GrassmannCalculus`
RawGrade  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
In[2]:=
★A;
Grade takes into account the dimension of the space, which is the maximum grade.
In[3]:=
FoldList[ExteriorProduct,1,{,,,x,y,z}]/.
[2,2]Grade[%]
e
1
e
2
e
3
 | ★ℜ |
Out[3]=
{1,,⋀,⋀⋀,⋀⋀⋀x,⋀⋀⋀x⋀y,⋀⋀⋀x⋀y⋀z}
e
1
e
1
e
2
e
1
e
2
e
3
e
1
e
2
e
3
e
1
e
2
e
3
e
1
e
2
e
3
Out[3]=
{0,1,2,3,★0,★0,★0}
RawGrade
In[4]:=
FoldList[ExteriorProduct,1,{,,,x,y,z}]/.
[2,2]
[%]
e
1
e
2
e
3
| ★ℜ |
 | RawGrade |
Out[4]=
{1,,⋀,⋀⋀,⋀⋀⋀x,⋀⋀⋀x⋀y,⋀⋀⋀x⋀y⋀z}
e
1
e
1
e
2
e
1
e
2
e
3
e
1
e
2
e
3
e
1
e
2
e
3
e
1
e
2
e
3
Out[4]=
{0,1,2,3,4,5,6}
In the following example, is zero (interior product of a scalar on the left and a vector on the right), hence returning a grade of zero .
a⊖x
★0
In[5]:=
★A;
a⊖x+1+2x+3⋀y+4⊖z
| RawGrade |
x
3
x
6
Out[5]=
{0,1,4,5,★0}
RawGrade
GrassmannSimplify
RawGrade
In[6]:=
| ★ℬ |
1
| RawGrade |
Out[6]=
{2,2}
Simplifying would yield zeros, the of which returns the symbol for the grade of zero .
RawGrade
★0
In[7]:=
| RawGrade |
| ★ |
Out[7]=
{★0,★0}
However, when encounters a need to know the dimension of the declared space when computing a grade, it uses the symbolic value .
RawGrade
★n
In[8]:=
| RawGrade |
x
z
Out[8]=
{2-★n,3-2★n,-1+★n,3-2★n}
You can also use new (undefined) symbols as long as you assert their grades, or you can override the grades of currently declared symbols.
In[9]:=
Clear[A];
{A+x⋀y,x⋁y},A∈,x∈,y∈
| RawGrade |
★Λ
5
★Λ
2
★Λ
3
Out[9]=
{5,5-★n}
The expression can contain powers (including reciprocals) of scalars.
In[10]:=
Raw
Grade
1
+
x⋁y⋁z
a
(
x⊖y)+
2
(a⋀b)
z
Out[10]=
{0,4-3★n}
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""