SamplePublisher`GrassmannCalculus`
BasisPushforward  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Establish routines to set and coordinate systems.
{x,y}
{u,v,w}
In[2]:=
setXYCoordinates:=
[2,{x,y},"Vector"]setUVWCoordinates:=
[3,{u,v,w},"Vector"]
 | SetCoordinateVectorSpace |
| SetCoordinateVectorSpace |
In[3]:=
setXYCoordinates;
[];setUVWCoordinates;
[];
| GrassmannSymbolsPalette |
| GrassmannSymbolsPalette |
Write a function to map points from space to space and immediately obtain the pullback rules for the coordinates.
{x,y}
{u,v,w}
{u,v}
In[4]:=
F:={x,y}{xy,x+y,x-y}uvwPullback=Thread[{u,v,w}F[x,y]]
Out[4]=
{uxy,vx+y,wx-y}
In[5]:=
setXYCoordinates;xyBasisPushforward=
[F,{,},{,,}]
| BasisPushforward |
e
x
e
y
e
u
e
v
e
w
Out[5]=
{y++,x+-}
e
x
e
u
e
v
e
w
e
y
e
u
e
v
e
w
These rules map the basis vectors at to the vectors at . The vectors are parametrized in terms of and , not in terms of , and . Of course, we know what and are so we can write the vectors, but they only exist on the parametrized surface in . The easy doesn't help us. It goes the wrong way.
{,}
e
x
e
y
{x,y}
{y++,x+-}
e
u
e
v
e
w
e
u
e
v
e
w
F[x,y]{xy,x+y,x-y}
x
y
u
v
w
x
y
3
uvwPullback
Now let's set a diffeomorphic mapping . We'll pick an easy case. Instead of defining a , we use Equal to obtain .
2
2
uvPullback
uvEquations
In[6]:=
setUVCoordinates:=
[2,{u,v},"Vector"]F:={x,y}{+y,x-2y}uvEquations=Thread[{u,v}F[x,y]]
| SetCoordinateVectorSpace |
2
x
Out[6]=
{u+y,vx-2y}
2
x
We can solve these equations to obtain an .
xyPullback
In[7]:=
xyPullback=Solve[uvEquations,{x,y}]〚1〛
Out[7]=
x(-1-(-1-4v-
1
4
1+16u+8v
),y1
8
1+16u+8v
)In[8]:=
setXYCoordinates;xyBasisPushforward=
[F,{,},{,},xyPullback]
| BasisPushforward |
e
x
e
y
e
u
e
v
Out[8]=
(-1-+,-2
e
x
1
2
1+16u+8v
)e
u
e
v
e
y
e
u
e
v
Now we can push forward a vector field from space to space.
{x,y}
{u,v}
In[9]:=
y-x%/.xyBasisPushforward/.xyPullback//Collect[#,{,},Simplify]&
e
x
e
y
e
u
e
v
Out[9]=
y-x
e
x
e
y
Out[9]=
1
16
1+16u+8v
)(5+4v+1+16u+8v
)e
u
1
8
1+16u+8v
))e
v
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""