Course: 2023/2024

Integration and Measure

(18265)

Requirements (Subjects that are assumed to be known)

Linear Algebra (Course : 1 Semester : 1),
Differential Calculus (Course : 1 Semester : 1),
Integral Calculus (Course : 1 Semester : 2),
Vector Calculus (Course : 1 Semester : 2).

To introduce the student in the study of modern integration methods, in particular the Lebesgue integral.
To know the convergence theorems on integration and the functional L^p spaces.
To apply these results to the differentiation of parametric integrals and in particular to the Fourier and Laplace transforms.

Skills and learning outcomes

Description of contents: programme

1. Integrals on curves and surfaces
2. Green's, Stokes' and Gauss' theorems
3. Set measure
4. The Lebesgue Integral
5. Monotone and dominated convergence
6. Lp spaces
7. Parametric integrals
8. Integral transforms: Laplace and Fourier

Learning activities and methodology

LEARNING ACTIVITIES AND METHDOLOGY
THEORETICAL-PRACTICAL CLASSES. [44 hours with 100% classroom instruction, 1.76 ECTS]
Knowledge and concepts students must acquire. Student receive course notes and
will have basic reference texts to facilitate following the classes and carrying
out follow up work. Students partake in exercises to resolve practical problems
and participate in workshops and evaluation tests, all geared towards
acquiring the necessary capabilities.
TUTORING SESSIONS. [4 hours of tutoring with 100% on-site attendance, 0.16 ECTS]
Individualized attendance (individual tutoring) or in-group (group tutoring)
for students with a teacher.
STUDENT INDIVIDUAL WORK OR GROUP WORK [98 hours with 0 % on-site, 3.92 ECTS]
FINAL EXAM. [4 hours with 100% on site, 0.16 ECTS]
Global assessment of knowledge, skills and capacities acquired throughout the
course.
METHODOLOGIES
THEORY CLASS. Classroom presentations by the teacher with IT and audiovisual
support in which the subject`s main concepts are developed, while providing
material and bibliography to complement student learning.
PRACTICAL CLASS. Resolution of practical cases and problems, posed by the
teacher, and carried out individually or in a group.
TUTORING SESSIONS. Individualized attendance (individual tutoring sessions) or
in-group (group tutoring sessions) for students with a teacher as tutor.

Assessment System

- % end-of-term-examination 50
- % of continuous assessment (assigments, laboratory, practicals...) 50

Basic Bibliography

- Folland, G.B.. Fourier Analysis and its Applications. Wadsforth & Brooks/Cole. 1992
- Marsden, J.E., Tromba, A,J.. Vector Calculus. W.H. Freeman and Company. 2003
- Rudin, W. . Real and complex Analysis. Mc Graw-Hill (International Student Edition). 1970

Additional Bibliography

- Apostol, T.M.. Mathematical Analysis. Addison-Wesley. 1974
- Bauer, H.. Measure and Integration Theory. Walter De Gruyter. 2001
- Beerends, R.J., ter Morsche, H.G., vanden Berg, J.C., van de Vrie, E.M.. Fourier and Laplace Transforms. Cambridge University Press. 2003
- Bogachev, V.I.. Measure Theory, Volume I. Springer. 2007
- Gamkrelidze (Ed.). Analysis I (Encyclopaedia of Mathematical Sciences, Volume 13). Springer-Vergal. 1989
- Guzmán, M., Rubio, B.. Integración, Teoría y Técnicas. Alhambra. 1979
- Leadbette, R., Cambanis, S., Pipiras, V.. A Basic Course in Measure and Probability. Cambridge University Press. 2014
- Pao, K., Soon, F., Marsden, J.E., Tromba, A.J.. Vector Calculus (Solved Problems). W.H.Freeman & Co Ltd. 1989
- Pestana, D., Rodriguez, J.M., Marcellán, F.. Curso Práctico de Variable Compleja y Teoría de Transformadas. Pearson. 2014

The course syllabus may change due academic events or other reasons.