Data on the course

Show instruction and examinations
TCM305 Mathematical Methods of Physics IIIb, 5 cr 
Code TCM305  Validity 01.01.2017 -
Name Mathematical Methods of Physics IIIb  Abbreviation Mathematical Me 
Scope5 cr   
TypeAdvanced studies
  GradingGeneral scale 
    Can be taken more than onceno
Unit Master's Programme in Theoretical and Computational Methods 

Esko Keski-Vakkuri 

Target group 

Master’s Programme in Theoretical and Computational Methods is responsible for the course.

Modules where the course belong to:

  • TCM300 Advanced Studies in Theoretical and Computational Methods
  • PAP300 Advanced Studies in Particle Physics and Astrophysical Sciences
    Optional for:
    1. Study Track in Particle Physics and Cosmology

The course is available to students from other degree programmes.


The course is offered in the autumn term, during II period.

Learning outcomes 

After the course, the student will be familiar with basic concepts of calculus on differentiable manifolds and Riemannian geometry, which are mathematical tools used in physics e.g. in the contexts of general relativity and gauge field theories. The student will also be familiar with basics of Lie algebra representation theory, which is used e.g. in particle physics and condensed matter theory. The student can work with differential forms, express metrics in different coordinates and compute metric tensors of general relativity. He also understands basic representations of Lie algebras used e.g. in the theory of strong interactions.

Completion methods 

The course is lectured as contact teaching, but it is also possible to pass the course by studying it independently (e.g. by a final exam), if so agreed with the lecturer.  In the course form, the completion is based on  a final exam and weekly homework performance.


The student should be familiar with and understand the concepts of the course Mathematical Methods of Physics IIIa. In addition it is recommended to be familiar with linear algebra, differential and integral calculus, and (partial) differential equations. It is also helpful to know basic physics such as classical mechanics, electrodynamics, some quantum mechanics, and theory of special relativity.


Differentiable manifolds and calculus on manifolds: differentiable manifolds, manifolds with boundary, differentiable maps, vector fields, 1-form fields, tensor fields, differentiable map and pullback, flow generated by a vector field, Lie derivative, differential forms, Stokes' theorem

Riemannian geometry: metric tensor, induced metric, connections, parallel transport, geodesics, curvature and torsion, covariant derivative, isometries, Killing vector fields

Semisimple Lie algebras and representation theory: SU(2), roots and weights, SU(3), introduction to their most common unitary irreducible representations

Study materials and literature 

The course follows "Mathematical methods of physics III", lecture notes by E. Keski-Vakkuri, C. Montonen and M. Panero. Supplementary reading is listed in the lecture notes. The students are encouraged to actively search for additional supplementary material from the Web (e.g., from Wikipedia and other such pages.)

Activities and teaching methods in support of learning 

Weekly contact lectures, independent work of the student including solving weekly homework problem sets. The solutions to the problem sets will be submitted weekly, graded by the teaching assistant and discussed in weekly exercise sessions. In these sessions the students may also discuss and get tutoring for next week's homework.

Assessment practices and criteria 

The grade is determined in a way agreed in the beginning of the course.


Current and future instruction
No instruction in WebOodi

Future examinations
Functions Name Type cr Teacher Schedule
Registration Mathematical Methods of Physics IIIb  General Examination  Esko Keski-Vakkuri 
14.08.20fri 10.00-14.00
You may enter WebOodi: