Mathematical Analysis 1

The course aims to provide the student with an introduction the beginnings of a deeper understanding of Mathematical Analysis (essential for a student of physics) which will continue in the courses Mathematical Analysis 2 and 3. Fundamental concepts will be treated with rigor and precision with particular attention dedicated to: the real number system viewed as a complete ordered field with the cardinality of the continuum, metric spaces (including the real numbers) viewed as an abstract and general setting in order to formulate in a robust way the notions of limits of sequences in metric spaces, limits and continuity of functions between metric spaces (fundamental for the development of differential and integral calculus in Euclidean spaces), and, reducing to the important special case of the real number field, convergence of sequences and series and differential calculus for real functions of a real variable.

1. Knowledge and understanding of the concept of a complete ordered field with the cardinality of the continuum.
2. Knowledge and understanding of the concept of a metric space and elements of the topology of metric spaces (classification of points and sets)
3. Knowledge and understanding of the concept of the limit of a sequence in a metric space (uniqueness of the limit, necessary conditions for the convergence).
4. Knowledge and understanding of the concept of the limit of a sequence of real numbers and techniques for the calculation of limits (regularity of monotone sequences, comparison criteria, algebraic properties, Landau symbols e comparison for vanishing and diverging sequences).
5. Knowledge and understanding of the concept of convergence and divergence for numerical series (modes of convergence and relations between them).
6. Knowledge and understanding of techniques for establishing the character of a series (Cauchy's criterion, ratio test, root test, Leibniz's test, condensation test, for example).
7. Knowledge and understanding of the concepts of limit and continuity for functions between metric spaces.
8. Knowledge and understanding of the consequences of continuity (existence for equations involving real valued functions, existence of maxima and minima for real value functions on compact sets, the intermediate value property for real valued functions).
9. Knowledge and understanding of the concept of differentiability for real valued functions of a real variable (geometric and physical significance for unidimensional motions of the first and second derivatives).
10. Knowledge and understanding of techniques for the calculation of derivatives (algebraic properties, compositions and the chain rule, derivative of inverse functions).
11. Knowledge and understanding of the applications of differential calculus for real valued functions of a real variable (monotonicity, convexity, local maximums and minimums, qualitative study of functions).
12. Ability to correctly state the principal definitions and theorems.
13. Ability to perform abstract reasoning in concrete situations.
14. Ability to perform quickly and correctly calculations in the resolution of exercises in the written exams.
15. Ability to think synthetically and critically with regard to the concepts studied and the relations between them.
16. Ability to think critically concerning the value of the mathematics studied in the context of the degree program in physics.
17. Ability to communicate in the discussions of the material studied during the final oral examination.
18. Ability to continue in an autonomous way a deeper understanding of mathematical analysis in future courses.
19. Possibility to work in groups during the recitation sections, tutorial and in preparation for the written and oral examinations.