METODI MATEMATICI PER LA FISICA
- Overview
- Assessment methods
- Learning objectives
- Contents
- Full programme
- Teaching methods
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Knowledge of calculus in one and more real variables. Basic elements of linear algebra and the theory of finite-dimensional vector spaces.
The final exam will consist of a written exam in which 4-5 exercises must be solved in 3 hours, followed by an oral exam in which the level of learning of the theory and the ability to connect the different conceptual aspects to practical ones and to physical applications will be investigated. A positive result in the written exam is a prerequisite for the oral exam.
In the written exam, within each exercise it may also be required to provide some notions of theory or some theoretical demonstration. Each exercise will consist of several questions. In the written exam, each exercise will be assigned a score (expressed in thirtieths) and explicitly indicated on the exam topic. The sum of the points assigned to each exercise will be equal to 30/30. For the purposes of evaluating the written exam, the following criteria will be considered, in order of priority: 1) the correctness and explanation of the procedures used to solve the problems; 2) the reliability of the results obtained; 3) the correctness of the calculations and of the final result in carrying them out; 4) the correct use of technical terminology. The written test will be considered positive with a grade greater than or equal to 16/30. During the oral exam, the student will be questioned on the program. First, he or she will be asked to comment on any errors made in the written test. Then the student will be asked 4 or 5 questions related to the content of the course. In particular, some demonstrations and the completion of some calculations will be asked. The final grade will be assigned by making a weighted average of the grades obtained in the written test and in the oral test. Please note that the weight assigned to the written test is 1/3, while that given to the oral test is 2/3. The exam will be considered passed if the outcome of this weighted average is at least 18/30.
To obtain honors, the student must produce a perfect written test and be able, during the oral exam, to answer questions with a high level of difficulty and to perform non-trivial calculations.
The central objective of the course is the introduction to complex analysis and functional analysis.
At the end of the course, the student is expected to be able to handle computational tools based on complex analysis and functional analysis that are frequently used in applied mathematics and physics.
Complex numbers. Holomorphic functions. Notion of a path in a metric space. Cauchy's theorem; Integrals of the Cauchy type, and Cauchy's integral formula. Remarkable series. Isolated singularities; Laurent expansion, classification of isolated singularities. Residue theorem, and its applications. Fundamental theorem on Analytic Extension; Ordinary, linear, 2nd order differential equations. Elementary theory of Hilbert spaces, measure theory and Lebesgue integral, the Fourier transform, introduction to the theory of tempered distributions, differential operators and Green's functions.
Review of basic notions on complex numbers. Definition of elementary transcendental functions of a complex variable. Holomorphic functions: Cauchy-Riemann conditions, and conformal representations. Rules of differentiation. Inverse functions, roots, and logarithms. The extended complex plane, and the Riemann sphere. Notion of a path in a metric space. Regular paths in the complex plane. The path integral for functions of n real variables, and for functions of a complex variable. Conservative fields. Holomorphic functions, as vector fields. Cauchy's theorem; Integrals of the Cauchy type, and Cauchy's integral formula. Harmonic functions. Principle of maximum modulus. Liouville's theorem. Power series; analytic functions. Remarkable series. Isolated singularities; Laurent expansion, classification of isolated singularities. Residue theorem, and its applications. Fundamental theorem on Analytic Extension; analytic extension along a path; monodromy, and polydromy; complete analytic functions, and Riemann surface. Integrals of polydromic functions. The Gamma function. Ordinary, linear, 2nd order differential equations. Theorem of existence and local uniqueness; analytic extension of solutions; structure of the space of solutions. Singular points. Euler equation. Fundamental solutions. Regular singularities, and behavior of solutions in their neighborhood. Bessel equation and functions. Equations of the Fuchs class; Gauss equation; Hypergeometric function; confluent hypergeometric equation.
For the functional analysis part:
- Linear algebra; - Measure theory; - Hilbert spaces; - Fourier transform; - Distribution theory.
The course consists of 88 hours of lectures and 20 hours of exercises. In the latter, complementary topics will be developed, and exercises and problems will be solved in preparation for the exam.
Professors
Borrowers
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Degree course in: MATHEMATICS
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Degree course in: MATHEMATICS