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. Limits and continuity of functions from a subset of R^n into R^m [3 h CLE]
- Directional, iterated and subset limits of scalar functions
- Continuity of scalar functions
- The case of vector valued functions
2. First order differentiability of scalar functions of several real variables [5h CLE]
- Partial and directional derivatives
- Differential of a function and differentiable functions
- Necessary and sufficient conditions of differentiability
- Gradient vector
3. First order differentiability of vector valued functions of several real variables [4h CLE]
- Jacobian matrix
- Differentiation rules
4. The mean value theorem [2h CLE]
- The mean value theorem for scalar functions
- A generalisation to vector fields
5. Higher order differentiability [4h CLE]
- Second order differential and Hessian matrix
- Simmetry of the second-order differential
- Class k functions
6. Taylor's theorem and its consequences [4h CLE]
- Taylor polynomials
- Approximation of regular functions by polynomials: Taylor's theorem
- Computation of the extrema of a scalar function
7. The implicit function theorem and the inverse function theorem [4h CLE]
- Solving equations locally: the implicit function theorem
- Inverting functions locally: the inverse function theorem
8. Applications of the implicit function theorem and the inverse function theorem [2h CLE]
- Change of variables
- Computation of constrained extrema of a scalar function
- Geometrical problems
Basic bibliography:
[1] Apostol, T. M. (1979). Análisis matemático (2ª ed.). Reverté.
[2] Fernández Viña, J. A. (1984). Análisis matemático (Vol. II). Tecnos.
[3] Rodríguez López, G. (2003). Diferenciación de funciones de varias variables reales. Universidade de Santiago de Compostela, Servicio de Publicacións e Intercambio Científico.
Complementary bibliography:
[4] Bartle, R. G. (1991). Introducción al análisis matemático (1ª ed.). Limusa.
[5] Besada Morais, M., García Cutrín, F. J., Mirás Calvo, M. A., & Vázquez Pampín, C. (2001). Cálculo de varias variables: cuestiones y ejercicios resueltos. Prentice Hall.
[6] Besada Morais, M., García Cutrín, F. J., & Mirás Calvo, M. A. (2011). Cálculo diferencial en varias variables: Problemas y ejercicios tipo test resueltos. Garceta.
[7] Bombal Gordon, F., Rodríguez Marín, L., & Vera Botí, G. (1994). Problemas de análisis matemático (2ª ed.). AC.
[8] Burgos, J. de (2008). Cálculo infinitesimal de varias variables (2a ed.). McGraw-Hill Interamericana.
[9] Fernández Viña, J. A., & Sánchez Mañes, E. (1992). Ejercicios y complementos de análisis matemático (Vol. II). Tecnos.
[10] Krantz, S. G., & Parks, H. R. (2003). The implicit function theorem: history, theory, and applications (1ª ed.). Springer Science+ Business Media, LLC. https://doi.org/10.1007/978-1-4612-0059-8
[11] Larson, R., Hostetler, R. P., & Edwards, B. H. (2006). Cálculo (8a ed.) (Vol II). McGraw Hill.
[12] Thomas, G. B. (2015). Cálculo. Varias variables (13ª ed.). Pearson Educación.
Our aim is to contribute to prepare the students in the competences mentioned for the Mathematics Degree 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.
The general methodological indications established in the Title of Degree of Mathematics of the USC will be followed.
Teaching is scheduled in theoretical and interactive lessons.
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). The participation of students will be promoted to its full extent on the various classes of interactive teaching laboratory. Indeed, the discussion and debate with the students on aspects of the subject and the resolution of the proposed tasks will aim to let them practice and strengthen their knowledge, and to work to achieve some of the above-mentioned competences.
Tutorials will be in person or through email.
The general criterion of assessment established in the Memory of the Title of Degree in Mathematics of the USC will be followed.
For the computation of the final grade (FG), we will take into account the mark of the continuous assessment (CA) and that of the final exam (FE). In fact, FG will be obtained by using the following formula:
FG=Maximum{0.7*FE+0.3*CA; FE}.
The continuous assessment mark is going to be based on
- the grade (0-10 points) obtained by the resolution of problems with notes (PRN), done individually or in groups, during lessons;
- the grade (0-10 points) obtained in a midterm test that will not reduce the amount of contents of the final exam and realised without notes (MT) in a theoretical lesson.
In particular, the continuous assessment mark will be computed via the following formula:
AC=Minimum{0.5*PRN+0.7*MT; 10}.
The activities of the continuous assessment will be similar in all the groups and their grade will be preserved for the second opportunity.
The attendance to the lessons is not going to be taken into account, except for those sessions when continuous assessment activities in face-to-face format take place. Such dates are going to be communicated to the students in advance. Concretely, not attending to a continuous assessment activity without proper justification shall mean that such an activity will be graded with 0 points.
The final exam (graded from 0 to 10 points) will consist of the resolution of theoretical and practice questions similar to those realised during the development of the subject and will be the same for all the blackboard-lesson groups.
Those students who do not take the final exam will be considered as not presented.
For the second opportunity, the final grade 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 h)
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 the subjects "Introduction to Mathematical Analysis", "Continuity and Differentiability 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 lessons (with special relevance to those in small groups). Moreover, the daily work is essential.
Daniel Cao Labora
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813174
- daniel.cao [at] usc.es
- Category
- Professor: University Lecturer
Sebastian Buedo Fernandez
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813160
- sebastian.buedo [at] usc.es
- Category
- Professor: LOU (Organic Law for Universities) PhD Assistant Professor
Victor Cora Calvo
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- victor.cora.calvo [at] usc.es
- Category
- Xunta Pre-doctoral Contract
| Monday | |||
|---|---|---|---|
| 16:00-17:00 | Grupo /CLE_02 | Galician | Classroom 03 |
| 17:00-18:00 | Grupo /CLIS_02 | Galician | Classroom 02 |
| 18:00-19:00 | Grupo /CLIS_01 | Galician | Classroom 09 |
| Tuesday | |||
| 18:00-19:00 | Grupo /CLE_01 | Galician | Classroom 02 |
| 18:00-19:00 | Grupo /CLE_02 | Galician | Classroom 03 |
| Wednesday | |||
| 17:00-18:00 | Grupo /CLIL_07 | Galician | Classroom 09 |
| 18:00-19:00 | Grupo /CLE_01 | Galician | Classroom 02 |
| 19:00-20:00 | Grupo /CLIL_06 | Spanish, Galician | Classroom 09 |
| Thursday | |||
| 15:00-16:00 | Grupo /CLIL_05 | Spanish, Galician | Classroom 03 |
| 16:00-17:00 | Grupo /CLIL_08 | Galician | Classroom 02 |
| 17:00-18:00 | Grupo /CLIL_02 | Galician | Classroom 09 |
| 19:00-20:00 | Grupo /CLIL_03 | Galician | Classroom 09 |
| Friday | |||
| 15:00-16:00 | Grupo /CLIL_01 | Galician | Classroom 09 |
| 16:00-17:00 | Grupo /CLIS_04 | Galician | Classroom 08 |
| 16:00-17:00 | Grupo /CLIL_04 | Galician | Classroom 09 |
| 17:00-18:00 | Grupo /CLIS_03 | Spanish, Galician | Classroom 08 |
| 01.12.2026 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |
| 06.12.2026 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |