GrassmannCalculus`
ComposeBivector  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Set the default book vector space.
In[2]:=
★A;
You can use the full form of , or its alias .
ComposeBivector[c]
★
c
In[3]:=
[c]
//Column
 | ComposeBivector |
,
 | ★ |
c
Out[3]=
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 2 e 3 |
You can compose bivectors in any dimension greater than 1.
In[4]:=
Table
;
,{i,2,4}//Column
| ★ℬ |
i
| ★ |
c
Out[4]=
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 bivectors.
In[5]:=
 | ★ℬ |
3
 | ★ |
{a,b}
Out[5]=
a 1 e 1 e 2 a 2 e 1 e 3 a 3 e 2 e 3 |
b 1 e 1 e 2 b 2 e 1 e 3 b 3 e 2 e 3 |
You can make the starting index of the coefficients whatever you want.
In[6]:=
| ★ |
{a,b},0
Out[6]=
a 0 e 1 e 2 a 1 e 1 e 3 a 2 e 2 e 3 |
b 0 e 1 e 2 b 1 e 1 e 3 b 2 e 2 e 3 |
In[7]:=
| ★ |
{a,b},{-3,α}
Out[7]=
a -3 e 1 e 2 a -2 e 1 e 3 a -1 e 2 e 3 |
b α e 1 e 2 b 1+α e 1 e 3 b 2+α 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[8]:=
ScalarSymbols
Out[8]=
{a,b,c,d,e,f,g,h,,,,,,,,,,,,,,}
a
-3
a
-2
a
-1
a
0
a
1
a
2
a
3
b
0
b
1
b
2
b
3
b
α
b
1+α
b
2+α
You can also generate templates for bivectors using the placeholder symbol. This allows you to tab through the composed result and enter your own values.
In[9]:=
 | ★ |
| |
Out[9]=
⋀+⋀+⋀
e
1
e
2
e
1
e
3
e
2
e
3
Then that could be copied, filled and assigned to a bivector.
In[10]:=
bivec1=-a⋀+b⋀+4⋀
e
1
e
2
e
1
e
3
e
2
e
3
Out[10]=
-a⋀+b⋀+4⋀
e
1
e
2
e
1
e
3
e
2
e
3
Subscripted scalars are convenient for generic derivations. For specific cases it will often be easier simply to dot with a List of coefficients.
GradeBasis[2]
In[11]:=
bivec1={-a,b,4}.
[2]
| GradeBasis |
Out[11]=
-a⋀+b⋀+4⋀
e
1
e
2
e
1
e
3
e
2
e
3
In[12]:=
Clear[bivec1]
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""