GrassmannCalculus`
GrassmannBases  |
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Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
In the algebra of 3-space, the bases are:
In[2]:=
★A;
 | GrassmannBases |
Out[2]=
{{1},{,,},{⋀,⋀,⋀},{⋀⋀}}
e
1
e
2
e
3
e
1
e
2
e
1
e
3
e
2
e
3
e
1
e
2
e
3
The basis of the algebra can be obtained by flattening this list.
In[3]:=
A=Flatten
 | GrassmannBases |
Out[3]=
{1,,,,⋀,⋀,⋀,⋀⋀}
e
1
e
2
e
3
e
1
e
2
e
1
e
3
e
2
e
3
e
1
e
2
e
3
You can obtain a general Grassmann number by taking the Dot product of this list of basis elements with a list of 8 scalar coefficients.
In[4]:=
A.{a,b,c,d,e,f,g,h}
Out[4]=
a+b+c+d+e⋀+f⋀+g⋀+h⋀⋀
e
1
e
2
e
3
e
1
e
2
e
1
e
3
e
2
e
3
e
1
e
2
e
3
All of the basis elements of various grades appear on the Basis Pallete.
In[5]:=
 | BasisPalette |
Out[5]=
Basis Palette | ||||||||||||||||
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The complete set of basis elements for the Grassmann Plane point space are:
In[6]:=
SetActiveAssociation
"Grassmann Plane"
Flatten[%]
| PublicGrassmannAtlas |
| GrassmannBases |
Out[6]=
{{1},{★,,},{★⋀,★⋀,⋀},{★⋀⋀}}
e
x
e
y
e
x
e
y
e
x
e
y
e
x
e
y
Out[6]=
{1,★,,,★⋀,★⋀,⋀,★⋀⋀}
e
x
e
y
e
x
e
y
e
x
e
y
e
x
e
y
They now appear on a regenerated Basis Palette.
In[7]:=
| BasisPalette |
Out[7]=
Basis Palette | ||||||||||||||||
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In[8]:=
Clear[A];
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""