GrassmannCalculus`
ComposeMElement  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
In[2]:=
savePreferences=
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 | AllPreferences |
Set a book 6-dimensional space.
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| ★ℬ |
6
Compose a general 4-element.
In[4]:=
 | ComposeMElement |
Out[4]=
c
1
e
1
e
2
e
3
e
4
c
2
e
1
e
2
e
3
e
5
c
3
e
1
e
2
e
3
e
6
c
4
e
1
e
2
e
4
e
5
c
5
e
1
e
2
e
4
e
6
c
6
e
1
e
2
e
5
e
6
c
7
e
1
e
3
e
4
e
5
c
8
e
1
e
3
e
4
e
6
c
9
e
1
e
3
e
5
e
6
c
10
e
1
e
4
e
5
e
6
c
11
e
2
e
3
e
4
e
5
c
12
e
2
e
3
e
4
e
6
c
13
e
2
e
3
e
5
e
6
c
14
e
2
e
4
e
5
e
6
c
15
e
3
e
4
e
5
e
6
Compare that to a simple 4-element.
In[5]:=
| ComposeSimpleMElement |
Out[5]=
(+++++)⋀(+++++)⋀(+++++)⋀(+++++)
c
1
e
1
c
2
e
2
c
3
e
3
c
4
e
4
c
5
e
5
c
6
e
6
c
7
e
1
c
8
e
2
c
9
e
3
c
10
e
4
c
11
e
5
c
12
e
6
c
13
e
1
c
14
e
2
c
15
e
3
c
16
e
4
c
17
e
5
c
18
e
6
c
19
e
1
c
20
e
2
c
21
e
3
c
22
e
4
c
23
e
5
c
24
e
6
You can compose m-elements in any dimension greater than or equal to m.
In[6]:=
★A;Table
;
[2,c],{i,2,4}//Column
 | ★ℬ |
i
| ComposeMElement |
Out[6]=
c e 1 e 2 |
c 1 e 1 e 2 c 2 e 1 e 3 c 3 e 2 e 3 |
c 1 e 1 e 2 c 2 e 1 e 3 c 3 e 1 e 4 c 4 e 2 e 3 c 5 e 2 e 4 c 6 e 3 e 4 |
That particular case is the same as ComposeBivector.
In[7]:=
★A;Table
;
[c],{i,2,4}//Column
| ★ℬ |
i
| ComposeBivector |
Out[7]=
c e 1 e 2 |
c 1 e 1 e 2 c 2 e 1 e 3 c 3 e 2 e 3 |
c 1 e 1 e 2 c 2 e 1 e 3 c 3 e 1 e 4 c 4 e 2 e 3 c 5 e 2 e 4 c 6 e 3 e 4 |
You can compose lists of m-elements.
In[8]:=
| ★ℬ |
3
 | ComposeMElement |
Out[8]=
c 1 e 1 c 2 e 2 c 3 e 3 |
c 1 e 1 e 2 c 2 e 1 e 3 c 3 e 2 e 3 |
In[9]:=
| ★ℬ |
3
| ComposeMElement |
Out[9]=
α 1 e 1 e 2 α 2 e 1 e 3 α 3 e 2 e 3 |
β 1 e 1 e 2 β 2 e 1 e 3 β 3 e 2 e 3 |
γ 1 e 1 e 2 γ 2 e 1 e 3 γ 3 e 2 e 3 |
In[10]:=
 | ComposeMElement |
Out[10]=
e 1 α 1 e 2 α 2 e 3 α 3 |
β 1 e 1 e 2 β 2 e 1 e 3 β 3 e 2 e 3 |
γ e 1 e 2 e 3 |
You can make the starting index of the coefficients whatever you want.
In[11]:=
| ComposeMElement |
Out[11]=
a 0 e 1 e 2 a 1 e 1 e 3 a 2 e 2 e 3 |
c k e 1 e 2 e 3 |
Note that all the scalar symbols generated as coefficients of the basis elements have been automatically declared as scalar symbols.
In[12]:=
ScalarSymbols
Out[12]=
{a,b,c,d,e,f,g,h,γ,,,,,,,,,,,,,,,,,,,}
a
0
a
1
a
2
c
1
c
2
c
3
c
4
c
5
c
6
c
k
α
1
α
2
α
3
β
1
β
2
β
3
γ
1
γ
2
γ
3
You can also generate templates for m-vectors using the placeholder symbol. This allows you to tab through the composed result and enter your own values.
In[13]:=
| ★ℬ |
4
| ComposeMElement |
| |
Out[13]=
⋀⋀+⋀⋀+⋀⋀+⋀⋀
e
1
e
2
e
3
e
1
e
2
e
4
e
1
e
3
e
4
e
2
e
3
e
4
You may also generate specific m-vectors by dotting a list of coefficients with the GradeBasis[m] basis.
In[14]:=
| GradeDimension |
| GradeBasis |
Out[14]=
4
Out[14]=
a⋀⋀+b⋀⋀+c⋀⋀+d⋀⋀
e
1
e
2
e
3
e
1
e
2
e
4
e
1
e
3
e
4
e
2
e
3
e
4
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| RestorePreferences |
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