1.1. DefBasis

This section will explain how to define and work with new bases, without assuming that they have an underlying coordinate chart.

DefBasis            Define a basis
VBundleOfBasis        VBundle on which a basis lives
PDOfBasis            Parallel derivative associated to the given basis
$Bases                List of currently defined bases
BasisQ                Check existence of a given basis name

Definition of a basis.

To define a basis we only have to provide a name, a vector bundle and a list of cnumbers (whose lenght must be the dimension of the bundle). The cnumbers can include 0 and even negative integers.

In[58]:=

DefBasis[polar, TangentM3, {0, 1, 2}]

** DefCovD: Defining parallel derivative PDpolar[-a] .

** DefTensor: Defining torsion tensor TorsionPDpolar[a, -b, -c] .

** DefTensor: Defining non-symmetric Christoffel tensor ChristoffelPDpolar[a, -b, -c] .

** DefTensor: Defining vanishing Riemann tensor RiemannPDpolar[-a, -b, -c, d] .

** DefTensor: Defining vanishing Ricci tensor RicciPDpolar[-a, -b] .

** DefTensor: Defining antisymmetric +1 density etaUppolar[a, b, c] .

** DefTensor: Defining antisymmetric -1 density etaDownpolar[-a, -b, -c] .

As we can see, several other objects are automatically defined: the parallel derivative and its torsion, Christoffel, Riemann and Ricci tensors. We shall say more about them in Section 3.

In[59]:=

PDOfBasis[polar]

Out[59]=

PDpolar

Each basis has a colour, used to identify its associated indices and objects in StandardForm. The default for new bases is red, but we can specify a different one:

In[60]:=

DefBasis[cartesian, TangentM3, {0, 1, 2}, BasisColor→ RGBColor[0, 1, 0]]

** DefCovD: Defining parallel derivative PDcartesian[-a] .

** DefTensor: Defining torsion tensor TorsionPDcartesian[a, -b, -c] .

** DefTensor: Defining non-symmetric Christoffel tensor ChristoffelPDcartesian[a, -b, -c] .

** DefTensor: Defining vanishing Riemann tensor RiemannPDcartesian[-a, -b, -c, d] .

** DefTensor: Defining vanishing Ricci tensor RicciPDcartesian[-a, -b] .

** DefTensor: Defining antisymmetric +1 density etaUpcartesian[a, b, c] .

** DefTensor: Defining antisymmetric -1 density etaDowncartesian[-a, -b, -c] .

We  can to give different cnumbers to two bases on the same vbundle. A basis has several associated definitions and UpValues:

In[61]:=

? polar

Global`polar

BasisColor[polar]^=RGBColor[1,0,0]
BasisQ[polar]^=True
CNumbersOf[polar]^={0,1,2}
Dagger[polar]^=polar
DependenciesOfBasis[polar]^={M3}
PDOfBasis[polar]^=PDpolar
ServantsOf[polar]^={PDpolar,etaUppolar,etaDownpolar}
VBundleOfBasis[polar]^=TangentM3

In[62]:=

$Bases

Out[62]=

{polar, cartesian}

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