xCobaDoc.nb_16.html

4.1. The η tensors and the Jacobian

EtaUp                  Define a basis
EtaDown                VBundle on which a basis lives
Jacobian            Parallel derivative associated to the given basis
WeightOf            Weight of an expression as a linear combination of basis names

Definition of a basis.

The η tensors represent the totally antisymmetric products of all basis (co)vectors:
$~ RowBox[{Cell[TextData[Cell[BoxData[η]]]], ^, (a_1··· a_n)}]$ = n! ··· , = n! ···
In xCoba`, these objects are automatically defined for each basis.

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${Overscript[η, ~] _ ^abc, Underscript[η, ~] _abc^ }$

The basis is marked with an over or undertilde, whose significance we shall see below. Their components in the natural basis are very simple:

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{etaUppolar[{1, polar}, {2, polar}, {3, polar}], etaUppolar[{1, polar}, {3, polar}, {2, polar}], etaUppolar[{3, polar}, {1, polar}, {2, polar}], etaUppolar[{1, polar}, {1, polar}, {1, polar}]}

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And they are related to the generalised Kronecker delta:

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The presence of a metric provides additional relations, as we shall see later. Each basis has a different pair of η tensors but they are all related through the Jacobians of the transformations. Because of this, η are generally considered not tensors, but tensor densities with weight = ± 1. In xCoba` we consider them tensors, but dependent on a reference basis. If we change the reference basis (e.g., from etaUppolar to etaUpcartesian), we have to include Jacobians in the transformation. However, if we don't change the reference basis (e.g., from etaUppolar[{a,polar},{b, polar}, {c,polar}] to etaUppolar[{a,cartesian}, {b,cartesian}, {c, cartesian}]) they transform as tensors. This is analogous to our treatment of Christoffel tensors. Consider this example

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This product is a density of weight +1 if we change the basis polar and of weight -1 if we change cartesian. In order to convey all this information, we represent weigths as linear combinations of the names of the bases:

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