GrassmannCalculus`
ToMetricElements  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Set a 4-dimensional vector space.
In[2]:=
★A;
;
 | ★ℬ |
4
The metric tensor gives values to scalar products of the basis elements. Here we have the Euclidean metric.
In[3]:=
Outer[#1⊖#2&,Basis,Basis]//MatrixForm%//
//MatrixForm
| ToMetricElements |
Out[3]//MatrixForm=
e 1 e 1 | e 1 e 2 | e 1 e 3 | e 1 e 4 |
e 2 e 1 | e 2 e 2 | e 2 e 3 | e 2 e 4 |
e 3 e 1 | e 3 e 2 | e 3 e 3 | e 3 e 4 |
e 4 e 1 | e 4 e 2 | e 4 e 3 | e 4 e 4 |
Out[3]//MatrixForm=
1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
The following does the same evaluation for a general metric.
In[4]:=
saveMetric=Metric;
[]
TableSubscript[g,Min[i,j],Max[i,j]],i,1,
,j,1,
;Outer[#1⊖#2&,Basis,Basis]//MatrixForm%//
//MatrixForm
 | ★★S |
g
_,_
| DeclareMetric |
 | ★D |
| ★D |
| ToMetricElements |
Out[4]//MatrixForm=
e 1 e 1 | e 1 e 2 | e 1 e 3 | e 1 e 4 |
e 2 e 1 | e 2 e 2 | e 2 e 3 | e 2 e 4 |
e 3 e 1 | e 3 e 2 | e 3 e 3 | e 3 e 4 |
e 4 e 1 | e 4 e 2 | e 4 e 3 | e 4 e 4 |
Out[4]//MatrixForm=
g 1,1 | g 1,2 | g 1,3 | g 1,4 |
g 1,2 | g 2,2 | g 2,3 | g 2,4 |
g 1,3 | g 2,3 | g 3,3 | g 3,4 |
g 1,4 | g 2,4 | g 3,4 | g 4,4 |
Using a diagonal metric, the following calculates the metric associated with the Grade[2] basis elements.
In[5]:=
| DeclareMetric |
g
1,1
g
2,2
g
3,3
g
4,4
 | GradeBasis |
| GradeBasis |
| ToMetricElements |
| DeclareMetric |
Out[5]//MatrixForm=
g 1,1 | 0 | 0 | 0 |
0 | g 2,2 | 0 | 0 |
0 | 0 | g 3,3 | 0 |
0 | 0 | 0 | g 4,4 |
Out[5]//MatrixForm=
e 1 e 2 e 1 e 2 | e 1 e 2 e 1 e 3 | e 1 e 2 e 1 e 4 | e 1 e 2 e 2 e 3 | e 1 e 2 e 2 e 4 | e 1 e 2 e 3 e 4 |
e 1 e 3 e 1 e 2 | e 1 e 3 e 1 e 3 | e 1 e 3 e 1 e 4 | e 1 e 3 e 2 e 3 | e 1 e 3 e 2 e 4 | e 1 e 3 e 3 e 4 |
e 1 e 4 e 1 e 2 | e 1 e 4 e 1 e 3 | e 1 e 4 e 1 e 4 | e 1 e 4 e 2 e 3 | e 1 e 4 e 2 e 4 | e 1 e 4 e 3 e 4 |
e 2 e 3 e 1 e 2 | e 2 e 3 e 1 e 3 | e 2 e 3 e 1 e 4 | e 2 e 3 e 2 e 3 | e 2 e 3 e 2 e 4 | e 2 e 3 e 3 e 4 |
e 2 e 4 e 1 e 2 | e 2 e 4 e 1 e 3 | e 2 e 4 e 1 e 4 | e 2 e 4 e 2 e 3 | e 2 e 4 e 2 e 4 | e 2 e 4 e 3 e 4 |
e 3 e 4 e 1 e 2 | e 3 e 4 e 1 e 3 | e 3 e 4 e 1 e 4 | e 3 e 4 e 2 e 3 | e 3 e 4 e 2 e 4 | e 3 e 4 e 3 e 4 |
Out[5]//MatrixForm=
g 1,1 g 2,2 | 0 | 0 | 0 | 0 | 0 |
0 | g 1,1 g 3,3 | 0 | 0 | 0 | 0 |
0 | 0 | g 1,1 g 4,4 | 0 | 0 | 0 |
0 | 0 | 0 | g 2,2 g 3,3 | 0 | 0 |
0 | 0 | 0 | 0 | g 2,2 g 4,4 | 0 |
0 | 0 | 0 | 0 | 0 | g 3,3 g 4,4 |
Here is the basic definition of a Clifford product of two vectors reduced to a scalar and an exterior product.
In[6]:=
x⋄y//
| ToMetricElements |
Out[6]=
x⊖y+x⋀y
Here is a more complicated expression.
In[7]:=
(x⋀y)⋄(u⋀v)//
| ToMetricElements |
Out[7]=
(u⊖y)(v⊖x)-(u⊖x)(v⊖y)+(v⊖y)u⋀x-(v⊖x)u⋀y-(u⊖y)v⋀x+(u⊖x)v⋀y+u⋀v⋀x⋀y
The following does the same for a hypercomplex product. It involves HypercomplexSign expressions, ★σ. Await Volume 2.
In[8]:=
x∘y//
| ToMetricElements |
Out[8]=
(x⊖y)+x⋀y
1,1,1
★σ
1,0,1
★σ
Here is an expression involving the Clifford product of two basis 2-elements on a Euclidean 4-space converted to scalar and exterior products.
In[9]:=
X=
[1+(⋀)⋄(⋀)]
| ★ |
 | ToScalarProducts |
e
1
e
2
e
3
e
4
Out[9]=
1+(⊖)(⊖)-(⊖)(⊖)-(⊖)⋀+(⊖)⋀+(⊖)⋀-(⊖)⋀+⋀⋀⋀
e
1
e
4
e
2
e
3
e
1
e
3
e
2
e
4
e
2
e
4
e
1
e
3
e
2
e
3
e
1
e
4
e
1
e
4
e
2
e
3
e
1
e
3
e
2
e
4
e
1
e
2
e
3
e
4
Applying to this expression evaluates all the scalar products on the metric.
ToMetricElements
In[10]:=
| ToMetricElements |
Out[10]=
1+⋀⋀⋀
e
1
e
2
e
3
e
4
We could have applied directly to the original expression.
ToMetricElements
In[11]:=
| ToMetricElements |
e
1
e
2
e
3
e
4
Out[11]=
1+⋀⋀⋀
e
1
e
2
e
3
e
4
In[12]:=
Clear[X]
 |
|
""