GrassmannCalculus`
ExpandAndSimplifyRegressiveProducts  |
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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;
ExpandRegressiveProducts
SimplifyRegressiveProducts
ExpandAndSimplifyRegressiveProducts
ToCommonFactor
In[3]:=
X=(a⋀+b⋀)⋁(c⋀+d⋀)X//
%//
X//
%//
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 | ExpandRegressiveProducts |
| SimplifyRegressiveProducts |
| ExpandAndSimplifyRegressiveProducts |
| ToCommonFactor |
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⋀⋁⋀+bd⋀⋁⋀
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Out[3]=
ac⋀⋁⋀+ad⋀⋁⋀+bd⋀⋁⋀
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Out[3]=
★c(ac+ad+bd)
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The following illustrates a more complicated symbolic case. expands only the regressive products in a Grassmann expression.
ExpandRegressiveProducts
In[4]:=
X=(a+b)⊖(z+w)+(ax+b+cy)⋁(ex⋀y+fy)X1=
[X]
 | ExpandRegressiveProducts |
Out[4]=
(a+b)⊖(w+z)+(b+ax+cy)⋁(fy+ex⋀y)
Out[4]=
(a+b)⊖(w+z)+b⋁(fy)+b⋁(ex⋀y)+(ax)⋁(fy)+(ax)⋁(ex⋀y)+(cy)⋁(fy)+(cy)⋁(ex⋀y)
Applying to gives
SimplifyRegressiveProducts
X1
In[5]:=
| SimplifyRegressiveProducts |
Out[5]=
(a+b)⊖(w+z)+aex⋀y⋁x+cex⋀y⋁y
However, applying directly to the expression gives the same result.
ExpandAndSimplifyRegressiveProducts
In[6]:=
 | ExpandAndSimplifyRegressiveProducts |
| ToCommonFactor |
Out[6]=
(a+b)⊖(w+z)+aex⋀y⋁x+cex⋀y⋁y
Out[6]=
0
Note that is neither expanded, nor simplified to zero, since it is not a regressive product. However, does perform some evaluation on interior products.
(a+b)⊖(z+w)
ToCommonFactor
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 x is of grade 2, making the regressive product non-zero. (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 a regressive product).
A
In[7]:=
| ★ℬ |
4
| ExpandAndSimplifyRegressiveProducts |
★Λ
5
★Λ
2
Out[7]=
A+(x+y)⋁(x+y)
Out[7]=
A+x⋁x
In[8]:=
Clear[X,X1]
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