# MATHEMATICAL METHODS FOR PHYSICS (MOD. 2)

## Learning outcomes of the course unit

Students are supposed to get to master the mathematical skills which are essential

for the study of the most advanced subjects in Physics, in particular Quantum

Physics. They should understand the power of mathematical tools for tackling a

variety of problems in different fields: what they learn should put them in a position

to solve problems in many different circumstances. Students will be involved in

solving problems in front of their colleagues during the lessons; they will be also

asked to present solutions to problems they will be assigned for being worked out at

home. All this is intended as a training of their communication skills (they should be

able to argue in public).

## Prerequisites

Basic notions of real analysis, calculus, geometry and algebra

## Course contents summary

The course aims at providing a final framework of the mathematical skills which are

fundamental for a physicist. We want to combine mathematical rigor and fluency in

acquiring new tools, with the final goal of ability in problem solving.

First goal is to complete a basic preparation in classical real and complex analysis,

with the theory of analytic functions (residues, power series, integration in the

complex plane).

Main part of the course is devoted to the theory of linear operators in finite

dimensional spaces (aiming at a sufficiently rigorous knowledge of the spectral

theory), with the due insight in algebra and metric topology.

## Recommended readings

There are many excellent books on the subjects covered. A (partial) list includes

V. Smirnov, Corso di Matematica superiore, vol.III,2 (MIR)

E. Onofri, Teoria degli Operatori lineari,

http://www.fis.unipr.it/home/enrico.onofri/#Lezioni

F.G.Tricomi, Metodi Matematici della Fisica (Cedam)

M.Spiegel, Variabili Complesse (Schaum, Etas)

E.Kolmogorov, S.Fomin, Elementi di teoria delle funzioni e dell'analisi funzionale

(ER)

## Teaching methods

Lessons and exercises in the classroom (with students involved in working out the

solutions). Exercises will be assigned for being worked out at home.

## Assessment methods and criteria

Final written and oral tests.

The written test consists in exercises aiming to check the skill in calculus, trating

problems which are variations of exercises already developed in the lessons.

The oral test consists in the discussion of some typical subjects, showing the

methodological and conceptual mastership of the tudent on the fundamental topics.