COURSE GOALS:
- acquire knowledge and understanding of the differential geometry (manifolds, differential forms, Lie derivative)
- understand the usage of differential geometry in physics
LEARNING OUTCOMES AT THE LEVEL OF THE PROGRAMME:
Upon completing the degree, students will be able to:
1. KNOWLEDGE AND UNDERSTANDING
1.1 formulate, discuss and explain the basic laws of physics including mechanics, electromagnetism and thermodynamics
1.2 demonstrate a thorough knowledge of advanced methods of theoretical physics including classical mechanics, classical electrodynamics, statistical physics and quantum physics
2. APPLYING KNOWLEDGE AND UNDERSTANDING
2.1 identify the essentials of a process/situation and set up a working model of the same or recognize and use the existing models
2.2 evaluate clearly the orders of magnitude in situations which are physically different, but show analogies, thus allowing the use of known solutions in new problems;
2.3 apply standard methods of mathematical physics, in particular mathematical analysis and linear algebra and corresponding numerical methods
4. COMMUNICATION SKILLS
4.3 develop the written and oral English language communication skills that are essential for pursuing a career in physics
5. LEARNING SKILLS
5.1 search for and use physical and other technical literature, as well as any other sources of information relevant to research work and technical project development (good knowledge of technical English is required)
5.2 remain informed of new developments and methods and provide professional advice on their possible range and applications
5.3 carry out research by undertaking a PhD
LEARNING OUTCOMES SPECIFIC FOR THE COURSE:
Upon passing the course on Differential Geometry in Physics, the student will be able to:
1. Understand the basic concepts in topology
2. Understand the basic properties of manifolds and their usage in physics
3. Understand and use the tensor calculus
4. Use differential forms in physical problems
5. Understand the geometrical meaning of Lie derivative and its usage in physics
6. Understand the integration on manifolds and the generalization of the Stokes' theorem in language of differential forms
7. Understand the usage of differential forms in Hamiltonian mechanics, mechanics of fluids, thermodynamics and classical electrodynamics
COURSE DESCRIPTION:
The Spring semester (15 weeks)
1st week: Introduction to topological spaces and metric spaces
2nd week: Manifolds
3rd week: Tangent vectors, vector fields, dual vectors (1-forms)
4th week: Tensors, coordinate transformations, metric tensor
5th week: Differential forms (definitions and basic operations)
6th week: Lie derivative, Killing vectors
7th week: Submanifolds, Frobenius' theorem and its usage
8th week: Integration on manifolds
9th week: Stokes' theorem in language of differential forms
10th week: Introduction to cohomology
11th week: Usage of differential forms in physics, examples from Hamiltonian mechanics, mechanics of fluids and thermodynamics
12th week: Usage of differential forms in physics, examples from classical electrodynamics
13th week: Introduction to fibre bundles
14th week: Fibre bundles in gauge theories
15th week: Student seminars
GRADING AND ASSESSING THE WORK OF STUDENTS:
During the semester each student gets several simpler problems which they have to solve and shortly present next week in front of the other colleagues. Students are encouraged to discuss these presentations and participate with additional questions. Also, each student gets one seminar topic which s/he has to present in a written and oral form at the end of the semester. The final grade is based upon the quality of this final seminar.
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