30543 - MATHEMATICAL ANALYSIS - MODULE 2
Course taught in English
Go to class group/s: 27
Lezioni della classe erogate in presenza
To feel comfortable with this course students should be familiar with the content of the courses of Mathematical Analysis 1 and Algebra and Geometry.
The aim of this Course is to lead the students to acquire the strong methodological, theoretical and practical techniques of Mathematical Analysis necessary in a challenging degree in Mathematical and Computing Sciences. In continuity with the first module of Mathematical Analysis, the course covers fundamental notions of topology in metric spaces, and of differential and integral calculus for functions of several variables.
The course will cover the following elements:
- Elements of topology in the framework of normed and metric spaces. Convergence of sequences, continuity of functions, compactness and completeness of metric spaces.
- Differentiability for functions of several variables. Continuity, differentiability, differential and Jacobian. Particular case of curves. Application to the search of extrema.
- Integral along a curve. Integral of a scalar field and of a vector function along a curve. Length.
- Integral of functions of several variables. Definition, Fubini's theorem and change of variables.
- Limit theorems for integrals. Monotone and dominated convergence theorem, differentiation under the integral. If time permits, introduction to the basic concepts of measure theory.
- Around Stokes theorem. Path integral of a gradient, conservative fields. Green's theorem. Closed and exact forms, simply connected sets.
- know the fundamental notions and results presented in the course concerning the techniques used in Mathematical Analysis.
- express these notions in a conceptually and formally correct way, using adequate definitions, theorems and proofs.
- understand the relevance of the mathematical language and the interplay between the abstract setting and a concrete mathematical problem.
- apply the mathematical knowledge to model problems.
- provide rigorous mathematical reasonings to solve exercicses.
- using the methodological tools of Mathematical Analysis to dialog with other Sciences such as Physics and Economics.
- 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.
Students will be evaluated on the basis of written and oral exams and continuous assignements.
Some individual assignement will be proposed during the course with the aim of stimulating partecipation and evaluating
- the ability to think on complex problems
- the ability to write proofs.
Both written and oral exams cover the whole syllabus of the course and contribute in a specific way to the assessment of the students’ acquired knowledge, in particular it has the aim of evaluating:
- the ability to articulate the knowledge of mathematical notions, definitions, theorems, and proofs in a conceptually and formally correct way
- 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 complex problems.
The written exam can be split in two partial exams (March and May) or taken as a general exam (May, June, August/September, January/February).
The oral exam can be taken after the written exam (in May after the second partial exam or after the general exam in May, June, August/September, January/February).
The final grade will be made: 10% assignements, 45% written exam and 45% oral exam.
We will take care to obtain final grades whose distribution follows the grade distribution that is recommended by Università Bocconi.
Giaquinta, Mariano; Modica, Giuseppe. Mathematical analysis. Linear and metric structures and continuity.
Birkhäuser Boston, Inc., Boston, MA, 2007. xx+465 pp.
ISBN: 978-0-8176-4374-4; 0-8176-4374-5
Giaquinta, Mariano; Modica, Giuseppe. Mathematical analysis. An introduction to functions of several variables.
Birkhäuser Boston, Inc., Boston, MA, 2009. xii+348 pp. ISBN: 978-0-8176-4509-0