27213 - Mathematical Analysis 2

Course contents

Metric spaces, nomea spaces, spaces with inner product. Complete metric spaces. Banach spaces and Hilbert spaces. Banach-Caccioppoli theorem. Uniform and total convergence. Differential equations: existence and uniqueness of the Cauchy (Peano-Picard Theorem). Differential calculus for function with several variables: limits, directional derivative, partial differential derivative, differential of a function, chain's rule, gradient of a function, Taylor's formula. Research free maximum and minimum: necessary condition and sufficient conditions. Infinitesimal calculus for curves: regular curves and integral of fist type. Differential calculus for function with several variables with vector values: limit, continuity and differentiability of a function, Jacobian matrix, Hessian matrix, Taylor's formula. Implicit function Theorem (Dini).  Inverse function Theorem.  k-dimensional manifolds. 
Maximum and minimum on manifold,  integral on curves, integral calculus in several variables. Tangent space and space at a point in a manifold. Lagrange Theorem,  maximum and minimum with constrains.  Integral calculus: multiple integral,  simple sets,  regular and measurable sets. Fubini Theorem.  Theorem on change of variables. Basics about measure theory, basics on abstract theory, Lebesgue integral, 
Fubini's theorem and Lebesgue Theorem,  L^p spaces. Vector fields Vector fields: integral of second type. Vector fields, potentials and differential forms. Integral on surfaces.
Gauss-Green Theorems, Gauss and Stokes Theorems.

Readings/Bibliography

Some books where the students can find the subjects of the course: 

C.D. Pagani, S. Salsa: Analisi Matematica 2 (Masson);

W. Rudin: Analisi Reale e Complessa (Boringhieri);

E. Giusti: Analisi Matematica 2 (Boringhieri).

S. Salsa, A. Squellati: Esercizi di Matematica volume 2 (Zanichelli);

T. Tao: An introduction to measure theory, GSM 126, AMS;

Marcellini Sbordone: Esercitazioni di Matematica, Secondo volume (Liguori Editore);

M. Bramanti: Esercitazioni di Analisi Matematica 2, Progetto Leonardo - Esculapio (2012).

Teaching methods

The course is taught in lessons in classroom. Some complementary material will be available on the personal web page of the teacher

Assessment methods

The grading is split in several  written  preparatory parts and ends with a final colloquium. The candidate must collect in an exercise book all the exercises that the teacher has assigned during all the lessons of the course. The exercise book has to be written by hand by the candidate itself (no papers written on computers or photocopies or scansions are accepted). Some queries could be posed about the assigned exercises. 

Example of the test A+B

Example of the test C

Further details about the final exam may be found in the parallel italian page of the course. In any case the candidate may ask to the teacher all the clarification about the structure of the tests and their grade.

The teacher plans to schedule also two tests during the lessons called tests "in itinere". For the details please check the explanation in the parallel italian version or ask to the teacher. 

Teaching tools

Books, and papers on the teacher web page. Tutor (if assigned).

Links to further information

http://www.unibo.it/SitoWebDocente/default.htm?UPN=fausto.ferrari@unibo.it

Office hours

See the website of Fausto Ferrari