Introduction to Mathematical Analysis

To introduce the student, with the support of examples and practice, in the comprehension of the first structure of the Mathematical Analysis: the ordered and complete field of the real numbers.

To introduce and consolidate, with examples and exercises, the notions of convergence of sequences and of numerical series.

To present, practising with the different notations, the operations with complex numbers.

1. REAL NUMBERS (approx. 8 lectures)
1.1 Natural numbers. Principle of induction.
1.2 Rational numbers. Countability.
1.3 Axiomatic of the real numbers (R). Supreme axiom and consequences.
1.4 Archimedian property of R. Density of Q in R. Topology of the real line.

2. SEQUENCES OF REAL NUMBERS (approx. 10 lectures)
2.1 Intuitive introduction to the concepts of sequence and limit. Generalities.
2.2 Convergent sequences and their limits. Properties.
2.3 Infinite limits.
2.4 Convergence and divergence of monotone sequences.
2.5 Subsequences. Bolzano-Weierstrass Theorem. Limits of oscillation.
2.6 Cauchy sequences. Completeness of R.
2.7 Calculus of limits. Stirling and Stolz criteria.

3. SERIES OF REAL NUMBERS (approx. 8 lectures)
3.1 Intuitive introduction to the concepts of series and their sum.
3.2 Numerical series. Convergence of series.
3.3 Series of non-negative terms. Convergence criteria.
3.4 Absolute and conditional convergence. Criteria of non-absolute convergence.
3.5 Real decimal expansion and other numerical systems.

4. COMPLEX NUMBERS (approx. 2 lectures)
4.1 Complex numbers. Expressions, operations and roots of complex numbers.
4.2 Exponential form and its consequences: powers, roots and Euler and de Moivre formulae.

BASIC:

[1] T.M. Apostol. Análisis Matemático (2ª Ed.). Reverté, 1979.

[2] R.G. Bartle and D.R. Sherbert. Introducción al Análisis Matemático de una Variable (3ª Ed.). Limusa Wiley, 2010.

[3] R. Figueroa Sestelo and Ó.A. Otero Zarraquiños. Números reais. Universidade de Santiago de Compostela, 2022.

[4] R. Figueroa Sestelo and Ó.A. Otero Zarraquiños. Números complexos. Universidade de Santiago de Compostela, 2022.

[5] R. Figueroa Sestelo and Ó.A. Otero Zarraquiños. Series de números reais. Universidade de Santiago de Compostela, 2022.

[6] R. Figueroa Sestelo and Ó.A. Otero Zarraquiños. Sucesións de números reais. Universidade de Santiago de Compostela, 2022.

COMPLEMENTARY:

[1] S. Behar Jequín, R. Roldán Inguanzo and A. Arredondo Soto. Análisis matemático real: ejercicios y problemas. Universidad de La Habana, 2021.
https://elibro-net.ezbusc.usc.gal/es/lc/busc/titulos/196988

[2] J. Casasayas and M.C. Cascante. Problemas de Análisis Matemático de una variable real. Edunsa, 1990.

[3] A. García López et al. Cálculo I. Teoría y problemas de Análisis Matemático en una variable (2ª Ed.). Clagsa, 1994.

[4] R. Magnus. Fundamental Mathematical Analysis. Springer Cham, 2020.
https://link.springer.com/book/10.1007/978-3-030-46321-2

[5] T. Radozycki. Solving problems in Mathematical Analysis, Part I. Springer Nature, 2020.
https://link.springer.com/book/10.1007/978-3-030-35844-0

[6] B.S.W. Schröder. Mathematical Analysis: A Concise Introduction, John Wiley & Sons, 2007.
https://onlinelibrary.wiley.com/doi/book/10.1002/9780470226773

[7] M. Spivak. Calculus (2ª Ed.). Reverté, 1994.

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

It will follow the general methodological instructions established in the Memory of the Title of Degree in Mathematics of the USC.

Teaching is programmed in lectures, small group practices and tutorials.

In the theoretical classes the essential contents of the subject will be presented, and they will allow the work of basic, general and transversal skills, in addition to the specific competences CE1, CE2, CE5 and CE6. Meanwhile, in interactive sessions some exercises and/or problems for more autonomous realization will be proposed. This will emphasize the acquisition of specific skills CE3 and CE4, as well as the transversal skills CT1, CT2, CT3 and CT5 . Finally, in the tutorials we will discuss with the students, and we solve some exercises for the students to practice and secure the knowledge and the transversal skills previously commented.

Moreover, teaching material will be available to students in the USC virtual classroom.

Lectures and small group practices will be face-to-face and will be complemented by the virtual course of the subject, in which the students will find notes, problem bulletins, etc.

The tutorials will be in person, through email or through MS teams.

The assessment will be carried out by combining a continuous formative evaluation and a final exam.

The continuous assessment will consist of a theoretical-practical test and at least one short activity carried out in class.

The test will include the contents of topics 1 and 2 and will have a weight of 70% of the continuous assessment grade. The activities carried out in class will have a weight of 30% of the grade of the continuous assessment.

In the final written exam, the knowledge obtained by the students in relation to the concepts and results of the subject will be measured both from a theoretical and a practical point of view. The clarity and logical rigor shown in their exposition will also be assessed.

Both in the continuous assessment and in the final exam the specific competences from CE1 to CE6 will be evaluated.

For the calculation of the final mark (CF), the continuous assessment grade (EC) and the final exam score (EF) will be taken into account, and the formula CF = EC/3 + (1-EC/30)*EF will be applied. For details of this formulation the following work can be consulted:
Xavier Bardina, Eduardo Liz, "Mathematics and evaluation", MATerials MATemàtics, 2011, 6, 19 pp.
http://www.mat.uab.cat/matmat/PDFv2011/v2011n06.pdf

The continuous assessment tests may not coincide for both lecture groups. However, both teachers agree to make coordinated exams in order to guarantee the educational equivalence for all the students. The final test for each opportunity will be the same for the two lecture groups.

Any student who has not taken the final exam will be considered as not presented.

The same evaluation system will be used for the second opportunity, but by considering the exam corresponding to this new opportunity as the final exam in the above-mentioned formula. This new final exam will be of the same type as that of the former.

Warning. In cases of fraudulent performance of exercises or exams (plagiarism or improper use of technologies), the provisions of the Regulations for the evaluation of student academic performance and review of grades will apply.

PRESENCE WORK IN THE CLASS
Plenary lectures (28 hours)
Reduced group lectures (14 hours)
Lab lectures (14 hours)
Tutorials in very reduced groups (2 hours)
Total hours of presence work in the class: 58

PERSONAL WORK OF THE STUDENT
Autonomous Individual Study or in group (57 hours)
Writing of exercises, conclusions or other works (20 hours)
Programming/experimentation or other works in computer/laboratory (10 hours)
Recommended readings, activities in library or similar (5 hours)
Total hours of personal work of the student: 92

To study every day by using the bibliographical material. To read carefully the theoretical part until it has been assimilated and then, to answer the corresponding questions, exercises or problems. To follow the suggestions that the teacher may give throughout the academic year.