# ADVANCED MATHEMATICAL METHODS IN PHYSICS

**PHYSICS 9012 (coorte 2019/2020)**- LAB OF BIOPHYSICS 62739
- LAB OF FUNDAMENTAL INTERACTIONS PHYSICS AND ASTROPHYSICS 61868
- THEORY OF FUNDAMENTAL INTERACTIONS 62422
- GENERAL RELATIVITY (6 CFU) 61875
- PHYSICS OF ASTROPARTICLES 61873
- FIELDS THEORY 61876
- STATISTICAL PHYSICS 61867
- LAB OF ANDVANCED THERMODYNAMICS 62424
- GROUP THEORY 63662
- FOUNDATIONS OF ASTROPHYSICS AND COSMOLOGY 61874
- APPLIED ELECTRONICS 68873
- MESOSCOPIC SYSTEMS AND NANOSTRUCTURES 66800
- NANOSTRUCTURES 62744
- MATERIALS AND DEVICES FOR ELECTRONICS 62421
- PHYSICS OF ELEMENTARY PARTICLES 61872
- LABORATORIO DI FISICA DELLA MATERIA (6 CFU) 61862
- SOLID STATE PHYSICS 61861
- APPLIED NUCLEAR PHYSICS 61871
- PHYSICS OF THE OCEAN 68875
- SOFT MATTER PHYSICS 61863

**PHYSICS 9012 (coorte 2020/2021)**- PHYSICS OF THE OCEAN 68875

OVERVIEW

Advanced mathematical methods of physics (code 61843) has credit value 6 and it is taught in the first semester of the first or second year of the LM.

Lectures are usually given in Italian. Should it be requested, they can be given in english.

## AIMS AND CONTENT

LEARNING OUTCOMES

Boundary and/or initial values problems for second order, linear, partial differential operators of Mathematical Physics (wave operator, diffusion operator, Laplace and Helmholtz operators.

AIMS AND LEARNING OUTCOMES

The course is divided in two parts:

the first one is about the theory of distributions and Green's functions.

The second one is on the applications of calculus of variations to classical field theory.

PREREQUISITES

Mathematical methods of Physics.

Teaching methods

Teaching is done in the traditional (chalk and blackboard) way. Students are (very) strongly suggested to attend the class.

SYLLABUS/CONTENT

Introduction to the theory of distributions.

Examples: Dirac's Delta function and its derivatives.

Green's formulae.

Dirichlet and Neumann problem.

The wave equation.

Vibrations of a circular membrane.

Wave equation in 3 dimensions.

Kirchoff's formula.

Functionals. Variation of a functional. Simple examples.

Euler-Lagrange equations.

Symmetries and Noether's theorem.

The vibrating string.

Klein-Gordon Lagrangian. Retarded, advanced and Feynmann Green's functions.

Gross-Pitaevskii Lagrangian. Gross-Pitaevskii equations.

U(1) symmetry and conserved current.

RECOMMENDED READING/BIBLIOGRAPHY

Suggested textbooks

Duffy, Green's functions.

Gelfand and Fomin, Calculus of variations.

Butkov, Mathematical physics.

## TEACHERS AND EXAM BOARD

**Ricevimento:** The teacher meets with students everyday from 14,00 to 16,00 (please contact the teacher by mail).

Exam Board

NICODEMO MAGNOLI (President)

ANDREA AMORETTI

PIERANTONIO ZANGHI' (President Substitute)

## LESSONS

Teaching methods

Teaching is done in the traditional (chalk and blackboard) way. Students are (very) strongly suggested to attend the class.

LESSONS START

First semester, usually last week of September

## EXAMS

Exam description

Compulsory written examination.Optional oral examination.

Assessment methods

Methodology of ranking

The explicit aim of the course is to carry the students to the level of making calculations and solving problems. This is why the essential part of the exam is the written one, where the student is asked to make calculations and to solve explicitly problems.

The text is divided into three levels, each level corresponds to a range of marks. (These ranges of marks are clearly indicated in the text given to each student). The first range is for marks up to 24, the second for marks up to 28, the third for marks over 28.

The first corresponds to an easy exercise that do not contain either conceptual or computational difficulties. The second corresponds to an exercise without difficulties, but containing some challenging calculation (as an example, a non elementary integral). Marks over 28 correspond to an exercise whose solution requires some skill, and it is not the immediate application of something seen during the course. In particular it contains a “difficult” part or question that is required to get the “LAUDE”.

It is my firm belief, that comes from many years of teaching, that the oral part of the exam can just be a small correction to the written part. It has to be stressed that this correction is not necessarily positive. This is why the student is allowed to retain the mark of the written part as the final one.