GrassmannCalculus`
ExpandAndSimplifyExteriorProducts  |
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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;
ExpandExteriorProducts
SimplifyExteriorProducts
ExpandAndSimplifyExteriorProducts
In[3]:=
X=(a+b)⋀(c+d)X//
%//
X//
e
1
e
2
e
2
e
3
 | ExpandExteriorProducts |
| SimplifyExteriorProducts |
| ExpandAndSimplifyExteriorProducts |
Out[3]=
(a+b)⋀(c+d)
e
1
e
2
e
2
e
3
Out[3]=
(a)⋀(c)+(a)⋀(d)+(b)⋀(c)+(b)⋀(d)
e
1
e
2
e
1
e
3
e
2
e
2
e
2
e
3
Out[3]=
ac⋀+ad⋀+bd⋀
e
1
e
2
e
1
e
3
e
2
e
3
Out[3]=
ac⋀+ad⋀+bd⋀
e
1
e
2
e
1
e
3
e
2
e
3
Here is another more symbolic example. expands only the exterior products in a Grassmann expression.
ExpandExteriorProducts
In[4]:=
X=(a+b)⊖(z+w)+(ax+b+cy)⋀(ex⋀y+fy)X1=
[X]
 | ExpandExteriorProducts |
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
SimplifyExteriorProducts
X1
In[5]:=
| SimplifyExteriorProducts |
Out[5]=
bfy+(a+b)⊖(w+z)+(be+af)x⋀y
However, applying directly to the expression gives the same result.
ExpandAndSimplifyExteriorProducts
In[6]:=
 | ExpandAndSimplifyExteriorProducts |
Out[6]=
bfy+(a+b)⊖(w+z)+(be+af)x⋀y
Note that is neither expanded, nor simplified to zero, since it is not an exterior product.
(a+b)⊖(z+w)
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 exterior 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 an exterior product). Also the ordering of ia as it is because the Grassmann algebra ordering function puts 1-elements before higher grade elements.
x
A
y⋀x
In[7]:=
| ★ℬ |
4
| ExpandAndSimplifyExteriorProducts |
★Λ
5
★Λ
2
Out[7]=
A+(x+y)⋀(x+y)
Out[7]=
A+x⋀x+2y⋀x
In[8]:=
Clear[X,X1]
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