GrassmannCalculus`
ConvertInnerToScalar  |
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Examples
(2)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Set a 5-dimensional Euclidean space.
In[2]:=
★A;
;
 | ★ℬ |
5
 | ★P |
We start with an inner product of basis 3-elements, convert it to scalar product of vectors and finally evaluate using .
ToMetricElements
In[3]:=
(⋀⋀)⊖(⋀⋀)%//
%//
e
1
e
2
e
5
e
1
e
2
e
5
| ConvertInnerToScalar |
| ToMetricElements |
Out[3]=
(⋀⋀)⊖(⋀⋀)
e
1
e
2
e
5
e
1
e
2
e
5
Out[3]=
-(⊖)(⊖)(⊖)+(⊖)(⊖)(⊖)+(⊖)(⊖)(⊖)-(⊖)(⊖)(⊖)-(⊖)(⊖)(⊖)+(⊖)(⊖)(⊖)
e
1
e
5
e
2
e
2
e
5
e
1
e
1
e
2
e
2
e
5
e
5
e
1
e
1
e
5
e
2
e
1
e
5
e
2
e
1
e
1
e
2
e
5
e
5
e
2
e
1
e
2
e
2
e
1
e
5
e
5
e
1
e
1
e
2
e
2
e
5
e
5
Out[3]=
1
The expression can be evaluated directly with .
ToMetricElements
In[4]:=
(⋀⋀)⊖(⋀⋀)%//
e
1
e
2
e
5
e
1
e
2
e
5
| ToMetricElements |
Out[4]=
(⋀⋀)⊖(⋀⋀)
e
1
e
2
e
5
e
1
e
2
e
5
Out[4]=
1
The following is zero because every term contains zero factors with a diagonal metric.
In[5]:=
(⋀⋀)⊖(⋀⋀)%//
%//
e
1
e
2
e
3
e
1
e
3
e
4
| ConvertInnerToScalar |
| ToMetricElements |
Out[5]=
(⋀⋀)⊖(⋀⋀)
e
1
e
2
e
3
e
1
e
3
e
4
Out[5]=
-(⊖)(⊖)(⊖)+(⊖)(⊖)(⊖)+(⊖)(⊖)(⊖)-(⊖)(⊖)(⊖)-(⊖)(⊖)(⊖)+(⊖)(⊖)(⊖)
e
1
e
4
e
2
e
3
e
3
e
1
e
1
e
3
e
2
e
4
e
3
e
1
e
1
e
4
e
2
e
1
e
3
e
3
e
1
e
1
e
2
e
4
e
3
e
3
e
1
e
3
e
2
e
1
e
3
e
4
e
1
e
1
e
2
e
3
e
3
e
4
Out[5]=
0
Use a diagonal non-Euclidean metric. Here only the last term is nonzero.
In[6]:=
| DeclareMetric |
In[7]:=
(⋀⋀)⊖(⋀⋀)%//
%//
e
1
e
2
e
5
e
1
e
2
e
5
| ConvertInnerToScalar |
| ToMetricElements |
Out[7]=
(⋀⋀)⊖(⋀⋀)
e
1
e
2
e
5
e
1
e
2
e
5
Out[7]=
-(⊖)(⊖)(⊖)+(⊖)(⊖)(⊖)+(⊖)(⊖)(⊖)-(⊖)(⊖)(⊖)-(⊖)(⊖)(⊖)+(⊖)(⊖)(⊖)
e
1
e
5
e
2
e
2
e
5
e
1
e
1
e
2
e
2
e
5
e
5
e
1
e
1
e
5
e
2
e
1
e
5
e
2
e
1
e
1
e
2
e
5
e
5
e
2
e
1
e
2
e
2
e
1
e
5
e
5
e
1
e
1
e
2
e
2
e
5
e
5
Out[7]=
10
Declare a nondiagonal metric in 3-space.
In[8]:=
A;
;
[{{1,2,0},{2,1,0},{0,0,1}}]
| ★ℬ |
3
| DeclareMetric |
In[9]:=
(⋀)⊖(⋀)%//
%//
e
1
e
2
e
1
e
2
| ConvertInnerToScalar |
| ToMetricElements |
Out[9]=
(⋀)⊖(⋀)
e
1
e
2
e
1
e
2
Out[9]=
-(⊖)(⊖)+(⊖)(⊖)
e
1
e
2
e
2
e
1
e
1
e
1
e
2
e
2
Out[9]=
-3
Using symbolic vectors:
In[10]:=
(p⋀q)⊖(x⋀y)%//
| ConvertInnerToScalar |
Out[10]=
(p⋀q)⊖(x⋀y)
Out[10]=
-(p⊖y)(q⊖x)+(p⊖x)(q⊖y)
Other Exterior Product Evaluations
(1)
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