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)
Basic bibliography:
APOSTOL, T. M., Análisis Matemático, Ed. Reverté, 1991.
FERNÁNDEZ VIÑA, J.A. Análisis Matemático II:Topologia y Cálculo diferencial. 2nd ed. Tecnos. 1993.
RODRÍGUEZ, G. Diferenciación de Funciones de Varias Variables Reales. Manuais Universitarios. Nº 4. Publicacións da Universidade de Santiago. 2003.
Complementary bibliography:
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.
BÚCARI, N. D., LANGONI, L., VALLEJO, D., Cálculo diferencial. https://openlibra.com/es/book/download/calculo-diferencial
BURGOS ROMAN, JUAN de. Cálculo Infinitesimal de Varias Variables. McGraw-Hill/ Interamericana de España. 2008.
FERNÁNDEZ VIÑA, J.A.; SÁNCHEZ MAÑÉS, E. Ejercicios y Complementos de análisis Matemático II. 2nd ed. Tecnos. 1993.
KRANTZ, S. G.; PARKS, H. R., The Implicit Function Theorem, 2013. https://link.springer.com/book/10.1007%2F978-1-4614-5981-1
LARSON; HOSTETLER; EDWARDS. Cálculo II. Ed. McGraw Hill. 2006
THOMAS, G. B. Cálculo de Varias Variables. Pearson. Addison Wesley. 2005.
TRENCH, W. F., Introduction to Real Analysis, 2013. http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF
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 degree of Mathematics of the USC.
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.
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=Maximum{Minimum{0.7*FE+0.3*CA; 10}; FE}.
The continuous assessment (assessed over 12 points) will consist of carrying out the following activities:
- resolution of problems, which might be indivual or in groups, during lessons, with notes (A);
- participation during the lessons by means of solving in the blackboard some previously proposed tasks (B);
- a midterm test (without notes) that will not reduce the amount of contents of the final exam (C).
This way, the continuous assessment will be computed with the following formula:
AC=0.6*A+0.2*B+0.4*C.
The activities of the continuous assessment will be similar in all the groups. 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. The final exam will be the same for all the groups.
Those students who do not take the final exam will be considered as not presented.
For the second opportunity, the final qualification will be computed using the same formula.
Warning: 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.
ON-SITE WORK AT CLASSROOM
Blackboard classes in big groups (28 hours)
Interactive classes in reduced groups (14 h)
Interactive classes of laboratory (14 h)
Tutorials in very small groups or individualized (2 h)
Total hours on-site work at classroom 58
PERSONAL WORK OF THE STUDENT
Personal work will depend on the students. On average, 92 hours per student are estimated.
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.
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: LOU (Organic Law for Universities) PhD Assistant Professor
| Monday | |||
|---|---|---|---|
| 16:00-17:00 | Grupo /CLE_02 | Spanish | Classroom 03 |
| 17:00-18:00 | Grupo /CLIS_02 | Spanish | Classroom 02 |
| 18:00-19:00 | Grupo /CLIS_01 | Spanish | Classroom 03 |
| Tuesday | |||
| 16:00-17:00 | Grupo /CLE_02 | Spanish | Classroom 03 |
| 18:00-19:00 | Grupo /CLE_01 | Spanish | Classroom 02 |
| Wednesday | |||
| 15:00-16:00 | Grupo /CLIL_06 | Spanish | Classroom 09 |
| 16:00-17:00 | Grupo /CLIL_04 | Spanish | Classroom 09 |
| 17:00-18:00 | Grupo /CLIL_05 | Spanish | Classroom 08 |
| 18:00-19:00 | Grupo /CLE_01 | Spanish | Classroom 02 |
| Thursday | |||
| 15:00-16:00 | Grupo /CLIS_03 | Spanish | Classroom 03 |
| 15:00-16:00 | Grupo /CLIL_03 | Spanish | Classroom 09 |
| 16:00-17:00 | Grupo /CLIS_04 | Spanish | Classroom 02 |
| 16:00-17:00 | Grupo /CLIL_01 | Spanish | Classroom 09 |
| 17:00-18:00 | Grupo /CLIL_02 | Spanish | Classroom 08 |
| 01.10.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |
| 06.17.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |