GrassmannCalculus`
HypercomplexProduct (∘)  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Set the book default and turn on Precedence showing.
In[2]:=
★A;
 | ★P |
These inputs are equivalent:
In[3]:=
{HypercomplexProduct[x,y],SmallCircle[x,y],x∘y}
Out[3]=
{x∘y,x∘y,x∘y}
The hypercomplex product is a binary operation. So for an expression involving hypercomplex products to be meaningful, it must involve them only in a binary way. For example the following expression is not meaningful.
In[4]:=
u∘v∘(w∘x)∘y∘z
Out[4]=
u∘v∘(w∘x)∘y∘z
However, the following expressions are meaningful (and different).
In[5]:=
{(u∘v)∘((w∘x)∘(y∘z)),((((u∘v)∘w)∘x)∘y)∘z}
Out[5]=
{(u∘v)∘((w∘x)∘(y∘z)),((((u∘v)∘w)∘x)∘y)∘z}
The hypercomplex product is listable.
In[6]:=
a∘{x,y}%//
| ToInnerProducts |
Out[6]=
{a∘x,a∘y}
Out[6]=
ax,ay
0,0,1
★σ
0,0,1
★σ
In[7]:=
{p,q,r}∘{x,y,z}
Out[7]=
{p∘x,q∘y,r∘z}
Expanding a hypercomplex product is similar to expanding a Clifford product except for the addition of hypercomplex signs.
In[8]:=
{p⋄q,p∘q}%//
| ToInnerProducts |
Out[8]=
{p⋄q,p∘q}
Out[8]=
p⊖q+p⋀q,(p⊖q)+p⋀q
1,1,1
★σ
1,0,1
★σ
A hypercomplex product in terms of basis vectors. Here we do a step by step reduction.
In[9]:=
(⋀)∘(⋀)%//
%//
%//
%//
e
1
e
2
e
2
e
3
| ConvertHypercomplexToGeneralized |
| ConvertGeneralizedToInterior |
| ToScalarProducts |
| ToMetricElements |
Out[9]=
(⋀)∘(⋀)
e
1
e
2
e
2
e
3
Out[9]=
2,0,2
★σ
e
1
e
2
△
0
e
2
e
3
2,1,2
★σ
e
1
e
2
△
1
e
2
e
3
2,2,2
★σ
e
1
e
2
△
2
e
2
e
3
Out[9]=
((⋀)⊖(⋀))+(((⋀)⊖)⋀-((⋀)⊖)⋀)+⋀⋀⋀
e
1
e
2
e
2
e
3
2,2,2
★σ
2,1,2
★σ
e
1
e
2
e
2
e
3
e
1
e
2
e
3
e
2
2,0,2
★σ
e
1
e
2
e
2
e
3
Out[9]=
(-(⊖)(⊖)+(⊖)(⊖))+(⊖)⋀-(⊖)⋀+(⊖)⋀
e
1
e
3
e
2
e
2
e
1
e
2
e
2
e
3
2,2,2
★σ
e
2
e
3
2,1,2
★σ
e
1
e
2
e
2
e
2
2,1,2
★σ
e
1
e
3
e
1
e
2
2,1,2
★σ
e
2
e
3
Out[9]=
-⋀
2,1,2
★σ
e
1
e
3
This could be processed in a single step.
In[10]:=
(⋀)∘(⋀)%//
e
1
e
2
e
2
e
3
| ToMetricElements |
Out[10]=
(⋀)∘(⋀)
e
1
e
2
e
2
e
3
Out[10]=
-⋀
2,1,2
★σ
e
1
e
3
The following expands a hypercomplex product of graded symbols.
In[11]:=
a∘b%//
%//
%//
%//
α
3
β
2
| SimplifyHypercomplexProducts |
| ConvertHypercomplexToGeneralized |
| ConvertGeneralizedToInterior |
| ToScalarProducts |
Out[11]=
(a)∘(b)
α
3
β
2
Out[11]=
ab∘
α
3
β
2
Out[11]=
ab++
3,0,2
★σ
α
3
△
0
β
2
3,1,2
★σ
α
3
△
1
β
2
3,2,2
★σ
α
3
△
2
β
2
Out[11]=
ab⊖+(⊖)⋀-(⊖)⋀+⋀
α
3
β
2
3,2,2
★σ
3,1,2
★σ
α
3
β
2
β
3
α
3
β
3
β
2
3,0,2
★σ
α
3
β
2
Out[11]=
ab(-(⊖)(⊖)+(⊖)(⊖))-ab(-(⊖)(⊖)+(⊖)(⊖))+ab(-(⊖)(⊖)+(⊖)(⊖))-ab(⊖)⋀⋀+ab(⊖)⋀⋀+ab(⊖)⋀⋀-ab(⊖)⋀⋀-ab(⊖)⋀⋀+ab(⊖)⋀⋀
β
2
α
6
β
3
α
5
β
2
α
5
β
3
α
6
3,2,2
★σ
α
4
β
2
α
6
β
3
α
4
β
2
α
4
β
3
α
6
3,2,2
★σ
α
5
β
2
α
5
β
3
α
4
β
2
α
4
β
3
α
5
3,2,2
★σ
α
6
α
6
β
3
3,1,2
★σ
α
4
α
5
β
2
α
6
β
2
3,1,2
★σ
α
4
α
5
β
3
α
5
β
3
3,1,2
★σ
α
4
α
6
β
2
α
5
β
2
3,1,2
★σ
α
4
α
6
β
3
α
4
β
3
3,1,2
★σ
α
5
α
6
β
2
α
4
β
2
3,1,2
★σ
α
5
α
6
β
3
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