GrassmannCalculus`
Grade (★G)  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Using the GrassmannPlane:
In[2]:=
 | SetActiveSpacePreferences |
 | 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}}
Grades of some expressions.
In[4]:=
Grade[(x+y)(+)]
e
x
e
y
Out[4]=
1
In[5]:=
Grade[1+xy★⋀]
e
x
Out[5]=
{0,2}
In[6]:=
Grade[(★⋀)⋁(⋀)]
e
y
e
x
e
y
Out[6]=
1
In[7]:=
Grade[★⋀⋀⋀⋀]
e
x
e
y
e
x
e
y
Out[7]=
★0
Set the book 3-dimensional space.
In[8]:=
★A
Grade will compute the grade of any Grassmann expression. This expression is multigraded.
In[9]:=
Grade[1+2x+3x⋀y+4(x⋀y)⊖z]
Out[9]=
{0,1,2}
Grade makes no attempt to simplify expressions. Each of the two expressions in this list would simplify to zero, but it is the role of to do this, not .
GrassmannSimplify
Grade
In[10]:=
Grade[{x⋀x,x⋀y+y⋀x}]
Out[10]=
{2,2}
Simplifying would yield zeros, the of which returns the symbol for the grade of .
Grade
★0
In[11]:=
Grade
[{x⋀x,x⋀y+y⋀x}]
| ★ |
Out[11]=
{★0,★0}
Only a scalar that is zero gives , otherwise the grade is .
GradeOdZero
0
In[12]:=
Grade[{0,1,2,3}]
Out[12]=
{★0,0,0,0}
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 .
Grade
★0
In[13]:=
Grade[,(1+x⋀y)⋀(1+y⋀x)]
x
4
Out[13]=
{★0,{0,2,★0}}
You can also use new symbols as long as you assert their grades or you can override the grades of currently declared symbols. This is done using a grade assertion from the T tab of the Common Operations palette.
★Λ
In[14]:=
| ★ℬ |
6
★Λ
5
★Λ
2
★Λ
3
Out[14]=
5
In[15]:=
| ★ℬ |
6
★Λ
3
Out[15]=
{3,6}
The expression can contain powers (including reciprocals) of scalars. Here, since and are scalars (declared by default), is also scalar.
a
b
a⋀b
In[16]:=
Grade
1
+
(x⋀y⋀z)⊖y
a
(
x⊖y)+
2
(a⋀b)
Out[16]=
{0,1}
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