Insegnamento a.a. 2020-2021

Classes: 27 (I sem.)

Instructors:
Class 27: GIUSEPPE SAVARE'

The aim of this course is to introduce students to a strong basic knowledge in Mathematical Analysis, which is necessary in a challenging degree in Mathematical and Computing Sciences. The course covers the fundamental elements of one-variable Mathematical Analysis: topology of the real line, sequences, series, limits and continuity, differential calculus, optimization, and integral calculus. The course also provides a first crucial approach to the language, structures and methods of Mathematics, and aims at presenting the general theoretical framework as an axiomatic-deductive system.

• The axiomatic approach to real numbers.

• Complex numbers.

• Sequences: convergence, subsequences, limsup and liminf. Cardinality: countable and uncountable sets. Discrete processes.
• Series: convergence and absolute convergence, elementary and advanced tests, infinite sums, operations on series.
• Basic metric and topological properties of the real line: open, closed, compact and connected sets.
• Functions. The concepts of limit and continuity of functions and their main applications. Uniform continuity.
• Differential calculus for real functions of one variable: derivatives, mean value theorems and their applications, l’Hopital’s rule, Taylor expansions, convexity.
• Cauchy-Riemann integral calculus: properties of the integral, integrability of monotonic and continuous functions, the Fundamental Theorem of Calculus, integration by parts, change of variables.
• Improper integrals, integrals and series, integral function.

At the end of the course student will be able to...

• Know the fundamental notions and results of one-variable mathematical analysis: limits, continuity, differential calculus, and integral calculus.
• Express these notions in a conceptually and formally correct way, using adequate definitions, theorems, and proofs.
• Understand the language and the formal aspects of Mathematics as an axiomatic-deductive system.

 

At the end of the course student will be able to...

• Apply the fundamental results of one-variable mathematical analysis to the solution of problems and exercises.
• Actively search for deductive ideas that are fit to prove possible links between the properties of mathematical objects and to solve assigned problems.
• Formulate simple problems through rigorous mathematical models, which can be analyzed with the help of calculus and analytical tools.  

 

• Face-to-face lectures
• Online lectures
• Exercises (exercises, database, software etc.)
• Individual assignments

Online lectures have the same conceptual role as face-to-face lectures. The actual blend of face-to-face lectures and online lectures will mainly depend on external constraints.

Exercise sessions (again: both face-to face and online) are dedicated to the application of the main theoretical results obtained during lectures to problems and exercises of various nature.

Individual assignments have the precise aim of calling for an active involvement of students. 

	Continuous assessment	Partial exams	General exam
Written individual exam (traditional/online)		x	x
Oral individual exam		x	x
Individual assignment (report, exercise, presentation, project work etc.)		x	x

Students will presumably be evaluated on the basis of written and oral exams, which can be taken in one of the two following ways.

• The exam can be split in two partial exams (October and January). Each partial may contain multiple-choice questions and open-answer questions; the second partial may also involve an oral exam; each partial weighs for at least one-third of the final mark. Presumably, there will also be two online tests/assignments; each one weighs for not more than one-sixth of the final mark. Each type of questions contributes in a specific way to the assessment of the students' acquired knowledge. Multiple-choice questions mainly aim at evaluating the knowledge of the fundamental mathematical notions and the ability to apply these notions to the solution of simple problems and exercises, while open-answer questions and oral exams mainly aim at evaluating:
  • The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs.
  • The ability to actively search for deductive ideas that are fit to prove possible links between the properties of mathematical objects.
  • The ability to apply mathematical notions to the solution of more complex problems and exercises.  
• The exam can also be taken as a single general exam, which contains both multiple-choice questions and open-answer questions, and may also involve an oral exam. The choice of a general exam may also include the consideration of online tests / assignments. The general exam covers the whole syllabus of the course and it can be taken in one of the four general sessions scheduled in the academic year (the two regular sessions in January and January/February, or the two make-up sessions in June and August/September). This option is mainly meant for students who have withdrawn from the two-partials procedure or could not follow it. Each type of questions contributes in a specific way to the assessment of the students' acquired knowledge. Multiple-choice questions mainly aim at evaluating the knowledge of the fundamental mathematical notions and the ability to apply these notions to the solution of simple problems and exercises, while open-answer questions and oral exams mainly aim at evaluating:
  • The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs. 
  • The ability to actively search for deductive ideas that are fit to prove possible links between the properties of mathematical objects.
  • The ability to apply mathematical notions to the solution of more complex problems and exercises.

We will take care to obtain final grades whose distribution follows the grade distribution that is recommended by Università Bocconi.

We underline that we prefer to maintain here a certain degree of freedom as for assessment methods: this is due to temporary external constraints, which may lead us to an unforecasted blend of teaching situations and therefore an unforecasted blend of assessment methods.

• Krantz, Steven G.: Real Analysis and Foundations

  Fourth edition. Textbooks in Mathematics, CRC Press, Boca Raton, FL. 2017, xxii+408 pp. ISBN: 978-1-4987-7768-1.
• Integrative teaching materials.