GrassmannCalculus`
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Entering gives all the rules in the database containing an exterior product symbol (). Note that the FullForm of is .
SearchRuleData[ExteriorProduct]
Wedge
x⋀y
Wedge[x,y]
In[2]:=
 | SearchRuleData |
Out[2]=
{{2,1},___⋀x_⋀___⋀x_⋀___/;OddGradeQ[x]0},{{2,2},a_?ScalarQ⋀x_ax},{{2,3},x_⋀a_?ScalarQax},{{2,4},x_⋀(y_a_?ScalarQ)ax⋀y},{{2,5},(x_a_?ScalarQ)⋀y_ax⋀y},{2,6},x_⋀y_★[x]★[y]⋁,{2,7},x_⋀y_★[x]★[y]⊖y,{{2,8},⋁},{2,9},x_⋀y_y⋀x,{3,6},x_⋁y_★[x]★[y]⋀,{{3,8},⋀},{{3,14},x_⋀y_⋁z_⋀y_/;RawGrade[x⋀y⋀z]★Dx⋀y⋀z⋁y},{{3,15},x_⋀y_⋀z_⋁y_/;RawGrade[x⋀y⋀z]★Dx⋀y⋁z⋀y},{{3,16},x_⋀y_⋁y_⋀z_/;RawGrade[x⋀y⋀z]★Dx⋀y⋀z⋁y},{{3,17},y_⋀x_⋁y_⋀z_/;RawGrade[x⋀y⋀z]★Dy⋀x⋀z⋁y},{{3,18},x_⋀y_⋁z_⋀y_/;RawGrade[x⋀y⋀z]★nx⋀y⋀z⋁y},{{3,19},x_⋀y_⋀z_⋁y_/;RawGrade[x⋀y⋀z]★nx⋀y⋁z⋀y},{{3,20},x_⋀y_⋁y_⋀z_/;RawGrade[x⋀y⋀z]★nx⋀y⋀z⋁y},{{3,21},y_⋀x_⋁y_⋀z_/;RawGrade[x⋀y⋀z]★ny⋀x⋀z⋁y},{6,5},x_⊖y_★[x]⋀y,{{6,6},★[y]⋀y},{10,1},x_y_x⋀y
x
y
x
x_⋀y_
x
y
RawGrade[x]RawGrade[y]
(-1)
x
y
x_⋁y_
x
y
x
x_⊖y_
x
△
0
Entering gives all the rules in the database containing the interior product of and .
SearchRuleData[x⊖y]
x
y
In[3]:=
| SearchRuleData |
Out[3]=
{{6,2},x_⊖y_a_?ScalarQa(x⊖y)},{{6,3},x_a_?ScalarQ⊖y_a(x⊖y)},{10,2},x_y_/;★λ★G[y]&&★G[x]≥★G[y]x⊖y
△
★λ_
Entering gives all the rules in the database containing the interior product of any two elements.
SearchRuleData[x_⊖y_]
In[4]:=
| SearchRuleData |
Out[4]=
{2,7},x_⋀y_★[x]★[y]⊖y,{{3,7},x_⋁y_★[y]x⊖},{{6,1},x_⊖a_?ScalarQax},{{6,2},x_⊖y_a_?ScalarQa(x⊖y)},{{6,3},x_a_?ScalarQ⊖y_a(x⊖y)},{{6,4},x_⊖y_x⋁},{6,5},x_⊖y_★[x]⋀y,{{6,6},★[y]⋀y},{10,2},x_y_/;★λ★G[y]&&★G[x]≥★G[y]x⊖y,{10,3},x_y_/;★λ★G[x]&&★G[y]≥★G[x]y⊖x
x
y
y
x
x_⊖y_
x
△
★λ_
△
★λ_
Entering selects those rules from the list above which contain an interior product in addition to an exterior product. Note that the FullForm of is .
SearchRuleData[ExteriorProduct&&InteriorProduct]
x⊖y
CircleMinus[x,y]
In[5]:=
| SearchRuleData |
Out[5]=
{2,7},x_⋀y_★[x]★[y]⊖y,{6,5},x_⊖y_★[x]⋀y,{{6,6},★[y]⋀y}
x
x
x_⊖y_
x
In the following example we are looking for rules containing either an exterior product or an interior product, but excluding any rules involving generalized products.
In[6]:=
| SearchRuleData |
 | GeneralizedProduct |
Out[6]=
{{2,1},___⋀x_⋀___⋀x_⋀___/;OddGradeQ[x]0},{{2,2},a_?ScalarQ⋀x_ax},{{2,3},x_⋀a_?ScalarQax},{{2,4},x_⋀(y_a_?ScalarQ)ax⋀y},{{2,5},(x_a_?ScalarQ)⋀y_ax⋀y},{2,6},x_⋀y_★[x]★[y]⋁,{2,7},x_⋀y_★[x]★[y]⊖y,{{2,8},⋁},{2,9},x_⋀y_y⋀x,{3,6},x_⋁y_★[x]★[y]⋀,{{3,7},x_⋁y_★[y]x⊖},{{3,8},⋀},{{3,14},x_⋀y_⋁z_⋀y_/;RawGrade[x⋀y⋀z]★Dx⋀y⋀z⋁y},{{3,15},x_⋀y_⋀z_⋁y_/;RawGrade[x⋀y⋀z]★Dx⋀y⋁z⋀y},{{3,16},x_⋀y_⋁y_⋀z_/;RawGrade[x⋀y⋀z]★Dx⋀y⋀z⋁y},{{3,17},y_⋀x_⋁y_⋀z_/;RawGrade[x⋀y⋀z]★Dy⋀x⋀z⋁y},{{3,18},x_⋀y_⋁z_⋀y_/;RawGrade[x⋀y⋀z]★nx⋀y⋀z⋁y},{{3,19},x_⋀y_⋀z_⋁y_/;RawGrade[x⋀y⋀z]★nx⋀y⋁z⋀y},{{3,20},x_⋀y_⋁y_⋀z_/;RawGrade[x⋀y⋀z]★nx⋀y⋀z⋁y},{{3,21},y_⋀x_⋁y_⋀z_/;RawGrade[x⋀y⋀z]★ny⋀x⋀z⋁y},{{6,1},x_⊖a_?ScalarQax},{{6,2},x_⊖y_a_?ScalarQa(x⊖y)},{{6,3},x_a_?ScalarQ⊖y_a(x⊖y)},{{6,4},x_⊖y_x⋁},{6,5},x_⊖y_★[x]⋀y,{{6,6},★[y]⋀y}
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RawGrade[x]RawGrade[y]
(-1)
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