27213 - Mathematical Analysis 2

Academic Year 2017/2018

  • Moduli: Nicola Arcozzi (Modulo 1) Giovanni Dore (Modulo 2)
  • Teaching Mode: Traditional lectures (Modulo 1) Traditional lectures (Modulo 2)
  • Campus: Bologna
  • Corso: First cycle degree programme (L) in Mathematics (cod. 8010)

Learning outcomes

At the end of the course the student will master the basic results as well as the basic tools of advanced calculus. He will master the notions of differentiability and integrability 'for functions of several real variables. He will be able to apply this knowledge to the solution of problems posed by the pure and applied sciences. He can solve practical problems of optimization and measure. He can formalize autonomously elementary problems raised by applied sciences.

Course contents

Function series: pointwise, uniform, total convergence. Power series, Taylor series. Multivariable differential calculus: mean value theorem, secon order Taylor's formula in several variables, convex functions and local extrema, local invertibility, and the implicit function theorem.

Lebesgue's measure theory in R^n. Lebesgue integral: elementary properties, theorems of Tonelli and Fubini, change of variables.

Fixed point theorem for contractions in metric spaces. 

Ordinary Differential Equations (ODEs) and systems thereof; Cauchy problem; local existence and continuation of solutions. Special ODEs and their solutions. Linear ODEs and sytstems thereof: general integral; solutions of linear ODEs and sytstems with constant coefficients.

Readings/Bibliography

Enrico Giusti: Analisi Matematica 2, Ed. Boringhieri

Ermanno Lanconelli, Analisi Matematica 2 (prima e seconda parte), Ed. Pitagora.

Walter Rudin: Principles of Mathematical Analysis, 3rd Edition, MAc Graw-Hill 2015

Teaching methods

Lectures and exercises at the blackboard

Assessment methods

Written and oral exam. The details will be discussed at the beginning of the course.

Teaching tools

Lecture notes and exercises will be made available on the web.

Office hours

See the website of Nicola Arcozzi

See the website of Giovanni Dore