Subject: MATHEMATICAL METHODS FOR PHYSICS (A.A. 2021/2022)
Unit Metodi matematici per la fisica
Theory and Foundations of Physics (lesson)
Knowledge and understanding
At the end of the course the studente should have acquired the basic knowledges of the mathematical tools that constitute the foundamental support of the modern theories of physics: the theory of analytic functions, linear vector spaces, the theory of linear operator, functional spaces, Hilbert spaces, Fourier series and transforms, and should be able to estimate their possible applications.
Applying knowledge and understanding
At the end of the course the student will develop the skill to apply the knowledges acquired to solve integral of complex variable, to diagonalize matrices in linear vector spaces, to deal with functional spaces, to evaluate Fourier series and transformes.
At the end of the course the student should be able to choose by himself the suitable mathematical approach to face the various problems arisen during the lessons.
At the end of the course the student should be able to describe with the appropriate technical language and the correct mathematical formalism the subject presented in the course.
At the end of the course the student should have developed the skill to deepen by himself the various topics treated, in a somehow limited way, during the lessons.
Elementary notions of mathematical analysis: integral and differential calculus. Elementary notions of matrices algebra.
1) Elements of vector analysis. Differential operators. Integration of vectors. Gauss, Green and Stokes theorems. Orthogonal curvilinear coordinates.
2) Complex Functions of complex variable. Analytic functions. Cauchy-Riemann conditions. Darbaux Inequality. Cauchy Theorem. Cauchy integral representation. Statement of Morera's theorem. Statement of Cauchy-Liuville Theorem. Taylor series. Laurent series. Classification of singularities. Method of residues. Jordan's lemma and its applications. Multivalued functions. Gamma and Beta functions.
3) Introduction to the theory of tempered distributions. Weak limit of divergent successions of functions. Representations of a distributions. The Dirac delta distribution and its properties.
4) linear finite-dimensional vector spaces. Scalar product. Dual spaces and inequalities of Cauchy-Swartz. Metric spaces. Linear operators. Right and left Inverse of an operator. Adjoint Operator. Projection operators. Linearly independent vectors. Orthonormalization process of Graham-Schmidt. Representation of a linear operator in a N-dimensional space. Change of basis vectors in a N-dimensional space. Covariant to contravariant components. Algebra of Matrices. Eigenvalue problem. Invariant subspaces. Characteristic equation and set out the Cayley-Hamilton theorem. Simultaneous diagonalization of Hermitian matrices.
5) space of continuous functions. Scalar product in functional spaces. The Lebesgue integral. Statement of the Riesz-Fisher Theorem. Square integrable functions. Bessel inequality. Formal development of Fourier series. Convergence on average. Closed systems and complete spaces. Hilbert spaces. Fourier transforms.
Lectures and numerical exercises carried out in class. Students receiving time: Friday 14.30-17 or by apointment
Written and oral examination. Two mid-term examinations will be done during the course. These examinations will substitute the written examination. The examination could be split into two parts in case of students with a certified handicap.
Knowledge and understanding:
Through lectures and course material possibly provided at the end of the course students will have basic knowledge of the theory of analytic functions, linear vector spaces and functional spaces.
Applying knowledge and understanding:
Through numerical exercises performed in the classroom at the end of the course the student will be able to apply this knowledge in physics problems that involves the content cited
Thanks to the variety of examples provided at the end of the course students will be able to recognize autonomously descriptive approaches and calculation methods appropriate to different types of problems of modern physics.
The discussions with the teacher and the final interview at the end of the course the student will be able to report orally on the arguments presented in the course with a rich technical language and a appropriate mathematical formalism.
The study, largely run on English texts, will allow the development of skills of independent learning and exploration
side of arguments to those presented in the course.
Arfken, Methematical Methods for physicists, Academic Press
Dennery and Krzywicki, Mathematics for physicists, Dover
Rossetti, Metodi matematici della fisica, Levrotto e Bella
Cicogna, Metodi matematici della Fisica, Springer
Barozzi, Matematica per l’ingegneria dell’informazione