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 general objective of this course is to understand, become familiar with, and handle the main concepts, results, and methods related to vector calculus and Lebesgue integration theory.
More specifically, the following objectives are set:
OB1 – to handle the concepts of flux, divergence, and curl of a vector field, as well as their physical interpretation;
OB2 – to understand the concepts and properties of line integrals of scalar and vector fields, as well as their applications;
OB3 – to understand the concepts and properties of surface integrals of scalar and vector fields, as well as their applications;
OB4 – to verify Green’s, Stokes’, and Gauss’ theorems through concrete examples;
OB5 – to understand the construction of Lebesgue’s measure and integration theory;
OB6 – to be able to conjecture and prove the Lebesgue measurability of sets and functions;
OB7 – to know the convergence theorems of the Lebesgue integral, as well as their usefulness and consequences;
OB8 – to understand the relationship between the Riemann and Lebesgue integration theories, as well as the advantages and necessity of the latter;
OB9 – to know Fubini’s theorem and the change of variables theorem for the Lebesgue integral.
1 - Vector Calculus (10 CLE hours)
1.1 - Curves in R^n. The concept of a curve in R^n. Length of a curve. Line integrals of scalar and vector fields. Conservative fields. Principle of conservation of energy. Curl and divergence. Green’s Theorem.
1.2 - Surfaces in R^3. The concept of a surface in R^3. Area of a surface. Surface integrals of scalar and vector fields. Solenoidal fields. Stokes’ Theorem. Gauss-Ostrogradsky Theorem.
2 - Lebesgue Integration (18 CLE hours)
2.1 - Measure theory(-ies)
Intervals in R^n. Measure of intervals in R^n. Elementary sets in R^n. Measure of elementary sets in R^n. Jordan-measurable sets in R^n. Measure of Jordan-measurable sets in R^n. Cantor set. Lebesgue outer measure. Vitali set. Lebesgue-measurable sets. Lebesgue measure in R^n.
2.2 - Integration theory(-ies)
Riemann integral. Lebesgue integral. Fundamental theorem for the transition from Lebesgue measure to Lebesgue integral. Properties of the Lebesgue integral. Convergence theorems: Monotone Convergence Theorem, Fatou’s Lemma, and Dominated Convergence Theorem. Lebesgue spaces. The space L^1. Lebesgue’s Theorem. Luzin’s Theorem. Egorov’s Theorem. Riesz–Fischer Theorem. Completeness of the Lebesgue space. Cavalieri’s Principle. Fubini’s Theorem. Tonelli’s Theorem.
Basic Bibliography
del Castillo F. Mathematical Analysis II. Madrid: Alhambra; 1980.
Kolmogorov AN, Fomin SV. Introductory Real Analysis. New York: Dover Publications; 1975.
Marsden JE, Tromba AJ. Vector Calculus. 5th ed. Madrid: Pearson; 2004.
Wilcox HJ, Myers DL. An Introduction to Lebesgue Integration and Fourier Series. New York: Dover Publications; 1978.
Complementary Bibliography
Boss V. Lessons in Mathematics. Volume 5. Functional Analysis. Moscow: URSS; 2009.
Chae SB. Lebesgue Integration. 2nd ed. New York: Springer-Verlag; 1995.
Fernández Viña J. A. Mathematical Analysis III. Integration and Exterior Calculus. Madrid: Tecnos; 1992.
Tao T. An Introduction to Measure Theory. Providence: American Mathematical Society; 2011.
In addition to contributing to the achievement of the basic, general, and transversal competencies outlined in the Bachelor's Degree in Mathematics Memory at the Universidade de Santiago de Compostela, which can be consulted at www.usc.gal/es/estudios/grados/ciencias/grado-matematicas, this course will contribute to the attainment of the following specific competencies:
CE1 – Understand and use mathematical language;
CE2 – Know rigorous proofs of some classical theorems in various areas of mathematics;
CE3 – Devise proofs of mathematical results, formulate conjectures, and imagine strategies to confirm or refute them;
CE4 – Identify errors in incorrect reasoning by proposing proofs or counterexamples;
CE5 – Assimilate the definition of a new mathematical object, relate it to other already known objects, and be capable of using it in different contexts;
CE6 – Be able to abstract the substantial properties and facts of a problem, distinguishing them from purely incidental or circumstantial ones;
CE9 – Use computer applications for statistical analysis, numerical and symbolic computation, graphical visualization, optimization, and general scientific software, to experiment in mathematics and solve problems.
The general methodological guidelines established in the Bachelor's Degree in Mathematics Memory at the Universidade de Santiago de Compostela will be followed.
Teaching is scheduled in both expository and interactive classes. In the expository classes, the essential content of the course will be presented, allowing for the development of basic, general, and transversal competencies, in addition to the specific competencies CE1, CE2, CE5, and CE6. Sometimes, the model will resemble a lecture, and at other times, there will be a greater focus on student involvement. In the interactive classes, problems and exercises will be proposed and corrected, emphasizing the acquisition of specific competencies CE3, CE4, and CE9.
Teaching will be in-person and complemented by the Virtual Campus, which will be used for the completion and submission of certain tasks related to continuous assessment.
The final grade (FG) will not be lower than the one obtained using the following formula: FG=max(FE,0.7FE+0.3CA), where FE is the grade of the final exam and CA is the grade of continuous assessment. Both FE and CA will take values between 0 and 10.
The grade for continuous assessment (CA) will not be lower than the one obtained using the following formula: CA=max(0.5P1+0.5P2, 0.6(0.5P1+0.5P2)+0.2E+0.2P), where P1 is the grade of the first midterm test, P2 is the grade of the second midterm test, E is the grade for completing the proposed exercises, and P is the grade for active participation during the course. P1, P2, E, and P will all take values between 0 and 10.
The final exam (both the first and second opportunities) may differ for the expository groups. Coordination and equivalence in teaching will be ensured for all groups in the course.
In the second opportunity, the same evaluation system will be used, maintaining the grade of continuous assessment and updating the grade for the final exam.
In cases of fraudulent completion of tasks or tests (plagiarism or misuse of technology), the regulations for the evaluation of academic performance and grade revision will apply.
TOTAL HOURS
150 hours: 58 hours of in-person classes and 92 hours of independent study.
IN-PERSON TEACHING IN THE CLASSROOM (26 hours CLE + 14 hours CLIS + 14 hours CLIL + 2 hours TGMR + 2 hours CLE for assessments)
(CLE) Expository classes (26 hours)
(CLE) Assessment tests (2 hours)
(CLIS) Interactive seminar classes (14 hours)
(CLIL) Interactive laboratory classes/small group tutorials (14 hours)
(TGMR) Very small group tutorials (2 hours)
INDEPENDENT STUDY TIME
On average, 92 hours 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.
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, Differentiation of Functions of Several Real Variables, and Functional Series and Riemann Integration in Several Variables.
It is recommended to study the subject regularly and try to complete the proposed exercises independently. It is also recommended to consult the teaching team for any doubts that may arise throughout the course.
Fernando Adrian Fernandez Tojo
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- fernandoadrian.fernandez [at] usc.es
- Category
- Professor: University Lecturer
Jorge Losada Rodriguez
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813215
- jorge.losada.rodriguez [at] usc.es
- Category
- PROFESOR/A PERMANENTE LABORAL
| Monday | |||
|---|---|---|---|
| 09:00-10:00 | Grupo /CLE_01 | Galician | Classroom 06 |
| 10:00-11:00 | Grupo /CLIL_06 | Spanish | Classroom 08 |
| 11:00-12:00 | Grupo /CLIL_05 | Spanish | Classroom 08 |
| 12:00-13:00 | Grupo /CLIL_04 | Spanish | Classroom 08 |
| Tuesday | |||
| 09:00-10:00 | Grupo /CLE_01 | Galician | Classroom 06 |
| Wednesday | |||
| 11:00-12:00 | Grupo /CLE_02 | Spanish | Classroom 06 |
| Thursday | |||
| 10:00-11:00 | Grupo /CLIL_02 | Galician | Classroom 05 |
| 11:00-12:00 | Grupo /CLE_02 | Spanish | Classroom 06 |
| 12:00-13:00 | Grupo /CLIL_01 | Galician | Classroom 05 |
| 13:00-14:00 | Grupo /CLIL_03 | Galician | Classroom 05 |
| Friday | |||
| 09:00-10:00 | Grupo /CLIS_02 | Galician | Classroom 06 |
| 09:00-10:00 | Grupo /CLIS_04 | Spanish | Classroom 08 |
| 10:00-11:00 | Grupo /CLIS_03 | Spanish | Classroom 08 |
| 11:00-12:00 | Grupo /CLIS_01 | Galician | Classroom 02 |
| 12.17.2025 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |
| 06.08.2026 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |