7.1. FoldedRule and ValID
FoldedRule List of lists of rules to be applied sequentially
ValID Identifier for each independent set of components stored for a tensor
TensorValIDs List of all the ValIDs of a given tensor
DateOfValID Time a given ValID was generated
Identifying sets of component values
Before introducing the general functions, let us consider a simple example. Let us suppose we have a rank two tensor 
, with no symmetries and only one basis. Then, the straightforward way of storing its components would be with a list of replacement rules:
{
→ a, 
→ b, 
→ c, ...}
But now consider, still with just one basis in play and no metric, that our tensor 
is symmetric. Then it would be a waste to store twice the component 
=
, for example. With higher rank tensors and higher symmetries the redundancy makes the process very inefficient. Instead, we could think of storing components as a pair of lists. The first one would give the relations due to symmetry and the second one the values of the independent components. We could have something like
{ {
→ 
, 
→ 
, 
→ 
}, { 
→ a, 
→ b, 
→ c, 
→ d, 
→ e, 
→ f} }
It may appear we have gained nothing, after all we still have 9 rules. However, all the information of the first list is determined by the symmetry group of the tensor, which xCoba` already knows since its definition. Also, the independent values may be very large expressions while the dependent rules are always very simple. This structure is represented in xCoba` with the concept of FoldedRule (defined in the auxiliary package xCore`):
In[311]:=
![FoldedRule[rules1, rules2, ...] contains a number of lists of rules which are applied sequentially (first rules1, then rules2, etc.) when called by ReplaceAll and ReplaceRepeated.](../HTMLFiles/xCobaDoc.nb_856.gif)
Example:
In[312]:=
![x + t/.FoldedRule[{x→y, t→x}, {y→z}]](../HTMLFiles/xCobaDoc.nb_857.gif)
Out[312]=
In the next subsection we shall see how xCoba` generates these lists. The next problem comes when we consider a more general case where we have a metric and several bases. In this situation we may want to store the values for a given tensor in different bases and with different index characters. However, not all those combinations are independent. Recalling our symmetric tensor 
, we see that the FoldedRules that would contain its values for 
and 
would have the same independent components (likewise for 
and 
). So among all sets of values for a tensor, only some will be independent. xCoba` identifies these independent sets with the concept of ValID:
In[313]:=
Continuing with our example, we would have for our tensor 
a single ValID, which would be
ValID[U, { {polar, polar}} ]
(both indices in the base polar). We have several for 
ValID[T, { {polar, -polar} , {-polar, polar}} ]
ValID[T, { {polar, cartesian}, {cartesian, polar}]
The first would represent the fact that 
and 
have the same independent rules and the second would represent the situation for 
and 
. Notice how even in the case of 
, where there is only one possible basis configuration in the ValID, we give a list of lists of bases. The command VTensorValIDs returns all the ValIDs we have stored for a given tensors (and its derivatives, as we shall see below).
All of this will be much clearer once we see how xCoba` generates the FoldedRules and their corresponding ValIDs for a tensor in the next subsection.
