70/0069-M - APPLIED MATHEMATICS
Academic Year 2016/2017
Free text for the University
LUISA FERMO (Tit.)
- Teaching style
- Lingua Insegnamento
|[70/77] CHEMICAL ENGINEERING||[77/00 - Ord. 2015] PERCORSO COMUNE||6||60|
|[70/78] MECHANICAL ENGINEERING||[78/00 - Ord. 2011] PERCORSO COMUNE||6||60|
1. Acquiring knowledge and understanding.
The course is devoted to students in the second year of the Bachelor's degree of Biomedical, Chemical, Electrical and Electronic, and Mechanical Engineering. It aims to provide a working knowledge of the fundamental methodologies of the 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 interested 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.
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 a basic expertise which is sufficient to understand advances 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 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 “Matematica 1” and “Fisica 1”.
1. Numerical linear algebra (20 hours)
Eigenvalues, eigenvectors. Characteristic equation and spectral radius. Hermitian matrices, positive definite and semidefinite matrices. Eigenvectors linearly independent and canonical forms. Vector norms and inner products. Norms with index 1, 2, and infinity. 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 and Green's functions. 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.
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. Simultaneoulsy with the course, a tutorial activity is furnished to students, to assist them while they study for the final grading. Teachers give constant assistance to students, during the whole year, both by personal interviews and by means of e-mail messages.
Verification of learning
Grading normally consists of a written test. It includes some exercises which require both operative skill and knowledge of the theory concering the whole programme. The student must demonstrate to know and have understood the algorithms described during the course and must be able to apply them to the solution of the exercises. 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 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.
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 mid term 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.
G. Rodriguez and S. Seatzu.
Introduzione alla Matematica Applicata e Computazionale.
Pitagora Editrice, Bologna, 2010.
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 tutors' web sites which contain solution of exercises and of grading tests.