04310 - Complementary Fundamentals of Mathematical Analysis

Learning outcomes

At the end of the course, the student knows the basic ideas and techniques of differential and integral calculus on manifolds. He acquires the main knowledge about the trigonometric series and their pointwise, uniform and quadratic mean convergence. He knows how to use the skills acquired in the mathematical models of applied sciences and engineering.

Course contents

The course explores some classic and central topics in mathematical analysis, particularly the theory of integration on manifolds with respect to Hausdorff measure and the Fourier analysis for series expansions of periodic functions.
Various applications of the theory to geometric and physical models will also be presented, such as the derivation of Maxwell's equations for the electromagnetic field, the study of electrostatic potential and heat diffusion problems.
The course is essentially self-contained: the necessary notions will be recalled or defined during the lectures, in any case the fundamental courses of the first two years Bachelor's Degree are assumed as prerequisites, particularly Mathematical Analysis 1 and Mathematical Analysis 2.


The course is divided into the following two modules:


Part 1 - Martino
- Outline of measure theory.
- Hausdorff measure.
- Hausdorff integral.
- The divergence theorem.
- Differential forms.
- Stokes' theorem.
- Applications.


Part 2 - Montanari
- Trigonometric polynomials.
- Fourier polynomials.
- Expansion in Fourier series.
- Gibbs phenomenon. Fejér series.
- Poisson kernel.
- Complex Fourier series.
- Applications.

Readings/Bibliography

Ermanno Lanconelli
Lezioni di Analisi Matematica 2, Seconda Parte.
Pitagora Editrice Bologna


Walter Rudin
Principles of Mathematical Analysis, Third Edition.
McGraw-Hill


Tom Apostol
Mathematical Analysis, second Edition.
Addison-Wesley Publishing Company

Teaching methods

Lectures in classroom.

Assessment methods

Final oral exam

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Students with learning disorders and\or temporary or permanent disabilities: please, contact the office responsible (https://site.unibo.it/studenti-con-disabilita-e-dsa/en/for-students ) as soon as possible so that they can propose acceptable adjustments. The request for adaptation must be submitted in advance (15 days before the exam date) to the lecturer, who will assess the appropriateness of the adjustments, taking into account the teaching objectives.

Teaching tools

Additional material can be found on Virtuale.

Office hours

See the website of Vittorio Martino

See the website of Annamaria Montanari