Vector Calculus and Lebesgue Integration

To understand, know and manage the main concepts, methods and results related to vector calculus and Lebesgue integral of several real variables:

• To manage the concepts of flow divergence and rotational of a vector field and its dynamic significance.
• To understand the concepts and properties of the line integral of scalar and vectorial fields, and its practical application to concrete examples.
• To understand the concepts and properties of the surface integral of scalar and vectorial fields, and its practical application to concrete examples.
• To check on verification examples of the theorems of Green, Stokes and Gauss.
• To know the construction of the measure and the Lebesgue integral for functions of several real variables.
• To have the ability to determine the nature of some examples of Lebesgue measurable sets and functions, and the integrability of functions on measurable sets.
• Mastering the convergence theorems of Lebesgue integral and have the ability to apply them in specific cases.
• To understand the relationship between the Riemann and Lebesgue integrals, and the importance of the extension process that involves consideration of the latter
• To know and use the theorems of Fubini and the changing variable in the Lebesgue integral.

These concepts are of fundamental importance in mathematical analysis as well as in other matters of the degree in mathematics, such as those relating to differential geometry, differential equations and applied mathematics.

Lebesgue integration (18 hours CLE*):

1.1 Outer measure of subsets of Rn. Lebesgue-measurable sets and Lebesgue measure. Null-measure sets. The sigma-algebra of Lebesgue measurable sets. Properties of the Lebesgue measure.
1.2 Measurable functions. Properties. Simple measurable functions. Approximation of measurable functions by simple measurable functions. Egorov’s Theorem. Luzin’s Theorem.
1.3 Integral of nonnegative simple measurable functions. Integral of nonnegative measurable functions. Properties. Monotone convergence theorem. Fatou’s Lemma. Lebesgue-integrable functions and Lebesgue integral. Properties of the Lebesgue integral. Dominated convergence theorem. The space L1.
1.4 Relationship between Riemann and Lebesgue integrals. Theorems of Tonelli and Fubini. Theorem of Change of variables.

Vector calculus (10):

2.1 Scalar and vector fields. Gradient, divergence, and rotational. Basic identities in vector analysis. Flow associated to a vector field. Gradient fields and potential functions.
2.2 Parametrized curves in Rn. Piecewise regular curves. Tangent vector. Line integral of a scalar field. Arc lenght. Oriented curves. Line integral of a vector field. Equivalence of curves and oriented curves. Fundamental theorems for line integrals. Characterization of conservative fields
2.3 Parametrized surfaces in R3. Regular surfaces. Normal vector. Orientable surfaces. Surface integral of a scalar field. Area of a regular surface. Surface integral of a vector field. Equivalent surfaces.
2.4 Theorems of Green, Stokes and Gauss. Consequences and aplications.

*CLE=Blackboard classes in big group

Basic Bibliography

Bartle, R.G.: "The elements of integration and Lebesgue Measure". Ed. Wisley. 1995.
Del Castillo, F.: "Análisis Matemático II". Ed. Alhambra. 1987.
Mardsen, J.E.; Tromba, A. J.: "Cálculo Vectorial". 5ª edición. Ed. Addison Wesley. 1987.

Complementary Bibliography

Apostol, T. M.: "Calculus, volumen 2". Ed. Reverté. 1973.
Bombal, F.; Marín, R.; Vera, G.: "Problemas de Análisis Matemático, 3. Cálculo Integral". Ed. AC. 1987.
Chae, S. B.: "Lebesgue Integration". Segunda edición, Springer-Verlag, 1995
Fernández Viña, J. A.: "Análisis Matemático III. Integración y cálculo exterior". Ed. Tecnos. 1992.
Franks, J.: "A (Terse) Introduction to Lebesgue Integration". AMS, 2009.
Kurtz, D. S., Swartz, Ch. W.: "Theories of integration. The integrals of Riemann, Lebesgue, Henstock-Kurzweil, and Mcshane". Series in Real Analysis, 9. World Scientific Publishing Co., Inc., River Edge, NJ, 2004.
Matthews, P.C.: "Vector Calculus". Springer, 1998
Spiegel, M. R.: "Análisis Vectorial". McGraw Hill, 1991.
Weaver, N.: "Measure Theory and Functional Analysis". World Scientific, 2013.

In addition to contribute to achieve the general and transverse competences taken up in the memory of the degree,

http://www.usc.es/export/sites/default/gl/servizos/sxopra/memorias_grao…,

this subject will allow the student to get the following specific competences:

CE1 - To understand and use mathematical language;
CE2 - TO know rigorous proofs of some classical theorems in different areas of mathematics;
CE3 - To devise demonstrations of mathematical results, formulate conjectures and imagine strategies to confirm or refute them;
CE4 - To Identify errors in faulty reasoning, proposing demonstrations or counterexamples;
CE5 - To assimilate the definition of a new mathematical object, and to be able to use it in different contexts;
CE6 - To identify the abstrac properties and material facts of a problem, distinguishing them from those purely occasional or incidental;
CE9 - To use statistical analysis applications, numerical and symbolic computation, graphical visualization, optimization and scientific software, to experience and solve problems in mathematics.

The general methodological indications established in the Report of the Degree in Mathematics of the USC will be followed.

Teaching is scheduled in expository, interactive and tutoring classes. In the expository classes the essential contents of the discipline will be presented, and will allow the work of the basic, general and transversal competences, in addition to the specific competences CE1, CE2, CE5 and CE6. On the other hand, in the interactive sessions, problems or exercises of more autonomous realization will be proposed, and that will allow to emphasize in the acquisition of the specific competences CE3 and CE4. The tutorials will be dedicated to the discussion and debate with the students, and to the resolution of the proposed tasks with which it is intended that the students practice and consolidate the knowledge.

The expository and interactive teaching will be face-to-face and will be complemented with the virtual course of the subject, in which the students will find bibliographical materials, bulletins of problems, explanatory videos, etc. The virtual course will also be used to perform tasks related to continuous assessment. Appropriate computer tools will be used to work on specific CE9 competence.

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.

TOTAL HOURS (150)

150 hours: 58 hours of presence work in the class and 92 hours of personal work of the student.

PRESENCE WORK IN THE CLASS (26 hours CLE + 14 hours CLIS + 14 hours CLIL + 2 hours TGMR + 2 hours CLE for assessment activities),

(CLE) Blackboard classes in big group (26 hours)

(CLE) Assessment Activities (2 hours)

(CLIS) Interactive classes in reduced group (14 hours)

(CLIL) Interactive classes of laboratory/tutorials in reduced group (14 hours)

(TGMR) Small group tutorials or individualized (2 hours)

PERSONAL WORK OF THE STUDENT (92 hours)

Personal work will depend on the students. On average, 92 hours per student 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.