GrassmannCalculus`
Cobasis  |
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Details and Options
Examples
(1)
Basic Examples
(1)
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<<GrassmannCalculus`
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 | SetActiveSpacePreferences |
 | PublicGrassmannAtlas |
The Cobasis of the GrassmannPlane are:
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 | Cobasis |
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{⋀,-(★⋀),★⋀}
e
x
e
y
e
y
e
x
Using the Listable attribute of the various Grassmann products, the exterior product of the Basis and Cobasis gives n copies of the basis n-element. The sign of each cobasis element cancels any negative sign produced by rearranging the factors to canonical order.
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Basis⋀
%//
| Cobasis |
| ★ |
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{★⋀⋀,⋀-(★⋀),⋀★⋀}
e
x
e
y
e
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e
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e
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Out[4]=
{★⋀⋀,★⋀⋀,★⋀⋀}
e
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e
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e
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e
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e
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e
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The following show the notation for the cobasis of (used mainly in the book) and one method for evaluation.
e
x
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e
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| CobasisElement |
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e
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-(★⋀)
e
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The following calculation illustrates equation 2.21 in the book. The cobasis of a basis element is obtained by replacing it in the basis n-element by a sign factor and factoring it out.
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★⋀⋀/.
%/.
%//
e
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e
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e
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 | |
2
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i_
i-1
(-1)
| ★ |
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★⋀⋀
2
e
y
Out[6]=
★⋀-1⋀
e
y
Out[6]=
-(★⋀)
e
y
The Cobasis is the same as:
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| GradeCobasis |
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{⋀,-(★⋀),★⋀}
e
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e
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e
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x
All the cobasis elements of all grades are given by:
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| GrassmannCobases |
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{{★⋀⋀},{⋀,-(★⋀),★⋀},{,-,★},{1}}
e
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e
y
e
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e
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e
y
e
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e
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