Introduction

Sections in this chapter:

TTC, what is it?
About this guide

For other chapters go to Index

TTC, what is it ?

TTC (Tools of Tensor Calculus)  is a Mathematica package for doing tensor and exterior calculus on differentiable manifolds
It can be of interest to the scientist working on any field where tensor calculus is needed. TTC can also be interesting for teaching purposes.
Among the capabilities of TTC can be mentioned:

There is no restriction about the number of coordinate systems (or manifolds...) defined simultaneously, their dimension or the type of the tensor fields.
TTC understands both intrinsic and index notation.
Tensors can be expanded using coordinate or noncoordinate basis.
Basis and coordinate changes.
Tensor and exterior calculus: tensor product, interior product, symmetrization and antisymmetrization, exterior product, Lie and exterior derivatives.
Riemannian manifold computations : volume form, Hodge or star operator, Christoffel symbols, Riemann and Ricci tensors, covariant derivative, etc.
Utilities to save and simplify objects in TTC notation.
TTC can handle explicit tensor calculus ( the tensors involved are known linear combinations of a known basis) and symbolic index tensor calculus ( the tensors involded are only known through its properties like symmetries of some index and others)

TTC works in the same way as Mathematica does: you enter an input and it gives an answer that can be used as a new input. In TTC each tensor is a single object expanded using some basis rather than a sample of components, and this is true even when the index notation is used. Therefore you may think that a tensor is a single expression like

TTC is provided free of charge and without guarantee of any kind. Everyone is allowed to copy and distribute copies of the software and the accompanying documentation.

Important

If you are running Mathematica 3 to 5 take the options in the Cell menu:

      Default Input FormatType = InputForm

      Default Output FormatType= OutputForm

For Mathematica 6 take the options in the Edit- Preferences menu -Evaluation tab:

       Default Input FormatType = RawInputForm

       Default Output FormatType= OutputForm

You can  load TTC  in a Mathematica session typing 

    In[1]:=<<path/ttc.m
 ----------------------------------------
 |TTC: Tools  of  Tensor  Calculus 4.3.1|
 |  A.Balfagon,P.Castellvi and X.Jaen   |
 |     http://baldufa.upc.edu/ttc       |
 |     e-mail:Xavier.Jaen@upc.edu       |
 |    version: june, 1, 2005            |
 ----------------------------------------
 |       TTC works correctly with       |
 |         Cell menu options:           |
 | Default Input FormatType=InputForm   |
 | Default Output FormatType=OutputForm |
 ----------------------------------------
 |     Session started on               |
 |     June,      5 , 2005              |
 |     at 17 h 50 min 6.981 s           |
 ----------------------------------------

 

 

 

About this guide

This guide assumes some familiarity with the concepts of differential geometry and tensor calculus at level of classical books:
Analysis, Manifolds and Physics
Manifolds, Tensor Analysis, and Applications.
Gravitation.
It Also assumes some familiarity with Mathematica
The examples in the guide are very simples in order to show the input syntax clearly.
IMPORTANT:All the examples in the guide asumes the TTC switch Compact[Off] , so if you want to mimetize the guide we sure that  Compact[]  is Off. If not enter Compact[Off] .
TTC allows both signs +1, -1 for the (riemannian) metric, so you can choose in every case what you want. On the other hand, in the definition of Christoffel symbols, Riemann tensors, covariant derivatives and so on, we have followed the conventions of Gravitation.

This page is maintained by XavierJaén.