GrassmannCalculus`
GrassmannRule (★ℜ)  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
To retrieve the first rule associated with Chapter 2 (rule 2,1) you would enter
In[2]:=
GrassmannRule
[2,1]Out[2]=
___⋀x_⋀___⋀x_⋀___/;OddGradeQ[x]0
You could use this rule to simplify an expression containing the pattern on the left.
In[3]:=
★A;p⋀⋀r⋀⋀s/.
q
3
q
3
GrassmannRule
[2,1]Out[3]=
0
To retrieve rule numbers 2,2 and 2,3 you could enter:
In[4]:=
GrassmannRule
[2,{2,3}]Out[4]=
{a_?ScalarQ⋀x_ax,x_⋀a_?ScalarQax}
You could use these rules (and the alias ★ℜ for GrassmannRule) to simplify an expression containing the patterns on the left.
In[5]:=
2⋀u+v⋀3/.
[2,{2,3}]
 | ★ℜ |
Out[5]=
2u+3v
To retrieve all the rules associated with Chapter 2 of the Grassmann Algebra book you could enter:
In[6]:=
GrassmannRule
[2]//ColumnOut[6]=
___⋀x_⋀___⋀x_⋀___/;OddGradeQ[x]0 |
a_?ScalarQ⋀x_ax |
x_⋀a_?ScalarQax |
x_⋀(y_a_?ScalarQ)ax⋀y |
(x_a_?ScalarQ)⋀y_ax⋀y |
x_⋀y_★[x]★[y] x y |
x_⋀y_★[x]★[y] x |
x_⋀y_ x y |
x_⋀y_ RawGrade[x]RawGrade[y] (-1) |
The function acting on a unigraded element x computes the factor that relates to the complement of the complement of in the currently declared space.
★
((-1)^(Grade[x](Dimension-Grade[x])))
x
x
For example, applying this list of rules to the expression ⋀ shows that the sixth rule fires.
x
k
y
m
In[7]:=
★A;⋀/.
x
k
y
m
GrassmannRule
[2]Out[7]=
(★n-k)k+(★n-m)m
(-1)
x
k
y
m
Of course, had you been in a space of dimension other than 3 (4, say) you would get:
In[8]:=
 | ★ℬ |
4
x
k
y
m
GrassmannRule
[2]Out[8]=
(★n-k)k+(★n-m)m
(-1)
x
k
y
m
To retrieve all the rules associated with Chapters 2, 3 and 5 of the Grassmann Algebra book you could enter:
In[9]:=
GrassmannRule
[{2,3,5}]Out[9]=
___⋀x_⋀___⋀x_⋀___/;OddGradeQ[x]0,a_?ScalarQ⋀x_ax,x_⋀a_?ScalarQax,x_⋀(y_a_?ScalarQ)ax⋀y,(x_a_?ScalarQ)⋀y_ax⋀y,x_⋀y_★[x]★[y]⋁,x_⋀y_★[x]★[y]⊖y,⋁,x_⋀y_y⋀x,___⋁x_⋁___⋁x_⋁___/;OddComplementaryGradeQ[x]0,___⋁x_⋁___⋁x_⋁___/;★G[x]≠★D&&(SimpleProductQ[x]||GradeQ[1][x])0,x_⋁(y_a_?ScalarQ)ax⋁y,(x_a_?ScalarQ)⋁y_ax⋁y,⋁,x_⋁y_★[x]★[y]⋀,x_⋁y_★[y]x⊖,⋀,x_⋁y_y⋁x,⋁x_x,x_⋁x,⋁x_/;d★Dx,x_⋁/;d★Dx,x_⋀y_⋁z_⋀y_/;RawGrade[x⋀y⋀z]★Dx⋀y⋀z⋁y,x_⋀y_⋀z_⋁y_/;RawGrade[x⋀y⋀z]★Dx⋀y⋁z⋀y,x_⋀y_⋁y_⋀z_/;RawGrade[x⋀y⋀z]★Dx⋀y⋀z⋁y,y_⋀x_⋁y_⋀z_/;RawGrade[x⋀y⋀z]★Dy⋀x⋀z⋁y,x_⋀y_⋁z_⋀y_/;RawGrade[x⋀y⋀z]★nx⋀y⋀z⋁y,x_⋀y_⋀z_⋁y_/;RawGrade[x⋀y⋀z]★nx⋀y⋁z⋀y,x_⋀y_⋁y_⋀z_/;RawGrade[x⋀y⋀z]★nx⋀y⋀z⋁y,y_⋀x_⋁y_⋀z_/;RawGrade[x⋀y⋀z]★ny⋀x⋀z⋁y,+,+,a,a,a,a,★[x]x,,
x
y
x
x_⋀y_
x
y
RawGrade[x]RawGrade[y]
(-1)
1
1
1
x
y
y
x_⋁y_
x
y
(★n-RawGrade[x])(★n-RawGrade[y])
(-1)
1
★n
1
★n
1
d_
1
d_
x_+y_
x
y
x_+y_
x
y
x_a_?ScalarQ
x
x_a_?ScalarQ
x
a_?ScalarQ
1
a_?ScalarQ
1
x_
1
1
★n
1
1
★D
To retrieve the first rule in each chapter of the Grassmann Algebra book you could enter:
In[10]:=
GrassmannRule
[All,1]Out[10]=
___⋀x_⋀___⋀x_⋀___/;OddGradeQ[x]0,___⋁x_⋁___⋁x_⋁___/;OddComplementaryGradeQ[x]0,+,x_⊖a_?ScalarQax,x_y_x⋀y,a_?ScalarQ∘x_ax,a_?ScalarQ⋄x_ax
x_+y_
x
y
△
0
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