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

English

Last update: prof. RNDr. Jiří Podolský, CSc., DSc. (21.12.2017)

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.

Czech

Last update: prof. RNDr. Jiří Podolský, CSc., DSc. (21.12.2017)

V přednášce bude ukázáno, jak mohou být tenzorové výpočty efektivně prováděny v systému Mathematica s použitím balíku xAct. Aplikace budou zejména z obecné relativity, případně z teorie kontinua.

Course completion requirements

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

Assignments made by the student during the course.

Literature

English

Last update: prof. RNDr. Jiří Podolský, CSc., DSc. (21.12.2017)

A. García-Parrado, J.M.Martín-García: Spinors: A Mathematica package for doing spinor calculus in General Relativity. https://doi.org/10.1016/j.cpc.2012.04.024

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

J. M. Martín-García: xAct: Efficient tensor computer algebra for the Wolfram Language http://www.xact.es

J.M.Martín-García: xPerm: fast index canonicalization for tensor computer algebra. https://doi.org/10.1016/j.cpc.2008.05.009

Requirements to the exam

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

Assignments made by the student during the course.

Syllabus

English

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

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.

See

http://www.xact.es/xActCourse_Prague/

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.

See

http://utf.mff.cuni.cz/seminare/semmf.pl