SamplePublisher`GrassmannCalculus`
ComplementaryGrade  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Using the GrassmannPlane:
In[2]:=
SetActiveAssociation
"Grassmann Plane"
 | PublicGrassmannAtlas |
Grades of all the basis elements.
In[3]:=
 | GrassmannBases |
Out[3]=
{{1},{★,,},{★⋀,★⋀,⋀},{★⋀⋀}}
e
x
e
y
e
x
e
y
e
x
e
y
e
x
e
y
Out[3]=
{{0},{1,1,1},{2,2,2},{3}}
The corresponding complementary grades.
In[4]:=
 | ComplementaryGrade |
| GrassmannBases |
Out[4]=
{{3},{2,2,2},{1,1,1},{0}}
Grades of some expressions.
In[5]:=
| ComplementaryGrade |
e
x
e
y
Out[5]=
2
In[6]:=
| ComplementaryGrade |
e
y
Out[6]=
{1,3}
In[7]:=
| ComplementaryGrade |
e
y
e
x
e
y
Out[7]=
2
In[8]:=
| ComplementaryGrade |
e
x
e
y
e
x
e
y
Out[8]=
★0
Using the book 3-space.
In[9]:=
| ★ℬ |
3
e
1
α
{0,2}
e
2
α
{1}
e
1
e
2
e
1
e
3
e
2
e
3
e
1
e
2
e
3
| ComplementaryGrade |
Out[9]=
0,{1},,,{},,xxx,{⋀,⋀,⋀},{⋀⋀}
e
1
α
{0,2}
e
2
α
{1}
e
1
e
2
e
1
e
3
e
2
e
3
e
1
e
2
e
3
Out[9]=
{★0,{0},{1,{0,2},{1},1,NotGraded[xxx]},{2,2,2},{3}}
Out[9]=
{★0,{3},{2,{1,3},{2},2,NotGraded[xxx]},{1,1,1},{0}}
Using an assertion:
In[10]:=
| ★ℬ |
3
e
1
α
{0,2}
e
2
α
{1}
e
1
e
2
e
1
e
3
e
2
e
3
e
1
e
2
e
3
★Λ
{1,2}
| ComplementaryGrade |
★Λ
{1,2}
Out[10]=
0,{1},,,{},,xxx,{⋀,⋀,⋀},{⋀⋀}
e
1
α
{0,2}
e
2
α
{1}
e
1
e
2
e
1
e
3
e
2
e
3
e
1
e
2
e
3
Out[10]=
{★0,{0},{1,{0,2},{1},1,{1,2}},{2,2,2},{3}}
Out[10]=
{★0,{3},{2,{1,3},{2},2,{1,2}},{1,1,1},{0}}
Using symbolic grades:
In[11]:=
step3=,,Grade[step3]
[step3]
α
m
α
{1,m}
α
{k,m}
| ComplementaryGrade |
Out[11]=
,,
α
m
α
{1,m}
α
{k,m}
Out[11]=
{m,{1,m},{k,m}}
Out[11]=
{3-m,{3-m,2},{3-m,3-k}}
ComplementaryGrade
GrassmannSimplify
Grade
In[12]:=
| ComplementaryGrade |
Out[12]=
{1,1}
Simplifying would yield zeros, the of which returns the symbol for the grade of .
ComplementaryGrade
★0
In[13]:=
| ComplementaryGrade |
| ★ |
Out[13]=
{★0,★0}
However, when determines a computed grade to be larger than the dimension of the declared space, it will assume the expression to be zero and return .
ComplementaryGrade
★0
In[14]:=
Grade[,(1+x⋀y)⋀(1+y⋀x)]
[,(1+x⋀y)⋀(1+y⋀x)]
x
4
| ComplementaryGrade |
x
4
Out[14]=
{★0,{0,2,★0}}
Out[14]=
{★0,{★0,1,3}}
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