xAct: tensor analysis by computer 2 - NTMF076

• Title: xAct: tensor analysis by computer 2
• Guaranteed by: Institute of Theoretical Physics (32-UTF)
• Faculty: Faculty of Mathematics and Physics
• Actual: from 2020
• Semester: summer
• E-Credits: 3
• Hours per week, examination: summer s.:2/0, Ex [HT]
• Capacity: unlimited
• Min. number of students: unlimited
• 4EU+: no
• Virtual mobility / capacity: no
• State of the course: not taught
• Language: English
• Teaching methods: full-time
• Teaching methods: full-time
• Additional information: http://xact.es/xActCourse_Prague/
• Guarantor: prof. RNDr. Pavel Krtouš, Ph.D.

Annotation

English

Last update: doc. RNDr. Karel Houfek, Ph.D. (11.05.2018)

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: doc. RNDr. Karel Houfek, Ph.D. (11.05.2018)

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. (11.06.2019)

Ústní zkouška

Literature

English

Last update: prof. RNDr. Pavel Krtouš, Ph.D. (17.11.2018)

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

A. García-Parrado: A course about the xAct system.
http://xact.es/xActCourse_Prague/

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

English

Last update: doc. RNDr. Karel Houfek, Ph.D. (11.05.2018)

The examination has an oral form.

Czech

Last update: doc. RNDr. Karel Houfek, Ph.D. (11.06.2019)

Zkouška je ústní, požadavky odpovídají sylabu, v detailech pak tomu, co bylo během semestru odpřednášeno.

Syllabus

English

Last update: prof. RNDr. Pavel Krtouš, Ph.D. (17.11.2018)

The xAct system: efficient tensor computer algebra for the Wolfram Language.

0. Introduction to the Wolfram Language.

1. xTensor: coordinate-free tensor analysis.

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

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

1.3. Canonicalization of expressions with covariant derivatives.

1.4. Pattern indices. Implementation of general tensorial rules. Constant symbols, inert heads, parameters and scalar functions.

1.5. Lie brackets and vector contraction of tensor slots.

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

2. xCoba: tensor analysis in coordinates.

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

2.3. The containers CTensor and CCovD and their converters. The xCoba cache system.

2.4. Application: curvature computations with xCoba.

3. xTerior: exterior calculus in the Wolfram Language and its applications.

3.1. The exterior algebra. Differential forms. Changes of coordinates. Basic operations with differential forms: the exterior derivative, the inner contraction and the Lie derivative.

3.2. Cartan structure equations.

3.3. Hodge duality. The co-differential. The Hodge Laplacian.

3.4. Tensor valued differential forms. The exterior covariant derivative.

3.5. Application: formulation of the Einstein's equations with differential forms.