SamplePublisher`GrassmannCalculus`DifferentialGeometry`
DifferentialExponentialOperator  |
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Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Set the GrassmannPlane.
In[2]:=
SetActiveAssociation
"Grassmann Plane"
[t,x0,y0]
 | PublicGrassmannAtlas |
 | ★★S |
The following establishes the 6th order Series in for the vector operator .
t
∂[(x+3y)+(x-y)]
e
x
e
y
In[3]:=
diffOp:=
[t,∂[(x+3y)+(x-y)],6];
 | DifferentialExponentialOperator |
e
x
e
y
The following evaluates on the and coordinates.
x
y
In[4]:=
xseries0=diffOp[x]yseries0=diffOp[y]
Out[4]=
x+(x+3y)t+2x++2y++(x+3y)++
2
t
2x
3
3
t
2x
4
t
3
2
15
5
t
4x
6
t
45
7
O[t]
Out[4]=
y+(x-y)t+2y+(x-y)++(x-y)++
2
t
2
3
3
t
2y
4
t
3
2
15
5
t
4y
6
t
45
7
O[t]
These are series in starting at the initial point . So we might write this as:
t
{x,y}
In[5]:=
xseries1=xseries0/.{xx0,yy0}yseries1=yseries0/.{xx0,yy0}
Out[5]=
x0+(x0+3y0)t+2x0++2y0++(x0+3y0)++
2
t
2x0
3
3
t
2x0
4
t
3
2
15
5
t
4x0
6
t
45
7
O[t]
Out[5]=
y0+(x0-y0)t+2y0+(x0-y0)++(x0-y0)++
2
t
2
3
3
t
2y0
4
t
3
2
15
5
t
4y0
6
t
45
7
O[t]
Now we solve the differential equations for the orbit through and take the Series for each coordinate function.
{x0,y0}
In[6]:=
step1=
[(x+3y)+(x-y),★+x0+y0,t]//
| DOrbitFunction |
e
x
e
y
e
x
e
y
| ToListCoordinates |
Out[6]=
(x0+3x0+3(-1+)y0),((-1+)x0+(3+)y0)
1
4
-2t
4t
4t
1
4
-2t
4t
4t
In[7]:=
xseries2=Series[step1〚1〛,{t,0,6}]yseries2=Series[step1〚2〛,{t,0,6}]
Out[7]=
x0+(x0+3y0)t+2x0++2y0++(x0+3y0)++
2
t
2x0
3
3
t
2x0
4
t
3
2
15
5
t
4x0
6
t
45
7
O[t]
Out[7]=
y0+(x0-y0)t+2y0+(x0-y0)++(x0-y0)++
2
t
2
3
3
t
2y0
4
t
3
2
15
5
t
4y0
6
t
45
7
O[t]
These are the same as the series produced by the exponential operator.
In[8]:=
{xseries1xseries2,yseries1yseries2}
Out[8]=
{True,True}
In[9]:=
Clear[step,xseries0,xseries1,xseries2,yseries0,yseries1,yseries2]
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