Ateneo - UPO

Partial differential equations, linear and non linear. Resolution methods and numerical methods. Power series, Series of functions and Fourier Series. Fourier and Laplace transform. Integration by several methods. Multiple integrals and parametric integrals. Fundamental basis of differential geometry, concepts as metric, curvature, geodesics and several applications. Some basic notions on distributions.

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

P. Olver, Introduction to Partial Differential Equations, Springer

Provide the knowledge on fundamental and basic mathematical methods and the analysis of the relevant properties; develop the student skills to a correct use of mathematical tools adequate to the physical problem to be analysed.

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. Excellence is reached with a written exam that has at most light calculations mistakes and misprints, and with an oral exams where questions are properly answered.

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 and understanding:
acquisition of the theoretical basis (theorems and definitions) of mathematical methods for applied sciences.
- Applying knowledge and understanding: Full ability to apply the computation techniques and analysis provided by the course. Developing needed software using already available automatic calculus
and implementation of new algorithms.
- Communicative Skills: Being able to provide either written or oral details of the computation and of the results for the problem. Communicative skill regarding the computation procedure by means of a detailed analysis of computations, either by manual techniques or with a computer.

- Learning skills: the student will have to acquire a certain mastery in the use of advanced mathematical tools to expand their knowledge
for advanced courses.