GrassmannCalculus`
OddComplementaryGradeQ  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Set the GrassmannPlane.
In[2]:=
SetActiveAssociation
"Grassmann Plane"
 | PublicGrassmannAtlas |
The dimension is:
In[3]:=
 | ★D |
Out[3]=
3
In[4]:=
Print["The GrassmannBases and their grades:"]
Grade
Print["The GrassmannComplements of the basis elements and their grades:"]
Grade
Print["Whether the complementary grades of the bases are odd:"]
| GrassmannBases |
 | GrassmannBases |
| GrassmannComplements |
| GrassmannComplements |
| OddComplementaryGradeQ |
| GrassmannBases |
The GrassmannBases and their grades:
Out[4]=
{{1},{★,,},{★⋀,★⋀,⋀},{★⋀⋀}}
e
x
e
y
e
x
e
y
e
x
e
y
e
x
e
y
Out[4]=
{{0},{1,1,1},{2,2,2},{3}}
The GrassmannComplements of the basis elements and their grades:
Out[4]=
{{★⋀⋀},{⋀,-(★⋀),★⋀},{,-,★},{1}}
e
x
e
y
e
x
e
y
e
y
e
x
e
y
e
x
Out[4]=
{{3},{2,2,2},{1,1,1},{0}}
Whether the complementary grades of the bases are odd:
Out[4]=
{{True},{False,False,False},{True,True,True},{False}}
Switching to the 4-dimensional book space.
EvenComplementaryGradeQ
In[5]:=
★A;
;
[1+3x⋀y+4u⋀v⋀x⋀y]
| ★ℬ |
5
 | OddComplementaryGradeQ |
Out[5]=
True
You can also use new symbols as long as you assert their grades, or you can override the grades of currently declared symbols.
In[6]:=
| ★ℬ |
4
| OddComplementaryGradeQ |
★Λ
3
★Λ
2
Out[6]=
True
The expression can contain powers (including reciprocals) of scalars. In a 5-dimensional space:
In[7]:=
Grade[(x⋀y)⋄z]
Out[7]=
{1,3}
In[8]:=
| ★ℬ |
5
| OddComplementaryGradeQ |
1
+
(x⋀y)⋄z
a
(
x⊖y)+
2
(a⋀b)
Out[8]=
True
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""