32087 - Complements Of Mathematical Methods Of Physics

Academic Year 2016/2017

  • Moduli: Francesco Ravanini (Modulo 1) Fiorenzo Bastianelli (Modulo 2) Roberto Zucchini (Modulo 3 )
  • Teaching Mode: Traditional lectures (Modulo 1) Traditional lectures (Modulo 2) Traditional lectures (Modulo 3 )
  • Campus: Bologna
  • Corso: First cycle degree programme (L) in Physics (cod. 8007)

Learning outcomes

At the end of the course the student learns some complementary mathematical knowledge and methods useful for an understanding of modern theoretical physics. He/she is able to use unlimited operators in a Hilbert space and deal with problems with generalized functions.

Course contents

Part 1 - Ordinary Linear Differential Equations (Prof. F. Ravanini)

  • Homogeneous linear equations with variable coefficients
  • Wronskian determinant and linear independence of solutions
  • Regular points, singular points of Fuchsian type or not
  • Frobenius recursive method of solutions
  • Hypergeometric equation and its solutions
  • Hypergeometric functions and their properties
  • Confluent hypergeometric eqaution
  • Elements of Monodromy theory and integral representations of solutions
  • Elements of non-homogeneous linear equations

Part 2 - Distribution theory (Prof. R. Zucchini)

  • Test functions and distributions
  • Derivative of a distribution
  • Operations with distributions
  • Tempered distributions
  • Fourier transform of a tempered distribution
  • Examples and applications

Part 3 - Partial differential equations and Green functions (Prof. F. Bastianelli)

  • Partial differential equations
  • Green functions
  • Applications to Poisson, heath and wave equations

Readings/Bibliography

Part 1 - C. Rossetti - Metodi matematici della Fisica (Ed. Levrotto & Bella, Torino): chapters 8 and 9

Part 2 - Teacher's notes and

  • G. Cicogna, Metodi Matematici della Fisica, Springer Italia 2008
  • L. Schwartz, Mathematics for the Physical Sciences, Dover 2008

Part 3 - Teacher's notes and

  • P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Cambridge University Press

 

Teaching methods

Lectures at the balckboard, sometimes with the help of projected slides.

Examples and exercises worked out during the lectures with the direct help of the teacher.

Further exercises proposed, useful for the preparation to the exam.

Assessment methods

The exam consists essentially in a written proof, with an optional oral colloqium in case the written exam mark is not satisfactory. To participate, the student must register on the Alma Esami site, in correspondence of "Appello a.a. 20xx/xx n. y - prova scritta".

Written exam

  • 3 hours
  • 3 problems, one for each part of the course
  • 3 theory questions, one for each part of the course
  • no test, electronic devices or other help is allowed. Possible "useful formulae" might be provided in the text of the problems.
  • The written exam is considered passed if a total of 18/30 is gained. This result will be available, as soon as the correction is done, on the Alma Esami site.
  •  The written exam can be repeated any number of times. However, the mark retained is the newer one, even if it is lower than the previous one.
  • It is a choice of the student to retreat at any moment without consequences
  • the solution of the problems (not of the theory questions, that are present in the texts) will be available soon after the correction.

As a consequence of the result of the written exam, in case the student has not gained at least 18/30, he/she will see on the Alma Esami site the wording "Respinto". He/she can retry the written exam at one of the subsequent dates.

For those who have passed the written exam, also in a previous date, an oral session is scheduled few days later than each written one. The student who passed the written exam has two options:

  • accept the mark of the written exam, that becomes therefore the final mark registered. To do that, the student has to register on Alma Esami to the oral session (see below) and declare his/her will to accept the mark via email to the teachers. The registration is completed officially on the date of the oral session.
  • if not satisfied with the mark of the written exam, the student can ask to undergo an oral examination. According to the performance, his/her mark will be modified up to a maximum of plus or minus 5 points w.r.t. the written exam mark. In case this results in a mark less than 18/30, the exam is registered as "Respinto" = not passed.

In both cases, it is necessary for the student to register on Alma Esami in correspondence of the wording "Appello a.a. 20xx/xx n. y - Prova orale - Registrazione".

Teaching tools

The lectures are of the traditioanl form at the blackboard and/or slide projectors.

Office hours

See the website of Francesco Ravanini

See the website of Fiorenzo Bastianelli

See the website of Roberto Zucchini