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Course, academic year 2023/2024
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xAct: tensor analysis by computer 1 - NTMF075
Title: xAct: tensor analysis by computer 1
Guaranteed by: Institute of Theoretical Physics (32-UTF)
Faculty: Faculty of Mathematics and Physics
Actual: from 2023
Semester: winter
E-Credits: 3
Hours per week, examination: winter s.:2/0, Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: English
Teaching methods: full-time
Teaching methods: full-time
Guarantor: Mgr. David Kofroň, Ph.D.
prof. RNDr. Pavel Krtouš, Ph.D.
Annotation -
It will be explained how tensor analysis can be carried out efficiently within Mathematica/Wolfram Language using xAct system. The applications are mostly tailored for Theoretical Physics and General Relativity but other applications to mechanics of continuous media are also possible.
Last update: Podolský Jiří, prof. RNDr., CSc., DSc. (21.12.2017)
Course completion requirements

Assignments made by the student during the course.

Last update: Houfek Karel, doc. RNDr., Ph.D. (12.05.2023)
Literature -

A. García-Parrado, J.M.Martín-García: Spinors: A Mathematica package for doing spinor calculus in General Relativity.

R. Maeder: Computer Science with Mathematica, Cambridge University Press, Cambridge (2000).

J. M. Martín-García: xAct: Efficient tensor computer algebra for the Wolfram Language

J.M.Martín-García: xPerm: fast index canonicalization for tensor computer algebra.

Last update: Podolský Jiří, prof. RNDr., CSc., DSc. (21.12.2017)
Requirements to the exam

Assignments made by the student during the course.

Last update: Parrado Gómez-Lobo Alfonso Garcia, Ph.D. (22.10.2019)
Syllabus -

Lecture 1. General description of xAct and some selected examples.

Lecture 2. Introduction to the Wolfram Language.

Lecture 3. xTensor and its data types: working with tensors and covariant derivatives. Canonicalization of tensorial expressions.

Lecture 4. Working with a single and multiple metric tensors. Canonicalization of expressions with a metric tensor.

Lecture 5. Canonicalization of expressions with covariant derivatives. Pattern indices.

Lecture 6. Implementation of general tensorial rules.

Lecture 7. Constant symbols, inert heads, parameters and scalar functions. Lie brackets and vector contraction of tensor slots.

Lecture 8. The variational derivative. Working examples with the Einstein-Hilbert action (Palatini formalism), f(R) theory and Lovelock gravity.

Lecture 9. The 1+3 decomposition. ADM formalism.

Lecture 10. Main differential identities of a Killing vector. The Mars-Simon tensor in vacuum.

Lecture 11. The conformal equations.

Lecture 12. Component computations with xCoba. Storage of components: the tensor values framework and the CTensor container.

Lecture 13. The containers CTensor and CCovD and their converters. The xCoba cache system.

Lecture 14. Curvature computations with xCoba.


for additional course details.

Some practical details about the course (schedule, etc) will be supplied during the meeting with students (Setkání se studenty) on Tuesday 8th October at 10:40.


Last update: Parrado Gómez-Lobo Alfonso Garcia, Ph.D. (02.10.2019)
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