70/0069-M - APPLIED MATHEMATICS
Academic Year 2021/2022
Free text for the University
LUISA FERMO (Tit.)
- Teaching style
- Lingua Insegnamento
|[70/75] BIOMEDICAL ENGINEERING||[75/00 - Ord. 2017] PERCORSO COMUNE||6||60|
|[70/77] CHEMICAL ENGINEERING||[77/00 - Ord. 2017] PERCORSO COMUNE||6||60|
1. Acquiring knowledge and understanding.
The course is devoted to students in the second year of the Bachelor's degree in Biomedical Engineering and to students in the third year of the Bachelor's degree in Chemical Engineering. It aims to provide a working knowledge of the fundamental methodologies of linear algebra and Fourier analysis, as well as the basic methods for the numerical solution of ordinary differential equations. These topics are presented by providing a rigorous theoretical justification, as far as possible.
2. Applying knowledge and understanding.
Possible applications of the methods treated during the course will be discussed, both for the solution of other mathematical problems, mainly differential ones, and for the resolution of applicative problems which are typical of the interesting branches of engineering.
3. Making informed judgements and choices.
This course allows assiduous students to achieve knowledge and understanding sufficient to:
- apply the methodologies described in the course to the solution of problems encountered in their own field of interest,
- identify the most important theoretical properties for a particular problem,
- choose the most suitable algorithm for its solution.
4. Communicating knowledge and understanding.
The evaluation of the written test keeps into account the ability of the student to give a methodical and consistent exposition of the program of the course. His communicating knowledge is further analyzed during the optional oral interview.
5. Capacities to continue learning.
This course allows assiduous students to acquire basic expertise which is sufficient to understand advanced mathematical texts for widening their knowledge autonomously.
1. Knowledge. The course requires a good knowledge of the basic concepts of linear algebra and of real and complex analysis which can be acquired during the first and second year of the course.
2. Skills. Students have to be able to apply the methods learned during the exams of the first year. In detail: the graphs of elementary functions, the computation of derivatives and integrals, performing operations with complex numbers, matrix and vector arithmetic, computation of eigenvalues.
3. Competence. Special skills are not required to access the course. The habit of mind of adopting a mathematical approach to problem solving, and a good ability in the manipulation of algebraic expressions are definitely useful. Previous experience in computer programming can help to have a deeper understanding of the algorithms and to be able to implement them efficiently.
Preparatory courses. According to the rules of the Engineering Faculty, students must have previously attended and passed the examinations of the courses "Analisi Matematica 1", "Geometria e Algebra", and "Fisica 1".
1. Numerical linear algebra (20 hours)
Review of linear algebra. Normed spaces and Hilbert spaces. Eigenvalues and eigenvectors. Structured matrices. Matrix norms. Condition number. Solution of linear systems by means of direct methods (Gaussian elimination) and iterative methods (Jacobi and Gauss Seidel).
2. Applied Fourier analysis (20 hours)
Periodic functions and trigonometric polynomials. Norm approximation and Fourier coefficients. Orthogonality and optimality properties. Periodic extension of a function and Fourier series. Convergence properties. Complex Fourier series. The Fourier transform. Inverse Fourier transform. Properties of the Fourier transform. Convolution. Applications.
3. Ordinary Differential Equations (20 hours)
Formulation of Cauchy problems. Existence and uniqueness of the solution. Systems of differential equations of the first order and equations of order larger than one. Finite differences methods. Implicit and explicit methods. Monostep and multistep methods. Explicit Runge-Kutta methods. Global and local discretization error. Convergence and stability. Consistency and order. General form for multistep methods. Zero-stability and roots condition. Dahlquist theorem. Local discretization error for multistep methods. Consistency and order.
Teaching will be delivered simultaneously both in person and online, outlining a mixed teaching activity that may be enjoyed both at the university and at home.
The course consists of 48 lecture hours and 12 practical hours. To guarantee the most efficacy of teaching, the theoretical and practical lessons, which also provide the solution of grading written tests, are integrated with each other without solution of continuity. Simultaneously with the course, a tutorial activity is furnished to students, to assist them while they study for the final grading. The teacher gives constant assistance to students, during the whole year, through personal interviews, e-mail messages, and teleconferencing systems. In accordance with the “Manifesto degli Studi per l'A.A. 2021-22” and compatibly to the pandemic situation, teaching will be delivered mainly in person, integrated and “augmented” by online strategies, with the aim of ensuring it may be enjoyed in an innovative and inclusive fashion.
Verification of learning
Grading normally consists of a written test. It includes some exercises which require both operative skill and knowledge of the theory concerning the whole program.
The proposed exercises aim to verify the ability of the student to solve standard problems in the context of the topics of the course. The ability to detect and apply the theoretical properties for a particular problem, and to choose consequently the most suitable algorithms for its solution, will also be assessed. Finally, student’s mastery of clearly explaining concepts will be kept into consideration. To pass the exam the student must reach a grade of at least 18/30. The student may ask for an oral interview to improve the final grade. An oral interview may also be required by the teacher if it is useful to correctly evaluate the performance of the student.
To pass the exam the student must attest a basic knowledge of all the topics covered in the course. In order to achieve the maximum score of 30/30, the student must demonstrate to know all the topics of the course in an excellent way and must be able to apply them to the solution of problems. The score 30/30 cum laude is reserved for students who show both a deep understanding of the course and an elegant exposition.
An alternative grading mode is available to students once a year. It consists of two written tests. The first one concerns the topics studied in the first 30 lecturing hours and takes place during the midterm interruption of the lectures. The second one concerns the remaining topics, but requires as well a global understanding of the program of the course, and coincides with the first official grading date. To pass the exam a student must reach a grade of at least 18/30 in both tests. Also, in this case, the student and the teacher may ask for a final oral interview.
Students have the opportunity to be aware of their level of preparation during the practical lectures carried out by the teacher or by the tutor. On these occasions, they can test their skills to solve exercises and grading tests, by comparing their results with those presented by the teacher and the tutor.
Due to the COVID-19 emergency, in the academic year 2021-2022, the exams will be in person or online according to the indications given by the University.
G. Rodriguez and S. Seatzu.
Introduzione alla Matematica Applicata e Computazionale.
Pitagora Editrice, Bologna, 2017.
The main tools to support teaching is the teacher's personal web site. It provides information updated in real time, including: lectures diary reporting the topics treated in each lecture, information on teaching activities, additional documents to support learning, grading tests, links to the tutor's web site, which contains solutions of exercises and of grading tests.