8. Charts

Thus far, everything is a field on a manifold or a function of a parameter. We shall want to express scalar fields as functions of coordinate fields. To do this we need a chart

DefChart            Define a basis
$Bases                List of currently defined bases
ChartQ                Check existence of a given basis name

Definition of a charts.

This whole section has not been implemented yet. At the moment, there is no easy way to restrict fields to points, for example. DefChart, however, does work. We need to supply a list of scalars (the coordinates)

In[403]:=

DefChart[cart, M3, {1, 2, 3}, {x[], y[], z[]}, BasisColor→ Green]

** DefTensor: Defining tensor x[] .

** DefTensor: Defining tensor y[] .

** DefTensor: Defining tensor z[] .

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

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

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

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

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

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

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

Notice that now the torsion tensor vanishes and the Christoffel is symmetric.

In[404]:=

TorsionPDcart[a, -b, -c]

Out[404]=

0

In[405]:=

Bracket[a][Basis[s, {1, -cart}], Basis[s, {2, -cart}]]

Out[405]=

0

In[406]:=

ChartsOfManifold[M3]

Out[406]=

{cart}

In[407]:=

ChartQ /@ {cart, polar}

Out[407]=

{True, False}

In[408]:=

BasisQ /@ {cart, polar}

Out[408]=

{True, True}

In[409]:=

PD[-a][z[]]

Out[409]=

e_a ^( 3)

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