Continuum Mathematics

A.Y. 2019/2020
12
Max ECTS
120
Overall hours
SSD
MAT/01 MAT/02 MAT/03 MAT/04 MAT/05 MAT/06 MAT/07 MAT/08 MAT/09
Language
Italian
Learning objectives
The aim of the curse is twofold. First, to provide students with a basic mathematical language, allowing them to formulate a problem in a correct way and to understand a problem formulated by other people. Secondly, to provide the necessary instruments to solve some specific problems, ranging from the behaviour of sequences to that of series and functions of a single variable.
Expected learning outcomes
Students have to correctly express a selected number of basic mathematical notions and instruments. Moreover, they must know which instrument is the most suitable to solve
some classical problem in Mathematical Analysis. Finally, they must be able to use such instrument to solve the problem itself, or at least have the appropriate know-how to understand some helpful mathematical text.
Single course

This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.

Course syllabus and organization

Single session

Responsible
Lesson period
First semester
Course syllabus
Complex numbers Algebraic and trigonometric representations, algebraic operations and roots of unity. The Fundamental Theorem of Algebra, factoring of polynomials.
Real numbers, rational numbers and integers. Comparison between rational and irrational numbers: countable and uncountable sets. Maximum and minimal elements of subsets of the real line, greatest lower bound and least upper bound.
Natural numbers: Induction over the integers and properties that holds eventually.
Sequences of real numbers: basic properties, boundedness and monotonicity.
Limits of sequences: the notion of limit, uniqueness, boundedness of convergent sequences, comparison theorems. Algebraic operations with limits and forms of indecision, comparison between infinite/infinitesimal sequences, ratio and root tests, Landau's symbol little-o and its use, the concept of asymptotic equivalence and its use. Regularity of monotone sequences, the number e (of Napier). Other Landau symbols: big-o, big-omega, big-theta and their use in the comparison of sequences.
Continuous functions: the notion of continuity and its graphical interpretation, points of discontinuity. The concept of limit for functions and its relation to continuity. Continuity and discontinuity of elementary functions: rational, exponential, logarithmic functions, the absolute value and step functions, the integer and fractional part functions. Change of variables in limits and the limit of compositions of functions. The theorem on zeros and Weirstrass' theorem for continuous functions.
Differential calculus: the concept of the derivative: linear approximations and the tangent to a curve. Calculation of derivatives for elementary functions. Angular points and cusps. Algebraic operations and the derivative. The theorems of Fermat, Rolle, Lagrange and applications. Cauchy's theorem. De l'Hôpital's theorem and comparison on the order of infinity. The formula of Taylor and its applications. Optimization problems (finding maxima and minima). Integral calculus: computing areas, approximation and the method of exhaustion. The integral of Riemann: definition of the definite integral, classes of integrable functions, properties of the definite integral.
The mean value theorem for integrals and the fundamental theorem of calculus.
Indefinite integrals and their calculation: integration by substitution and by parts,
integration of rational functions. Improper integrals: definition and fundamental examples.
Finite sums: shifts, inresions and other algebraic manipulations. Fundamental examples: powers of integers and geometric progressions.
The concept of series: fundamental examples, the geometric series and telescopic series.
Necessary condition for convergence, regularity, comparison test, absolute convergence and simple convergence. Estimates and asymptotic estimates on the rate of conver-gence/divergence of a series: comparison test, limit comparison test. The generalized harmonic series (p-series).
Real power series and the radius of convergence. Taylor series and analytic functions. Algebraic operations on power series, derivative and integrals of series.
Introduction to recursion: problems of counting and recurrence equations. Solving recurrence equations by the method of the (ordinary) generating functions: formal manipulations of power series.
Prerequisites for admission
Part of the ministerial program for the high schools, namely:
- basic algebra: monomials, polinomials, rational functions, powers, roots, exponentials and logarithms
- solving basic equations and inequalities
- basic theory of functions, standard functions and their graphs, graphic interpretation of inequalities
- basic analytical geometry on the plane: lineas, cicrcles and parabolas
- basic trigonometry: sinus, cosinus and tangent, addition formulas
- basic set theory
- basic elements of logic
Teaching methods
During the intial 4 weeks, besides the lessons and the exercise session of the course, it takes places a project aimed to recover the prerequisites, that consists in:
- a tightly scheduled and monitored work on a e-learning platform
- some exercise sessions to prepare the work
Attending to the project is on voluntary basis. The projects ends with a test in presence that, if passed, exempt from the part 0 of the written examinations of the current academic year.
The course consists in lessons and exercise sessions, that alternate according to the time-table published on the Ariel website.
Classroom lessons and exercise sessions are grouped by topics. Druing the classroom lessons devoted to a topic, on the Ariel website the teacher publishes:
- a draft of the lessons themselves, with links to the bibliography
- a list of exercises on the same argument
The exercises are aimed to verify the real theoretical understanding of the topic and its concrete use in solving related problems.
Usually, the exercise sessions on a topic take place a week later than the related exercises are published in the website: the delay is aimed to allow the student to face the exercices autonomously, in order to better exploit the exercise sessions, where most of the same exercises are solved by the teacher.
The written examinations are made by exercices taken from the same lists.
Attending to classroom lessons and exercise sessions is strongly suggested. However, in case of necessity, the videos of a past edition (2010/11) of the same activities are available on the website.
Teaching Resources
Ariel web site: Continuum Mathematics - F1X http://mtarallomc.ariel.ctu.unimi.it
Videos of the 1010/11 edition: http://vc.dsi.unimi.it
Bibliography:
- P. Marcellini e C. Sbordone, Calcolo, Liguori
- H.S. Wilf, Generatingfunctionology, Academic Press (freely available for didactical use)
- teacher's notes on some specific argument
A lot of material is available on the Ariel web site of the course:
- detailed Course Syllabus
- time-table of lessons and problem classes, grouped by topics and with links to the bibliography
- one or more list of exercises for every topic
- notes by the teacher and by some brilliant students from the past editions
- all the texts of the written examinations in the last 10 years
Assessment methods and Criteria
The exam consists of a written part and oral part, both of which are compulsary.
The written examination takes 3.5 hours and consists of three different parts with open answers:
0) test on prerequisites
1) calculus skills
2) theoretical skills on the main notions
Passing the part 0) allows the get the correction of parts 1) and 2), apart those students that are exempted because they attended the specific project.
Parts 1) and 2) contribute together to a maximum score of 30/30. This score depends on the arguments and the specialized lexicon used in the provided answer, and it is either sent by email to the single students and published on the Ariel website of the class.
Passing the written examination admits to oral one, that starts from a discussion about the written part and then covers the main topics of the class. The score of the oral examination may be positive or negative: the final score of the exam is given by the algebraic sum of the scores of the written and oral parts.
During the exam, no computers and books are admitted, apart a text book which is available at the teaching desk during the written part.
The text of all the written examinations of the last ten years are available on the Ariel website of the class.
MAT/01 - MATHEMATICAL LOGIC
MAT/02 - ALGEBRA
MAT/03 - GEOMETRY
MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS
MAT/05 - MATHEMATICAL ANALYSIS
MAT/06 - PROBABILITY AND STATISTICS
MAT/07 - MATHEMATICAL PHYSICS
MAT/08 - NUMERICAL ANALYSIS
MAT/09 - OPERATIONS RESEARCH
Practicals: 72 hours
Lessons: 48 hours
Professors: Gori Anna, Tarallo Massimo Emilio
Shifts:
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Professors: Gori Anna, Tarallo Massimo Emilio
Professor(s)