GrassmannCalculus`
ToInnerProducts  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Here are the Clifford and hypercomplex products of two 1-elements in a 3-space converted to inner products. In this case the resulting inner product is a scalar product.
In[2]:=
★A;
[{x⋄y,x∘y}]
 | ToInnerProducts |
Out[2]=
x⊖y+x⋀y,(x⊖y)+x⋀y
1,1,1
★σ
1,0,1
★σ
You can convert expressions involving unigraded or multigraded symbols to the inner product form. In such cases uses to convert the unigraded or multigraded symbols to simple form, automatically adding the new symbols generated to the list of declared vector symbols. Here we convert the Clifford product of two simple trivectors in a 4-space to inner (and exterior) products.
ToInnerProducts
ComposeSimpleForm
In[3]:=
★A;
;
⋄
| ★ℬ |
4
| ★ |
 | ToInnerProducts |
x
3
y
3
Out[3]=
-⊖-(⋀⊖⋀)⋀+(⋀⊖⋀)⋀-(⋀⊖⋀)⋀+(⋀⊖⋀)⋀-(⋀⊖⋀)⋀+(⋀⊖⋀)⋀-(⋀⊖⋀)⋀+(⋀⊖⋀)⋀-(⋀⊖⋀)⋀+(⊖)⋀⋀⋀-(⊖)⋀⋀⋀+(⊖)⋀⋀⋀-(⊖)⋀⋀⋀+(⊖)⋀⋀⋀-(⊖)⋀⋀⋀+(⊖)⋀⋀⋀-(⊖)⋀⋀⋀+(⊖)⋀⋀⋀
x
3
y
3
x
5
x
6
y
5
y
6
x
4
y
4
x
5
x
6
y
4
y
6
x
4
y
5
x
5
x
6
y
4
y
5
x
4
y
6
x
4
x
6
y
5
y
6
x
5
y
4
x
4
x
6
y
4
y
6
x
5
y
5
x
4
x
6
y
4
y
5
x
5
y
6
x
4
x
5
y
5
y
6
x
6
y
4
x
4
x
5
y
4
y
6
x
6
y
5
x
4
x
5
y
4
y
5
x
6
y
6
x
6
y
6
x
4
x
5
y
4
y
5
x
6
y
5
x
4
x
5
y
4
y
6
x
6
y
4
x
4
x
5
y
5
y
6
x
5
y
6
x
4
x
6
y
4
y
5
x
5
y
5
x
4
x
6
y
4
y
6
x
5
y
4
x
4
x
6
y
5
y
6
x
4
y
6
x
5
x
6
y
4
y
5
x
4
y
5
x
5
x
6
y
4
y
6
x
4
y
4
x
5
x
6
y
5
y
6
Note that:the first term is the interior product of the two trivectors,the next 9 terms involve inner products of bivectors,the last 9 terms involve inner products of vectors, which may thus also be called scalar products, andthe expected 6-element exterior product ⋀ does not appear, since it is zero in the 4-space.
x
3
y
3
The Clifford product corresponds to what is called the 'geometric product' in a number of texts. The following corresponds to its standard decomposition.
In[4]:=
x⋄y
[%]
 | ToInnerProducts |
Out[4]=
x⋄y
Out[4]=
x⊖y+x⋀y
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