GrassmannCalculus`
ExpandAndSimplifyInteriorProducts  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Set the book default to establish scalars and vectors.
In[2]:=
★A;
ExpandInteriorProducts
SimplifyInteriorProducts
ExpandAndSimplifyInteriorProducts
ToMetricElements
In[3]:=
 | ★P |
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 | ExpandInteriorProducts |
| SimplifyInteriorProducts |
| ExpandAndSimplifyInteriorProducts |
| ToMetricElements |
Out[3]=
(a⋀+b⋀)⊖(c+d)
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Out[3]=
(a⋀)⊖(c)+(a⋀)⊖(d)+(b⋀)⊖(c)+(b⋀)⊖(d)
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Out[3]=
ac((⋀)⊖)+ad((⋀)⊖)+bc((⋀)⊖)+bd((⋀)⊖)
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Out[3]=
ac((⋀)⊖)+ad((⋀)⊖)+bc((⋀)⊖)+bd((⋀)⊖)
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Out[3]=
-ac-bd
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ExpandInteriorProducts
In[4]:=
X=(z+w)⋀(z+w)+(ax+b+cy)⊖(ex⋀y+fy)X1=
[X]
 | ExpandInteriorProducts |
Out[4]=
(b+ax+cy)⊖(fy+ex⋀y)+(w+z)⋀(w+z)
Out[4]=
b⊖(fy)+b⊖(ex⋀y)+(ax)⊖(fy)+(ax)⊖(ex⋀y)+(cy)⊖(fy)+(cy)⊖(ex⋀y)+w⋀w+w⋀z+z⋀w+z⋀z
Applying to gives
SimplifyInteriorProducts
X1
In[5]:=
| SimplifyInteriorProducts |
Out[5]=
af(x⊖y)+cf(y⊖y)+w⋀w+w⋀z+z⋀w+z⋀z
However, applying directly to the expression gives the same result.
ExpandAndSimplifyInteriorProducts
In[6]:=
 | ExpandAndSimplifyInteriorProducts |
Out[6]=
af(x⊖y)+cf(y⊖y)+w⋀w+w⋀z+z⋀w+z⋀z
Note that is neither expanded, nor simplified to zero, since it is not an interior product.
(w+z)⋀(w+z)
You can also use new symbols as long as you assert their grades, or you can override the grades of currently declared symbols. Here we assert that is of grade 2, making the interior product with a scalar product, and hence symmetrical. (Note also that although has been asserted to be a 5-element, it has not been simplified to zero in this 4-space, since it is not an interior product).
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A
In[7]:=
| ★ℬ |
4
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| ExpandAndSimplifyInteriorProducts |
★Λ
5
★Λ
2
Out[7]=
A+(x+)⊖(x+)
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Out[7]=
A+x⊖x+2x⊖+⊖
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In[8]:=
★PP;Clear[X,X1]
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