máx(FE; 0.7*FE+0.3*CA).
The continuous assessment will consist of the resolution and exposition of problems (P) proposed by the professor, which might be indivual or in groups (and which will be evaluated over 6 points) and a midterm test (E) that will not reduce the amount of contents of the final exam (which will be evaluated over 4 points). This way, the continuous assessment will be computed with the following formula:
CA=0.6*P+0.4*E.
The continuous assessment will be preserved for the second opportunity.
The final exam will consist of the resolution of theoretical and practice questions similar to the ones studied during the development of the subject.
It will be understood as not presented who does not take the final test of the subject.
In cases of fraudulent performance of exercises or tests, the provisions of the "Normativa de avaliación do rendemento académico dos estudantes e de revisión de cualificacións" will apply.
This assessment system will be applied in the three scenarios mentioned in the document “Directrices para o desenvolvemento dunha docencia presencial segura. Curso 2020-21”, with the following adaptations:
SCENARIO 1 (adapted normality)
Part of the problems proposed by the professor (P) that will be evaluated as part of the continuous assessment will be raised in some interactive session, in which the students must solve the problems (individually or in groups established by the professor) and deliver them at the end of that session. Students will be informed in advance of the sessions in which these problems will be carried out.
For the rest of the problems, students will have more time (which will be established based on the difficulty of each of the exercises), allowing deliveries both on paper and by email or through the virtual campus.
The completion of the midterm test (E) will be face-to-face, in one of the expository sessions of the subject. The date of the same will be communicated to the students in advance.
SCENARIO 2 (distancing)
All the problems proposed by the professor (P) will be done at home, never requiring the delivery of an exercise on the same day that it is proposed. Deliveries will be made through the virtual campus.
The midterm exam (E) will be done electronically through the virtual campus.
For the final exam, the indications established at the time by the Faculty will be followed, being carried out in person if possible.
SCENARIO 3 (closure of the facilities)
All the problems proposed by the professor (P) will be done at home, never requiring the delivery of an exercise on the same day that it is proposed. Deliveries will be made through the virtual campus.
The midterm exam (E) will be done electronically through the virtual campus.
For the final test, the indications established by the Faculty at the time will be followed, carried out in person if possible.
ON-SITE WORK AT CLASSROOM
Blackboard classes in big groups (26 hours)
Interactive classes in reduced groups (13 h)
Interactive classes of laboratory/tutorials in reduced group (13 h)
Tutorials in very small groups or individualized (2 h)
Total hours on-site work at classroom 54
PERSONAL WORK OF THE STUDENT
Autonomous individual study or in group (54 hours)
Writing exercises, conclusions and other works (18 h)
Programming / experimentation and other works at computer (9 h)
Total hours personal work of the student 81
It is advised to handle with fluency the basic elementary concepts of: Introduction to Mathematical Analysis, Continuity and derivability of functions of one real variable, Topology of Euclidean spaces and Vector spaces and matrix calculus. Furthermore, it is important to take part actively in the learning process of the subject as well as attending regularly to the theoretical and practical classes (with special relevance to those in small groups). The daily work is essential.
CONTINGENCY PLAN
Adaptation of the methodology to Scenarios 2 and 3:
Scenario 2: Written material will be provided so that students can follow the subject online. Students will also have explanatory videos of the contents of the subject, which will be accessible from the Virtual Campus.
Scenario 3: Written material will be provided so that students can follow the subject online. The training will be exclusively through videos accessible from the Virtual Campus. Regular interactive sessions will be held in Microsoft Teams.
Adaptation of the evaluation system to Scenarios 2 and 3:
Same procedure as in Scenario 1 except that all problems will be proposed and delivered through the Virtual Campus and that the midterm exam will also be carried out through the Virtual Campus.
Lucia Lopez Somoza
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- lucia.lopez.somoza [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 |
| 16:00-17:00 | Grupo /CLIS_02 | Spanish | Classroom 03 |
| 17:00-18:00 | Grupo /CLIS_01 | Spanish | Classroom 03 |
| Tuesday | |||
| 15:00-16:00 | Grupo /CLE_02 | Galician | Classroom 08 |
| 19:00-20:00 | Grupo /CLE_01 | Spanish | Classroom 07 |
| Wednesday | |||
| 15:00-16:00 | Grupo /CLIS_04 | Spanish | Classroom 03 |
| 16:00-17:00 | Grupo /CLIS_03 | Spanish | Classroom 03 |
| 17:00-18:00 | Grupo /CLE_01 | Spanish | Classroom 08 |
| 18:00-19:00 | Grupo /CLIL_04 | Spanish | Classroom 03 |
| 19:00-20:00 | Grupo /CLIL_03 | Spanish | Classroom 02 |
| Thursday | |||
| 17:00-18:00 | Grupo /CLIL_02 | Spanish | Classroom 02 |
| 17:00-18:00 | Grupo /CLIL_06 | Spanish | Classroom 03 |
| 18:00-19:00 | Grupo /CLIL_01 | Spanish | Classroom 03 |
| 18:00-19:00 | Grupo /CLIL_05 | Spanish | Graduation Hall |
| 01.12.2021 16:00-20:00 | Grupo /CLE_01 | Classroom 02 |
| 01.12.2021 16:00-20:00 | Grupo /CLE_01 | Classroom 03 |
| 01.12.2021 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |
| 01.12.2021 16:00-20:00 | Grupo /CLE_01 | Ramón María Aller Ulloa Main Hall |
| 06.23.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |