SamplePublisher`GrassmannCalculus`
ScalarAtoms  |
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Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Set a coordinate system and declare some extra scalar symbols.
In[2]:=
 | SetActiveSpacePreferences |
 | PublicGrassmannAtlas |
| ★★S |
The ScalarAtoms include many special constructions in the Grassmann algebra that are also regarded as scalars.
In[3]:=
UserScalarSymbolsScalarSymbols
 | SpecialScalars |
| ScalarAtoms |
Out[3]=
{a,b,c,d,,,}
Out[3]=
{a,b,c,d,x,y,z,,,}
Out[3]=
★g,★n,,★λ,★0,,★c,_,
_,_,_
★σ
★t
(-1)
Out[3]=
,a,b,c,★c,d,★n,★λ,★0,x,y,z,,,,★g,_,,,,
(-1)
_,_,_
★σ
★t
_
0
_
{0}
The meaning of some of these special scalars are:
In[4]:=
?
| ★c |
★c is a symbolic scalar factor used to convert congruence to equality. Its FullForm is CongruenceFactor.
In[5]:=
?
| ★n |
★n is a symbolic scalar symbol used to represent the dimension of a space. Its FullForm is DimensionSymbol.
In[6]:=
?
| ★λ |
★λ is a symbol used in the expression of the Generalized Grassmann product. Its FullForm is GeneralizedProductOrder.
In[7]:=
?
| ★0 |
★0 is a symbol which represents the grade of 0 or the grade of an expression which evaluates to zero. Its FullForm is GradeOfZero.
In[8]:=
?
| ★g |
★g is an alias for MetricDeterminant. MetricDeterminant is a symbol representing the determinant of the metric tensor. It has no set value, but may be interpreted as the determinant of the metric tensor by some GrassmannAlgebra functions.
In[9]:=
?
| ★σ |
★σ is a symbol used in its overscripted form to represent the sign of a generalized product in a hypercomplex expression. Its FullForm is HypercomplexSign.
In[10]:=
?
| ★t |
★t is a symbol whose subscripted versions represent scalar parameters for use in various output expressions, for example the output of ExteriorQuotient and SignedOctonionProduct. Its FullForm is ScalarParameter.
〈expr〉
The following are zero grade elements and hence scalars.
In[11]:=
,
[%]
α
0
β
{0}
| ScalarQ |
Out[11]=
,
α
0
β
{0}
Out[11]=
{True,True}
 |
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""