GrassmannCalculus`
ComposeSimpleMElement  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
In[2]:=
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Set a book 6-dimensional space.
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| ★ℬ |
6
Compose a simple 4-element.
In[4]:=
 | ComposeSimpleMElement |
Out[4]=
(+++++)⋀(+++++)⋀(+++++)⋀(+++++)
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
Compare that to a general 4-element.
In[5]:=
| ComposeMElement |
Out[5]=
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
You can compose simple m-elements in any dimension (although some might simplify to zero).
In[6]:=
★A;Table
;
[3,c],{i,1,4}//Column
 | ★ℬ |
i
| ComposeSimpleMElement |
Out[6]=
( c 1 e 1 c 2 e 1 c 3 e 1 |
( c 1 e 1 c 2 e 2 c 3 e 1 c 4 e 2 c 5 e 1 c 6 e 2 |
( c 1 e 1 c 2 e 2 c 3 e 3 c 4 e 1 c 5 e 2 c 6 e 3 c 7 e 1 c 8 e 2 c 9 e 3 |
( c 1 e 1 c 2 e 2 c 3 e 3 c 4 e 4 c 5 e 1 c 6 e 2 c 7 e 3 c 8 e 4 c 9 e 1 c 10 e 2 c 11 e 3 c 12 e 4 |
You can compose lists of simple m-elements.
In[7]:=
| ★ |
3
 | ComposeSimpleMElement |
Out[7]=
c |
★ c 1 c 2 e 1 c 3 e 2 c 4 e 3 |
(★ c 1 c 2 e 1 c 3 e 2 c 4 e 3 c 5 c 6 e 1 c 7 e 2 c 8 e 3 c 9 c 10 e 1 c 11 e 2 c 12 e 3 |
In[8]:=
 | ComposeSimpleMElement |
Out[8]=
(★ α 1 e 1 α 2 e 2 α 3 e 3 α 4 α 5 e 1 α 6 e 2 α 7 e 3 α 8 |
(★ β 1 e 1 β 2 e 2 β 3 e 3 β 4 β 5 e 1 β 6 e 2 β 7 e 3 β 8 |
(★ γ 1 e 1 γ 2 e 2 γ 3 e 3 γ 4 γ 5 e 1 γ 6 e 2 γ 7 e 3 γ 8 |
In[9]:=
| ComposeSimpleMElement |
Out[9]=
★ α 1 e 1 α 2 e 2 α 3 e 3 α 4 |
(★ β 1 e 1 β 2 e 2 β 3 e 3 β 4 β 5 e 1 β 6 e 2 β 7 e 3 β 8 |
(★ γ 1 e 1 γ 2 e 2 γ 3 e 3 γ 4 γ 5 e 1 γ 6 e 2 γ 7 e 3 γ 8 γ 9 e 1 γ 10 e 2 γ 11 e 3 γ 12 |
You can make the starting index of the coefficients whatever you want.
In[10]:=
| ★ℬ |
3
| ComposeSimpleMElement |
Out[10]=
( a 0 e 1 a 1 e 2 a 2 e 3 a 3 e 1 a 4 e 2 a 5 e 3 |
( c k e 1 c 1+k e 2 c 2+k e 3 c 3+k e 1 c 4+k e 2 c 5+k e 3 c 6+k e 1 c 7+k e 2 c 8+k e 3 |
Note that all the scalar symbols generated as coefficients of the basis elements have been automatically declared as scalar symbols.
In[11]:=
ScalarSymbols
Out[11]=
{a,b,c,d,e,f,g,h,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,}
a
0
a
1
a
2
a
3
a
4
a
5
c
1
c
2
c
3
c
4
c
5
c
6
c
7
c
8
c
9
c
10
c
11
c
12
c
k
c
1+k
c
2+k
c
3+k
c
4+k
c
5+k
c
6+k
c
7+k
c
8+k
α
1
α
2
α
3
α
4
α
5
α
6
α
7
α
8
β
1
β
2
β
3
β
4
β
5
β
6
β
7
β
8
γ
1
γ
2
γ
3
γ
4
γ
5
γ
6
γ
7
γ
8
γ
9
γ
10
γ
11
γ
12
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[12]:=
| ★ℬ |
4
| ComposeSimpleMElement |
| |
Out[12]=
(+++)⋀(+++)⋀(+++)
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
4
It's also possible to compose simple m-elements by dotting lists of coefficients with the Basis vectors within an exterior product.
In[13]:=
{1,2,3,4}.Basis⋀{a,b,c,d}.Basis⋀{5,6,-7,-8}.Basis
Out[13]=
(+2+3+4)⋀(a+b+c+d)⋀(5+6-7-8)
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
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e
1
e
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e
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4
In[14]:=
Fold[#1⋀#2.Basis&,1,{{1,2,3,4},{a,b,c,d},{5,6,-7,-8}}]%/.
[2,2]
| ★ℜ |
Out[14]=
1⋀(+2+3+4)⋀(a+b+c+d)⋀(5+6-7-8)
e
1
e
2
e
3
e
4
e
1
e
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e
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e
4
e
1
e
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e
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e
4
Out[14]=
(+2+3+4)⋀(a+b+c+d)⋀(5+6-7-8)
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
4
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