1. SylabUZ
2. Faculty of Physics and Astronomy
3. winter term 2019/2020
4. Physics - Second-cycle studies leading to MS degree
5. Mathematical methods in physics

Generate PDF for this page

Mathematical methods in physics - course description

Aim of the course

To teach the students basic mathematical tools of differential geometry and tensor analysis
necessary to study general relativity.

Prerequisites

Save changes

Mathematical analysis I and II, and algebraic and geometric methods in physics.

Scope

- Elements of multivariable functions analysis: functions from R^n to R^m, continuity, limits,
differentiability, Jacobi matrix of transformation, inverse and implicit function theorems.
- Elements of differential geometry: Cartesian and curvilinear coordinate systems, in R^n and in a
domain of R^n, Curves in Euclidean space, length of curve, Riemannian metrics, natural
parametrisation of curve, curvature and torsions, Serret-Frenet formulae, surfaces in R^3, first and
second fundamental form of surfaces, mean and Gauss curvatures, hypersurfaces immersed in
higher-dimensional flat spaces, notion of differential manifold, coordinates on differential manifold,
tangent and cotangent spaces.
- Elements of tensor algebra. Space dual to a vector space, multilinear mapping, transformation
laws for tensor and tensor fields, algebraic operations on tensors, differential forms as skew-
symmetric tensors, examples of applications of tensors in physics.
- Elements of tensor analysis: affine connection, covariant derivative, Christoffel symbols, torsion,
Riemannian connection, parallel displacement, equation of parallel displacement, geodesics,
curvature tensor, Euclidean coordinate, properties of the Riemann curvature tensor, curvature
scalar.

Teaching methods

Conventional lecture with emphasis on contents useful for studies of general relativity
During class students solve exercises illustrating the content of the lecture with examples related to
general relativity

Learning outcomes and methods of theirs verification

Outcome description	Outcome symbols	Methods of verification	The class form

Assignment conditions

Lecture:
The course credit is obtained by passing a final written exam composed of tasks of varying degrees
of difficulty.
Class:
A student is required to obtain at least the lowest passing grade from tests organized during class.
To be admitted to the exam a student must receive a credit for the class
Final grade: average of grades from the class and the exam.

Recommended reading

[1] L. M. Sokołowski, Elementy analizy tensorowej, Wydawnictwo Uniwersytetu Warszawskiego, 2010.

[2] M. Spivak, Analiza na rozmaitościach, Wydawnictwo Naukowe PWN, Warszawa 2006.

[3] A. Goetz i inni, Zewnętrzne formy różniczkowe, WNT, Warszawa 1965.

[4] S. Lovett, Differential geometry of Manifolds, A K Peters, Ltd, Natick, Massachusetts 2010.

[5] A. S. Mishchenko, A. Fomenko, A course of Differential Geometry and Topology, Mir Publishers Moscow 1988.

[6] B. A. Dubrovin, A.T. Fomenko, S.P. Novikov, Modern Geometry – Methods and Applications,

Springer 1992.

[7] A. S. Mishchenko, Yu. P. Solovyev,, A.T. Fomenko, Problems in Differential Geometry and

Topology, Mir Publishers, Moscow 1985.

Further reading

[1] P. M. Gadea, J. Munoz Masque, Analysis and Algebra on Differentiable Manifolds, Springer, 2009.
[2] T. Banchoff, S. Lovett, Differential Geometry of Curves and Surfaces, A K Peters, Ltd, Natick,
Massachusetts 2010.
[3] S. Chandrasekhar, The Mathematical Theory of Black Holes, Clarendon Press, Oxford 1983.
[4] E. Karaśkiewicz, Zarys teorii wektorów i tensorów, Państwowe Wydawnictwo Naukowe, Warszawa
1964.

Notes

Modified by dr hab. Piotr Lubiński, prof. UZ (last modification: 05-03-2020 13:38)