GrassmannCalculus`
ZeroQ  |
Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Use the two-dimensional Grassmann plane with an Origin.
In[2]:=
SetActiveAssociation
"Grassmann Plane"
 | PublicGrassmannAtlas |
The following are some zero expressions. Repeated elements in an exterior product.
In[3]:=
p⋀p
[%]
 | ZeroQ |
Out[3]=
p⋀p
Out[3]=
True
A raw grade that exceeds the underlying dimension.
In[4]:=
★⋀⋀⋀p
[%]
e
x
e
y
| ZeroQ |
Out[4]=
★⋀⋀⋀p
e
x
e
y
Out[4]=
True
Even with the Euclidean metric the following expression is not recognized as zero unless metric simplification is also used.
In[5]:=
step1=⊖
[step1]
step1//
e
x
e
y
| ZeroQ |
| ZeroQ |
| ToMetricElements |
Out[5]=
e
x
e
y
Out[5]=
False
Out[5]=
True
Interior products that are zero independent of the specific metric are recognized. The first of the following is recognized as zero because of the general rule on grades with interior products. Remember that here is a scalar coordinate.
x
In[6]:=
step1=x⊖
[step1]
[Reverse[step1]]
e
x
| ZeroQ |
| ZeroQ |
Out[6]=
x⊖
e
x
Out[6]=
True
Out[6]=
False
In[7]:=
e
x
| ZeroQ |
Out[7]=
e
x
Out[7]=
True
The following is zero because the resulting grade would be .
1+1-3-1
In[8]:=
p⋁q//
| ZeroQ |
Out[8]=
True
The following is zero because the resulting product would have a grade of .
-1
In[9]:=
(★⋀)⋁(★⋀)//
e
x
e
x
| ZeroQ |
Out[9]=
True
The following is not zero because it fills the space three times with additional overlap.
In[10]:=
(★⋀⋀)⋁(★⋀⋀)⋁(★⋀⋀)⋁(⋀)%//
%%//
e
x
e
y
e
x
e
y
e
x
e
y
e
x
e
y
| ZeroQ |
| ToCommonFactor |
Out[10]=
★⋀⋀⋁★⋀⋀⋁★⋀⋀⋁⋀
e
x
e
y
e
x
e
y
e
x
e
y
e
x
e
y
Out[10]=
False
Out[10]=
3
★c
e
x
e
y
ZeroQ
In[11]:=
★A;Clear[A];
0,x⋀x,,a⊖x,1,x⋀y,,x⊖a,A
| ZeroQ |
x
4
x
3
Out[11]=
{{True,True,True,True},{False,False,False,False,False}}
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|
""