GrassmannCalculus`
CobasisElement  |
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Details and Options
Examples
(3)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Set the Grassmann Plane:
In[2]:=
 | SetActiveSpacePreferences |
 | PublicGrassmannAtlas |
Float the Cobasis palette. (This could also be obtained from the Palettes button on the GrassmannCalculus palette.)
In[3]:=
| FloatPalette |
| CobasisPalette |
The cobasis of is:
e
x
In[4]:=
| CobasisElement |
e
x
Out[4]=
-(★⋀)
e
y
CobasisElement
In[5]:=
| CobasisElement |
e
x
e
y
Out[5]=
{⋀,-(★⋀),★⋀}
e
x
e
y
e
y
e
x
The cobasis of is the basis n-element:
1
In[6]:=
| CobasisElement |
Out[6]=
★⋀⋀
e
x
e
y
CobasisElement
In[7]:=
| CobasisElement |
e
x
Out[7]=
-
e
y
Taking advantage of Listability we can show that all the products give the basis n-element.
basis⋀cobasis
In[8]:=
| GrassmannBases |
| CobasisElement |
| GrassmannBases |
| ★ |
Out[8]=
{{1⋀★⋀⋀},{★⋀⋀,⋀-(★⋀),⋀★⋀},{★⋀⋀,★⋀⋀-,⋀⋀★},{★⋀⋀⋀1}}
e
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e
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e
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Out[8]=
{{★⋀⋀},{★⋀⋀,★⋀⋀,★⋀⋀},{★⋀⋀,★⋀⋀,★⋀⋀},{★⋀⋀}}
e
x
e
y
e
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For each grade of the algebra the exterior product of a basis element and a cobasis element forms a Kronecker delta times the basis n-element.
In[9]:=
Outer
#1⋀
[#2]&,
[0],
[0]//MatrixForm
 | ★ |
 | CobasisElement |
 | GradeBasis |
| GradeBasis |
Out[9]//MatrixForm=
(
)
★⋀ e x e y |
In[10]:=
Outer
#1⋀
[#2]&,Basis,Basis//MatrixForm
| ★ |
| CobasisElement |
Out[10]//MatrixForm=
★⋀ e x e y | 0 | 0 |
0 | ★⋀ e x e y | 0 |
0 | 0 | ★⋀ e x e y |
In[11]:=
Outer
#1⋀
[#2]&,
[2],
[2]//MatrixForm
| ★ |
| CobasisElement |
| GradeBasis |
| GradeBasis |
Out[11]//MatrixForm=
★⋀ e x e y | 0 | 0 |
0 | ★⋀ e x e y | 0 |
0 | 0 | ★⋀ e x e y |
In[12]:=
Outer
#1⋀
[#2]&,
[3],
[3]//MatrixForm
| ★ |
| CobasisElement |
| GradeBasis |
| GradeBasis |
Out[12]//MatrixForm=
(
)
★⋀ e x e y |
Detailed Calculation of a Specific Case
(1)
Cobasis of a Cobasis
(1)
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