16314 - Mathematical Analysis B

Learning outcomes

The integrated Course in Mathematical Analysis provides students with the fundamental knowledge and methodological tools required for the understanding and quantitative description of engineering problems. Students become familiar with the concepts, properties, and techniques of mathematical analysis, learning how to use them to formulate and solve increasingly complex applied problems.

In this module, students further develop the methodological and applied aspects of mathematical analysis, with particular emphasis on functions of several real variables, vector fields, multiple integrals, and ordinary differential equations. They learn to employ these tools to interpret, model, and solve typical engineering problems, developing a solid operational command of the principal analytical techniques.

Course contents

EUCLIDEAN SPACE ( \mathbb{R}^n )

Vector space structure, inner product, and Euclidean norm. Elements of topology: open balls, open sets, closed sets, bounded sets, compact sets, and arcwise connected subsets of ( \mathbb{R}^n ).

LIMITS, CONTINUITY, AND DIFFERENTIAL CALCULUS

Real-valued and vector-valued functions of several real variables. Accumulation points. Limits of functions. Continuous functions. Weierstrass theorem for functions of several variables. Partial derivatives and directional derivatives. Differentiable functions and ( C^1 ) functions. Chain rule. Higher-order partial derivatives. Hessian matrix and Schwarz's theorem. Extrema of functions. Taylor's formula for functions of several variables. Lagrange Mean Value Theorem. Fermat's theorem. Classification of local extrema using the Hessian test.

DIFFERENTIAL EQUATIONS

The Cauchy initial value problem for ordinary differential equations. Existence, uniqueness, and continuation theorems. Solution methods for separable first-order nonlinear differential equations, first-order linear differential equations, and second-order linear differential equations with constant coefficients.

MULTIPLE INTEGRALS

Definition of the Riemann double integral over a normal domain. Properties of the double integral. Computation of double integrals over normal domains through iterated integrals. Change of variables theorem for double integrals. Extension to triple integrals.

LINE AND SURFACE INTEGRALS

Regular and piecewise regular curves. Arc length. Line integrals of scalar functions.

If time permits

Line integrals of vector fields along oriented curves. Irrotational and conservative vector fields: determination of scalar potentials.

Poincaré's theorem on simply connected domains. Green–Gauss theorem, Divergence theorem, and Stokes' theorem.

Readings/Bibliography

Lecture notes prepared by the instructor and available online through the Virtuale platform.

Reference texts for the lecture notes:

1. E. Giusti, Analisi Matematica 2, Third Edition, Bollati Boringhieri.
2. N. Fusco, P. Marcellini, C. Sbordone, Elementi di Analisi Matematica Due. Versione semplificata per i nuovi corsi di laurea, Liguori Editore.
3. G. C. Barozzi, G. Dore, E. Obrecht, Elementi di Analisi Matematica, Vol. 2, Zanichelli, 2015.

Exercise Books

• E. Giusti, Esercizi e Complementi di Analisi Matematica, Vol. 2, Bollati Boringhieri, 1992.
• P. Marcellini, C. Sbordone, Esercitazioni di Matematica, Volume II, Parts I and II, Liguori Editore.

 

Teaching methods

Mathematical Analysis B is offered during the second semester and constitutes the second part of the Integrated Course in Mathematical Analysis (12 ECTS credits).

The course consists of classroom lectures, preferably delivered at the blackboard. The theoretical foundations of each topic are presented first. After introducing the basic concepts, the main theorems and results concerning differential and integral calculus for functions of several real variables and ordinary differential equations are stated and rigorously proved.

A substantial portion of the course is devoted to the solution of exercises and applications.

 

Assessment methods

Student learning is assessed through a written examination lasting two hours. Admission to the examination requires successful completion of the written examination for Mathematical Analysis A within the same academic year.

The written examination for Mathematical Analysis B consists of two consecutive parts: a theory section and a problem-solving section, both of which must be passed sequentially. The assessment criteria are aligned with the learning objectives described below.

Students must register for the examination through the AlmaEsami system at least three days before the examination date.

After passing the written examination, students may request an optional oral examination to improve their final grade.

The oral examination consists of theoretical questions (definitions, theorem statements, and proofs), discussion of the connections among different topics, and interpretation of the solutions presented in the written examination.

Once the oral examination date has been agreed upon with the instructor, students must contact the instructor one week in advance to receive, by email, a selection of topics from the syllabus to be discussed.

No midterm examinations are scheduled for Module B.

The final grade for the Integrated Course in Mathematical Analysis is calculated as the arithmetic mean of the grades obtained in Mathematical Analysis A and Mathematical Analysis B. The grade obtained for Module B remains valid only during the examination session in which it is earned (the written examination of Mathematical Analysis A remains valid for twelve months). If multiple grades are obtained for either module, only the most recent grade will be considered.

Students may choose either to accept the averaged grade or to repeat the examination. Examination results are published through AlmaEsami and Virtuale, and students are given at least one week to accept the grade by registering for a subsequent online examination session on AlmaEsami.

Learning Outcomes (Dublin Descriptors)

Students are expected to demonstrate achievement of the following learning outcomes through the assessment activities.

1. Knowledge and Understanding

Students acquire the theoretical foundations and mathematical tools of multivariable differential and integral calculus, ordinary differential equations, and vector calculus.

In particular, they will be able to:

• understand the structure of Euclidean n-dimensional space and its fundamental topological properties;
• understand the concepts of limits, continuity, differentiability, and Taylor expansions for multivariable functions;
• understand the theory of ordinary differential equations, including existence and uniqueness theorems and the principal solution methods;
• understand the theory of multiple integrals, line and surface integrals, and the fundamental theorems of vector calculus (Green–Gauss, Divergence, and Stokes).

2. Applying Knowledge and Understanding

Students are able to apply theoretical concepts to solve mathematical and engineering problems, particularly those arising in mechanical and aerospace engineering.

Specifically, they will be able to:

• compute limits, partial derivatives, gradients, Hessian matrices, and Taylor expansions;
• determine unconstrained extrema of multivariable functions;
• solve first- and second-order ordinary differential equations using the methods studied;
• evaluate double and triple integrals, including by means of change of variables;
• compute line and surface integrals;
• determine whether a vector field is conservative and compute its scalar potential;
• apply the fundamental theorems of vector calculus to solve geometric and physical problems.

These competencies enable students to address mathematical models arising in engineering disciplines such as mechanics, fluid dynamics, heat transfer, and dynamical systems.

3. Decision Making  

Students develop the ability to select the most appropriate mathematical tools for solving a problem and to critically assess both the correctness of the adopted procedures and the validity of the results obtained.

In particular, they will be able to:

• identify the most appropriate solution method according to the characteristics of the problem;
• verify the mathematical consistency of results and interpret their meaning;
• recognize the assumptions required for applying the principal theorems and computational methods;
• critically compare alternative solution strategies and justify the chosen approach.

This autonomy provides the basis for addressing more advanced mathematical and modelling problems in subsequent engineering courses.

4. Communication Skills

Students acquire the ability to present theoretical concepts, essential proofs, and solution procedures rigorously and using appropriate mathematical language.

In particular, they will be able to:

• use mathematical notation correctly;
• clearly state the assumptions and conclusions of the principal theorems;
• present solutions to exercises and problems in an orderly and logically coherent manner;
• justify the methodological choices adopted during problem solving;
• communicate mathematical results both symbolically and through appropriate interpretation of the obtained solutions.

5. Learning Skills

Students develop an independent study method that enables them to deepen their understanding of the topics covered and to acquire new mathematical tools required in subsequent engineering courses.

Upon completion of the course, they will be able to:

• integrate acquired knowledge with new topics in mathematical analysis and mathematical modelling;
• consult university-level textbooks and scientific literature independently;
• apply learned mathematical techniques to new and interdisciplinary problems;
• successfully undertake subsequent courses requiring advanced mathematical analysis, including rational mechanics, fluid dynamics, continuum mechanics, automatic control, and aerospace and mechanical engineering subjects.

Attendance and participation in the course are intended to foster the development of all five learning outcomes, with particular emphasis on communication skills.

Teaching tools

• Course webpage on the Virtuale platform.
  
• teacher's lecture notes available through Virtuale.
• Student office hours (Tuesday afternoons, in person or online, by appointment).
• Tutorial exercise sessions.

Office hours

See the website of Simone Ciani