xTensorDoc.nb

8.1. MakeRule

Defining rules is one of the most important components of a tensor package. The function IndexRule takes care of dummy generation, but there are other points to consider when defining a rule:
    1. Checking consistency between left and right hand sides.
    2. Defining rules that work only for a given vbundle.
    3. Extending the rule to pattern expressions with equivalent index-structure under symmetries.
    4. Extendig the rule to patterns with a different up/down character if metrics are present.
We define the function MakeRule, which takes care of all these things. Note that there is not a parallel function for Set, but below we present a way to automatize a collection of rules.

MakeRule Construct rules between xTensor` expressions
AutomaticRules Automate rules

Construction of rules.

In[778]:=

Out[778]=

The simplest way to define a rule would easily give corrupted answers:

In[779]:=

Out[779]=

In[780]:=

Out[780]=

In[781]:=

Out[781]=

We have seen that changing to IndexRule we can solve those problems (underlined indices denote patterns):

In[782]:=

Out[782]=

$HoldPattern[v_ ^Underscript[a, _]] :→Module[{b}, T_ ^ab v_b^ ]$

In[783]:=

Out[783]=

In[784]:=

Out[784]=

We can also define the same rule using MakeRule. Note that there is no pattern on the lhs; by default all indices on the LHS are converted into patterns:

In[785]:=

Out[785]=

${HoldPattern[v_ ^Underscript[Underscript[a, _], _]] :→Module[{b}, T_ ( b)^a v_ ^b]}$

In[786]:=

Out[786]=

In[787]:=

Out[787]//InputForm=

{HoldPattern[v[(a_Symbol)?TangentM3`Q]] :> Module[{h$41203}, T[a, -h$41203]*v[h$41203]]}

Due to the option TestIndices→True, membership of the indices of the LHS is checked. The function M3`Q checks that an up-index belongs to M3, but it does not accept down-indices (this is due to the option MetricOn->None):

In[788]:=

Out[788]=

In[789]:=

Out[789]=

In[790]:=

Out[790]=

In order to allow both up/down-indices of a given vbundle we use MetricOn. We need to have a metric first. (If the vbundle of the index did not have a metric MakeRule would refuse to change the up/down character):

In[791]:=