SamplePublisher`GrassmannCalculus`
ComputeNSpaceFactoringData  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
The following sets a vector 4-space and launches a Keys palette for accessing the various symbolic expressions at the lowest level of the Association.
In[2]:=
 | ★ℬ |
4
 | SFAKeysPalette |
The following computes grades 2 and 3 in 4-space.
In[3]:=
| ComputeNSpaceFactoringData |
Out[3]=
SFA[ 4 ] Data |
Once computed the icon can be copied and pasted into an initialization statement (such as the following) in a notebook and doesn't have to be computed again. is the conventional symbol used to represent the top level symbolic factoring data Association.
SFA
In[4]:=
 | SFA |
SFA[ 4 ] Data |
| UnpackIcon |
Information can be extracted from the Association with the command.
IconExtract
In[5]:=
| GCphrase2 |
| IconExtract |
| SFA |
| GCphrase2 |
| IconExtract |
| SFA |
| GCphrase2 |
| IconExtract |
| SFA |
e
1
e
2
| GCphrase2 |
| IconExtract |
| SFA |
e
1
e
2
Out[5]=
Top level with grades as the main keys
Out[5]=
Dimension4,Basis{,,,},Symbolc,2
,3
e
1
e
2
e
3
e
4
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Out[5]=
Middle level with GradeBasis elements as the main keys. Only the first one is initially included.
Out[5]=
⋀
e
1
e
2
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Out[5]=
The third level is the Association of symbolic expressions.
Out[5]=
SymbolicMProduct(++)⋀(++),SymbolicMVector⋀+⋀+⋀-⋀-⋀+(-+)⋀,SymbolicCoefficients{,,,-,-,(-+)},SymbolicVariables{,,,,},EquationFunction(Thread[{,,,-,-,(-+)}##1]&)
c
0,0
e
1
e
3
c
1,3
e
4
c
1,4
e
2
e
3
c
2,3
e
4
c
2,4
c
0,0
e
1
e
2
c
0,0
c
2,3
e
1
e
3
c
0,0
c
2,4
e
1
e
4
c
0,0
c
1,3
e
2
e
3
c
0,0
c
1,4
e
2
e
4
c
0,0
c
1,4
c
2,3
c
1,3
c
2,4
e
3
e
4
c
0,0
c
0,0
c
2,3
c
0,0
c
2,4
c
0,0
c
1,3
c
0,0
c
1,4
c
0,0
c
1,4
c
2,3
c
1,3
c
2,4
c
0,0
c
1,3
c
1,4
c
2,3
c
2,4
c
0,0
c
0,0
c
2,3
c
0,0
c
2,4
c
0,0
c
1,3
c
0,0
c
1,4
c
0,0
c
1,4
c
2,3
c
1,3
c
2,4
Out[5]=
The final level is a specific symbolic expression from the symbolic Association. These are the directly useful objects in the entire structure.
Out[5]=
{,,,,}
c
0,0
c
1,3
c
1,4
c
2,3
c
2,4
Often we will simply want to get a specific expression Association for a specific grade and GradeBasis element. Then we can work with those expressions in specific cases. Here we use the symbolic expressions to factor a 3-vector using the ⋀⋀ pivots.
e
1
e
2
e
3
In[6]:=
expressions=
[4],{3,⋀⋀};mvector=
[3].{a,b,c,d}variables=
[expressions,{"SymbolicVariables"}]equations=
[expressions,{"EquationFunction"}][{a,b,c,d}]solutions=Solve[equations,variables]〚1〛factoredExpression=
[expressions,{"SymbolicMProduct"}]/.solutions
[expressions,{"SymbolicMVector"}]/.solutions
| IconExtract |
 | SFA |
e
1
e
2
e
3
| GradeBasis |
| IconExtract |
| IconExtract |
| IconExtract |
| IconExtract |
Out[6]=
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
Out[6]=
{,,,}
c
0,0
c
1,4
c
2,4
c
3,4
Out[6]=
{a,b,-c,d}
c
0,0
c
0,0
c
3,4
c
0,0
c
2,4
c
0,0
c
1,4
Out[6]=
a,,-,
c
0,0
c
1,4
d
a
c
2,4
c
a
c
3,4
b
a
Out[6]=
a+⋀-⋀+
e
1
d
e
4
a
e
2
c
e
4
a
e
3
b
e
4
a
Out[6]=
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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