Functional Series and Riemann Integration in Several Real Variables

To understand, know and handle the main concepts, results and methods related to sequences and function's series, which have a basic importance in the mathematical analysis. More immediately for the students, it is equally relevant in other subjects of undergraduate mathematics such as subjects on complex, functional analysis, Lebesgue's integration and Fourier series.

The mentioned contents can be divided in two contents blocks. In the first one, sequences and series of functions are introduced and different ways of convergence for them are studied. In the second one, the knowledge on integrals of funcions of one real variable are enlarged, presenting the basics of improper integrals of functions of one real variable and Riemann multiple integrals.

The achievement of the objectives will be obtained if the theoretical and practical contents are known and can be applied in concrete problems of different nature, occasionally, maybe, by means of the computer. If some software is needed, then Maple will be used.

I) IMPROPER INTEGRALS OF FUNCTIONS OF ONE REAL VARIABLE (6 h. expositivas apprx.)

1.1 Improper integrals.
Integration in non compact intervals. Convergent and divergent integrals. Properties of improper integrals . Cauchy's condition for the convergence of an integral.

1.2 Criteria of convergence.
Characterization of convergence of integrals of non negative functions. Comparison test, ratio and comparison limit test. Study of some integrals of relevance. Convergence and absolute convergence of integrals. Dirichlet's criteria.

1.3 Improper integrals and numerical series.

II) SEQUENCES AND SERIES OF FUNCTIONS (12 h. expositivas apprx.)

2.1 Sequences of functions.
Point-wise and uniform convergence. Cauchy's condition for uniform convergence. Results on continuity, derivability and integrability of a limit.

2.2 Series of functions.
Point-wise, absolute and uniform convergence of a series of functions. Cauchy's condition for uniform convergence of a series. Weierstrass's test for convergence. Results on continuity, derivability and integrability of a sum.

2.3 Power series.
Radius of convergence. Cauchy-Hadamard's theorem. Uniform convergence. Properties on continuity, derivability and integrability of a sum. Taylor series. Analytic functions.

III) RIEMANN MULTIPLE INTEGRALS (10 h. expositivas apprx.)

3.1 Riemann integral in R^n.
Partitions of an interval. Darboux's sums. Lower and higher integrals. Riemann integrable functions and their integrals. Properties of Riemann integral. Riemann sums for integrals. Riemann theorem.

3.2 Riemann integrable functions on an interval.
Zero content and zero measure sets. Oscillation of a function arround a point. Lebesgue's characterization of integrability in Riemann sense.

3.3 Integration in Jordan measurable sets.
Jordan measurable sets. Integration in Jordan measurable sets. Riemann integrable functions in measurable sets. Properties of Riemann integral. Fubini's theorem.

3.4 Integral calculus.
The Theorem for change of variable. Polar, spherical and cylindrical coordinates.

COMPLEMENTARY BIBLIOGRAPHY

T. M. Apostol: "Análisis Matemático". Ed. Reverté. 1977.
T. M. Apostol: "Calculus, volumen 2". Ed. Reverté. 1973.
R. G. Bartle, D. R. Sherbert: "Introducción al Análisis Matemático de una variable". Ed. Limusa.
R. G. Bartle: "Introducción al Análisis Matemático". Ed. Limusa.
F. Bombal, L .R. Marín, G.Vera: "Problemas de Análisis Matemático. Vol. 3: Cálculo integral". Ed. AC.
J. de Burgos: "Cálculo infinitesimal de una variable". Ed. McGraw-Hill.
J. de Burgos: "Cálculo infinitesimal de varias variables". Ed. McGraw-Hill.
F. del Castillo: "Análisis Matemático II". Ed. Alhambra. 1980.
J.A. Fernández Viña: "Análisis Matemático III, Integración y Cálculo exterior". Ed. Tecnos 1992.
J.A. Fernández Viña, Eva Sánchez Mañes: "Ejercicios y Complementos de Análisis Matematico III". Ed. Tecnos, 1994.
M. Spivak: "Cálculo en Variedades". Ed. Reverté. 1988.

The competences mentioned in the Guidelines for the Bachelor of Mathematics at USC for this subject: CG1, CG2, CG3, CG4 y CG5; CT1, CT2, CT3 y CT5; CE1, CE2, CE3, CE4, CE5, CE6 y CE9.

Such competences will be achieved by doing the following activities:
• Analysis of sequences and function's series, distinguishing the notions of point-wise and uniform convergence, and showing conditions that allow the functional limit (the sum) to inherit the properties of regularity of corresponding series (sequences) of functions.
• Study of the convergence of improper integrals, calculating its value when it is posible.
• Construction of Riemann integral of functions of several variables in Jordan measurable sets.
• Study of the integrability in Riemann sense for functions of several variables in measurable sets in the sense of Jordan.
• Calculation of multiple integrals in measurable sets in the sense of Jordan, use of Fubini's theorem and the theorem for change of variable with the most frequent transformations (polar, cylindrical and spherical).
• Use certain softwares to support the visualizations and the calculations.

In general, the resources needed for the developing of the subject (notes, explanatory videos, folders...) will be provided to the students by means of the virtual classroom.

The contents of the subject can be presented in different orders. The order given above may be modified if the circumstances advise so, during the face-to-face lessons destined by the Faculty of Mathematics to this end.

The developing of the subject will promote the students' learning as well as the continuous assessment, by means of different (voluntary) proposals of exercises. Moreover, the participation during the lessons will be also encouraged.

MAPLE program will be used for computer classes.

To make easier the learning of the subject, instructional materials (in galician) will be prepared. This includes the following: notes about the contents of the subject, explanatory videos, Maple practices for students use and others. All this material will be available for students in the virtual classroom of the subject.

In general, an evaluation will be carried out in which a continuous evaluation is combined with a final test.

The continuous evaluation will allow to check the degree of achievement of the competences specified above, with emphasis on the transversal competences CT1, CT2, CT3 and CT5.

In the final and second opportunity test, the knowledge acquired by the students in relation to the concepts and results of the subject will be measured, both from a theoretical and practical point of view, also assessing the clarity and logical rigor shown in their presentation. The achievement of the basic, general and specific competences referred to in the Memory of the Degree in Mathematics of the USC, which have been indicated above, will be evaluated.

In the development of the subject, an attempt will be made to favor, to a large extent, continuous assessment (which will be face-to-face) for those students who wish to do so, so that, being usually assistants, participants and workers, they will have the opportunity to reach a percentage of their final mark through the different activities (voluntary) that they have carried out (individually or in groups, in the classrooms or outside them, as appropriate) and, where appropriate, delivered or exhibited in the appropriate terms.

In this evaluation modality (which we will call Modality 1 and which presupposes the active presence in the classrooms and the completion of at least 80% of the proposed activities throughout the course) the final exam (which will be face-to-face) is considered. as one more activity, whose performance will be fundamental for the qualification of the students. These activities will serve to assess both the knowledge and the general, specific and transversal skills acquired by the students.

The corresponding final mark will be obtained by respecting the indications of the Degree Report. In any case, under the most favorable conditions, the percentage of the grade corresponding to the students' work during the course (excluding the final test), may reach 25% of the maximum final grade (CF), using a formula such as following, where E represents the final exam grade and T is the grade obtained for the rest of the activities carried out in the course:

CF = E + min {T / 4, 10 - E}. (Both E and T can take values ​​between zero and ten).

In order to try to respect the autonomy and pace of work of the students, a second evaluation modality will be offered (which we will call Modality 2), consisting of, at least, one intermediate test with prior notice. In this case, the final grade will be obtained with the formula CF = max {E, 0.7E + 0.3PI}, where PI designates the average grade of the intermediate tests which, like E, will take values ​​between zero and ten

As in Modality 1, it will be essential to take the final exam to be able to opt for this evaluation modality.
At the beginning of the semester, students will have the opportunity to choose the evaluation modality they wish, through a choice they will make through the Virtual Course, within the deadlines established for this purpose. If the election is not made within the corresponding periods, it will be understood that Mode 2 is chosen.

For the students of the CLE02 group, only evaluation modality 2 will be accessible, consisting of carrying out two intermediate tests with prior notice. The final mark will be obtained with the formula CF=máx{E, 0'7E+0'3PI}, where PI designates the average mark of the two intermediate tests that, like E, will take values between zero and ten. Training coordination and equivalence of all groups is guaranteed.

However, in the final exam any student will have the possibility of obtaining the highest numerical grade, whether or not they have completed the activities or the intermediate test during the course. Students who do not appear for the final exam will receive the grade of Not Presented.

The final exam may be different for the expository groups. Coordination and educational equivalence of all subject groups are guaranteed.

In the second opportunity, the same evaluation system will be used, but with the test corresponding to the second opportunity, which will be an exam of the same type as the first.

The exam corresponding to the second opportunity may be different for the exhibition groups. Coordination and educational equivalence of all subject groups are guaranteed.

Caveat. For cases of fraudulent completion of tasks or tests (plagiarism or improper use of technology), the provisions of the Regulations for the evaluation of the academic performance of students and review of grades will apply.

150 hours: 58 face-to-face hours and 92 hours of independent study.

It is important to understant the contents of the following topics to study this subject: Introduction to mathematical analysis, Continuity and derivability of functions of one real variable; Integration of functions of one real variable; Differentiation of functions of several real variables.

On the other hand, it is recommended to study regularly, and to make all the activities that are proposed during the (face-to-face or virtual) lessons. It is also very important to clarify all the doubts a student may face during the course.