Ateneo - UPO

Elements of Rational Mechanics. Harmonic analysis: Fourier series, Fourier transform and Laplace transform. Functional spaces and generalized functions. Applications to some classical differential equations of physics.

C. Bernardini, O. Ragnisco, P.M. Santini: Metodi Matematici della Fisica, La Nuova Italia Scientifica;

O. Luongo, S. Mancini: Introduzione ai Metodi matematici delle Scienze fisiche, MC Graw Hill 2022;

N. Zanghì Appunti di Metodi Matematici per la Fisica https://www.ge.infn.it/~zanghi/metodi/ZUL.pdf

M. Bramanti, C.D. Pagani, S. Salsa: Analisi matematica 2, Zanichelli

Acquire some mathematical methods frequently used in physics and integrate them with the skills developped in the previous courses in mathematics. Ability in text reading and understanding disentangling basic concepts and definitions from the follow up consequences, frequently organized as theorems. Ability in recognising hypothesis, thesis and proofs in theorems. Good command of the technical language both written and oral. Good command of the logic and deductive skills that allow for scientific reasoning, i.e. that allow, especially during problem solving, for explanations of and relations between the different themes and techniques presented in the course. Analytic skills in problem solving, by pointing out the key issues and by knowingly applying the acquired techniques.

The mathematics courses of the previous year

Blackboard teaching, theory and exercise sessions. The blackboard teaching complements the textbooks: topics are presented developping and relating different key concepts, and underplaying learning by heart. Methods and concepts given as prerequisites are thus naturally recalled and are contextualized. This brings to a view on the new concepts and methods that is integrated and organic with the previous notions (ex. complex numbers and vectors). We favour an experimental approach to problem solving, based on trial and error. Checks on the correctness of results are encouraged and created by considering trivial subcases of the problems at hand. Homework exercises are proposed and highly recommended.

TIME REQUIRED The amount of time required to successfully pass the exam is highly dependent upon the soundness of the mathematical knowledge acquired during the previous years. LEARNING CHECKS During every lesson the preceding one is recalled and summarized also through questions and sometimes the explanation of take home problems previously assigned. Students and teacher have therefore the possibility to assess the progresses in the learning process during the teaching of the course. See also the section: Modalita' di verifica dell'apprendimento.

Students with physical disabilities, Learning Disabilities or Special Education Needs can request specific services and tools via the "Staff Sviluppo e Coordinamento Carriere e Servizi alle Studentesse e agli Studenti", consulting the University webpage: https://www.uniupo.it/en/services/services-students-physical-or-learning-disabilities. Students with disabilities, learning disabilities or special education needs, once they have contacted the University Staff, can refer to the tutor in charge of the course to define the examination modalities, concerning academic aspects.

Written exam. Oral exam right after. There will be possible unexpected written exams during the course that can total up to 4/30 and sum up to the final written and oral exam that total up to 30/30. The written final exam lasts about two hours; a first part (6/30) of 6 questions (to be handed in after 30 minutes) is in order to check a minimum level of sudy and acquaintance with mathematical reasoning. Necessary condition to pass the exam is that of totalizing at least 4 out of the 6 points of this first part. The second part (24/30) is a set of 5-7 longer and more complex exercises. The oral part of the exam concerns the critical issues emerged in the written exam, and the logic analytic and deductive skills assessed by listening to the student presentation and the student ability to dialogue with the teacher.

Rational Mechanics. Lagrangian formalism. Symmetries and conservaiton laws. Hamiltonian formalism and canonical transformations.

Function spaces. Fourier series. Fourier and Laplace transform. Generalized functions and Dirac delta function. Fourier analysis of generalized functions.
Applications to partial differential equations.

Knowledge of the topics exposed in class and of the language used, including definitions and statements of the theorems. Integration of the topics of the course with the mathematics topics of the previous years. Ability to reproduce proofs exposed in class and to argue by deduction also in the resolution of exercises. Ability to use and consult at least one of the reference textbooks. Ability in understanding proposed exercises and in devising methods of solutions. Ability in performing independent checks of the correctness of the solutions obtained (ex. Fourier series of a function and its first approximating terms). See also the "Risultati di apprendimento attesi" described in the section "Obiettivi Formativi".