SamplePublisher`GrassmannCalculus`DifferentialGeometry`
FormBasis  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Set a 2-dimensional Point space with a form basis.
In[2]:=
 | SetCoordinateBoundVectorSpace |
The various types of basis vectors are:
In[3]:=
 | OrthonormalBasis |
| VectorBasis |
| FormBasis |
Out[3]=
,
e
x
e
y
Out[3]=
{,}
e
x
e
y
Out[3]=
{dx,dy}
The active basis is currently the .
FormBasis
In[4]:=
{BasisType,Basis}
Out[4]=
{Form,{dx,dy}}
Since we have a Euclidean metric we can easily transition a Grassmann expression between the various bases types with the built-in rules.
In[5]:=
adx+bdy+cdx⋀dy%/.
%/.
| FormToVector |
| VectorToForm |
Out[5]=
adx+bdy+cdx⋀dy
Out[5]=
a+b+c⋀
e
x
e
y
e
x
e
y
Out[5]=
adx+bdy+cdx⋀dy
Set cylindrical coordinates from the public atlas.
In[6]:=
| SetActiveSpacePreferences |
 | PublicGrassmannAtlas |
In[7]:=
{BasisType,Basis}
Out[7]=
{Vector,{,,}}
e
ρ
e
φ
e
ζ
The transitions between bases are determined by the .
GrassmannScaleFactors
In[8]:=
GrassmannScaleFactors
Out[8]=
{1,ρ,1}
In[9]:=
a+b+c%/.
%/.
e
ρ
e
φ
e
ζ
| VectorToOrthonormal |
| OrthonormalToVector |
Out[9]=
c+a+b
e
ζ
e
ρ
e
φ
Out[9]=
c+a+bρ
e
ζ
e
ρ
e
φ
Out[9]=
c+a+b
e
ζ
e
ρ
e
φ
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