GrassmannCalculus`
SimplifyExteriorProducts  |
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Details and Options
Examples
(2)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
For comparison, here is an expression which simplifies to zero.
GrassmannSimplify
In[2]:=
★A;X=u⋀v+v⋀u+x⋁y+a⊖x+x⋄0+x∘+x
x++;GrassmannSimplify[X]
z
4
 | △ |
0
α
5
0
Out[2]=
0
However, only attempts to simplify the exterior products in the expression.
SimplifyExteriorProducts
In[3]:=
X
 | SimplifyExteriorProducts |
Out[3]=
a⊖x+x⋄0+++x∘+x⋁y+u⋀v+v⋀u+xx
α
5
0
z
4
△
0
Out[3]=
a⊖x+x⋄0+++x∘+x⋁y+xx
α
5
0
z
4
△
0
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 exterior product with itself non-zero. (Note also that although A is zero in a 4-space, since it is not an exterior product, it has not been simplified to zero).
In[4]:=
| ★ℬ |
4
| SimplifyExteriorProducts |
★Λ
5
★Λ
2
Out[4]=
A+x⋀x+y⋀y
Out[4]=
A+x⋀x
In[5]:=
Wedge@@(Plus@@@Table[RandomChoice[ScalarSymbols]Basis〚j〛,{i,4},{j,4}])GrassmannExpandAndSimplify[%]
Out[5]=
(c+a+g+b)⋀(c+f+b+g)⋀(a+d+h+b)⋀(c+d+a+a)
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
4
Out[5]=
(b+bc-ac-ad-2abcd+2cd+bf-abcf-g+2acdg-bcdg-fg+bcfg+ad-cd-ch+bcdh+acfh-bcfh+acgh-cdgh)⋀⋀⋀
3
a
2
a
2
b
2
b
2
b
2
a
3
a
2
a
2
g
2
g
2
a
e
1
e
2
e
3
e
4
In[6]:=
Wedge@@(Plus@@@Table[RandomChoice[ScalarSymbols]Basis〚j〛,{i,4},{j,4}])%//
[Wedge,Alternatives@@Basis];(%/(⋀⋀⋀))//
//Timing
 | GrassmannBreakout |
e
1
e
2
e
3
e
4
| SimplifyExteriorProducts |
Out[6]=
(a+a+c+g)⋀(f+g+c+a)⋀(a+e+e+h)⋀(f+a+g+b)
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
4
Out[6]=
{1.154407,c+bc-e-abce+ef-abef-acef+bcef-g-cg-abcg+eg+abeg+aefg+cefg-2ef+a+ch-2acfh+afgh+cfgh-ah}
3
a
2
a
3
a
2
a
3
a
2
a
2
a
2
g
3
g
2
a
2
g
Properties & Relations
(1)
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