35143 - Mathematical Methods for Engineering

Learning outcomes

The student acquires competences in advanced mathematical methods and tools, with applications to aerospace and mechanical engineering.

Course contents

Prerequisites

Basic knowledge of Mathematical Analysis and Linear Algebra acquired during an undergraduate degree program.
In particular, students are expected to have a solid understanding of:
- functions of several real variables,
- limits and continuity,
- differential and integral calculus in multiple variables.

Course Program

1. Complex Analysis

Review of the topology of the Euclidean plane and the theory of integration of differential 1-forms.
Gauss–Green and divergence theorems in the plane.
Introduction to the theory of holomorphic functions: Cauchy–Riemann equations, mean value property, maximum principle.
Complex integration: Cauchy’s integral theorem, Cauchy’s integral formula and representation theorem.
Complex power series and analytic functions.
Zeros of analytic functions and the analytic continuation theorem.
Residue theorem and main applications, particularly to the computation of improper integrals.

2. Elements of Functional Analysis

Overview of abstract integration theory.
Banach and Hilbert spaces; L^p spaces and spaces of continuous functions.
Projections and orthonormal bases in Hilbert spaces.
Fourier series and their properties: convergence theorems for periodic functions.

3. Fourier and Laplace Transforms

Fourier transform in L^1 and L^2: algebraic and differential properties, relation with convolution.
Introduction to rapidly decreasing functions and distributions.
Laplace transform: definition, region of convergence, and main computation rules.

4. Introduction to Partial Differential Equations (PDEs)

Applications of Fourier and Laplace transforms to solving differential equations.
Examples of transport, diffusion, and wave propagation models.

5. Elements of Probability Theory

Probability calculus: combinatorial methods, Bayes’ theorem, conditional probability, and independent events.
Discrete and continuous random variables; independence of random variables.
Expectation, variance, and covariance.

Readings/Bibliography

Recommended textbooks: (being updated)
[1] C. Barozzi — Matematica per l'ingegneria dell'informazione, updated reprint, Zanichelli, 2005.
[2] M. Cadegone, L. Lussardi — Metodi matematici per l’ingegneria, second edition, 2021.
[3] K. F. Riley, M. P. Hobson, S. J. Bence — Mathematical Methods for Physics and Engineering, Cambridge University Press, 2006.
[4] F. Gazzola, F. Tomarelli, M. Zanotti — Analytic functions Integral transforms Differential equations, Esculapio, 2023.
[Italian language edition] F. Gazzola, F. Tomarelli, M. Zanotti — Analisi complessa trasformate Equazioni differenziali, Esculapio, 2023.

Recommended for further study:
[1] S. Salsa, G. Verzini — Partial Differential Equations in Action, UNITEX Springer, 2022.
[2] A. Pascucci — Probability Theory I, UNITEX Springer, 2024.
[Italian language edition] A. Pascucci — Teoria della Probabilità variabili aleatorie e distribuzioni, UNITEX Springer, 2020.

To cover possible gaps in Mathematical Analysis prerequisites and for introductory review:
[1] N. Fusco, P. Marcellini, C. Sbordone — Mathematical Analysis, UNITEX Springer, 2022.
[Italian language edition] N. Fusco, P. Marcellini, C. Sbordone — Lezioni di Analisi Matematica Due, Zanichelli, 2020.

Teaching methods

Lectures and public discussion of the exercises.

Assessment methods

The exam consists of a written test including exercises and questions designed to assess the understanding of the topics covered and the ability to apply the acquired knowledge.
During the exam, the use of any supporting materials (notes, books, handouts) as well as calculators or any electronic devices is strictly prohibited.

 

The exam dates can be found on the AlmaEsami web platform of the University of Bologna.To take part in the exam, each student must register well in advance on the lists available on the platform.On the day of the exam, students will be allowed to take the test only if they present a valid identification document.

 

For students who attended the course in academic years prior to 2025/2026, the exam syllabus and assessment rules remain unchanged.The exam consists of two parts: a written test with exercises and an oral examination on the theoretical part of the course, which must be taken in this order.
Further details and instructions are available on the official course webpage.

 

Students with special educational needs related to learning disorders and/or disabilities are advised to contact the competent University Office (https://site.unibo.it/studenti-con-disabilita-e-dsa/it) in advance so that possible exam accommodations can be arranged. Any proposed accommodations must be submitted for the instructor’s approval at least 15 days before the exam date, who will assess their appropriateness in relation to the learning objectives of the course.

 

Clarifications on the recording of the exam grade for those who have the  course MATHEMATICAL METHODS FOR ENGINEERING (6 CFU) as a module of the integrated course of NUMERICAL AND MATHEMATICAL METHODS FOR ENGINEERING (12 CFU)

If the MATHEMATICAL METHODS FOR ENGINEERING course (6 CFU) is one of the two modules that, together with NUMERICAL ANALYSIS (6 CFU), constitutes the integrated course of NUMERICAL AND MATHEMATICAL METHODS FOR ENGINEERING (12 CFU), the grade that will be recorded will be calculated with the arithmetic average of the single grades that the student has obtained in the two modules. It should be noted that the result of the average will be rounded to the nearest integer. Only if the resulting average is exactly equidistant between two integers, the grade will be obtained by rounding up to the next highest integer. Lastly, in order to obtain the "30 cum laude" final evaluation, the student must be in one of the two following cases:

- receiving "30 cum laude" in both modules;

- obtaining "30 cum laude" in one module and 30 in the other one.

Lastly, it should be noted that the recording of the final evaluation requires the passing of both the exam of NUMERICAL ANALYSIS (6 CFU) and the exam of MATHEMATICAL METHODS FOR ENGINEERING (6 CFU) in a time interval not exceeding 12 months.

Teaching tools

Pdf files uploaded in the institutional site.

Links to further information

https://virtuale.unibo.it/course/view.php?id=76178

Office hours

See the website of Enzo Maria Merlino