GrassmannCalculus`
GrassmannComplement  |
|
| | ||||
Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
These inputs are equivalent:
In[2]:=
[x],OverBar[x],
 | GrassmannComplement |
x
Out[2]=
{,,}
x
x
x
To convert complements of basis elements according to the currently declared metric, you can use For example in a 3-dimensional Euclidean space with basis you would get:
ConvertComplements
{,,}
e
1
e
2
e
3
In[3]:=
★A;
[{,⋀}]
 | ConvertComplements |
e
1
e
1
e
3
Out[3]=
{⋀,-}
e
2
e
3
e
2
In Grassmann algebra the GrassmannComplement and InteriorProduct have been constructed in such a way that the complement element is orthogonal to the element. So above the ⋀ direction is orthogonal to the direction and the direction is orthogonal to the ⋀ direction.
e
2
e
3
e
1
e
2
e
1
e
3
In[4]:=
{⋀⊖,⋀⊖}//
e
2
e
3
e
1
e
1
e
3
e
2
 | ToMetricElements |
Out[4]=
{0,0}
In a 3-space, the Grassmann complement of the Grassmann complement of an element is just the element itself. You can check this by applying (or its alias .
GrassmannSimplify
★
) to x
m
In[5]:=
| ★ |
a
x
x
2
x
3
Out[5]=
{a,x,,}
x
2
x
3
In general, the Grassmann complement of the Grassmann complement of an element may differ from the element by a sign. For example in a 4-space you will get:
In[6]:=
| ★ℬ |
4
| ★ |
a
x
x
2
x
3
x
4
Out[6]=
{a,-x,,-,}
x
2
x
3
x
4
The complement depends on the current metric. In a 2-space with default metric we get
In[7]:=
★A;
;
[{,}]
| ★ℬ |
2
| ConvertComplements |
e
1
e
2
Out[7]=
{,-}
e
2
e
1
But with a general (symmetric) metric we get
In[8]:=
| DeclareMetric |
In[9]:=
| ConvertComplements |
e
1
e
2
Out[9]=
-+,-+
b
e
1
★g
a
e
2
★g
c
e
1
★g
b
e
2
★g
The symbol stands for the determinant of the metric tensor.
★g
In[10]:=
| ToMetricElements |
| ★g |
Out[10]=
-+ac
2
b
 |
|
""