SamplePublisher`GrassmannCalculus`
GrassmannVectorLaplacian  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Set Cartesian coordinates.
In[2]:=
 | SetCoordinateVectorSpace |
| ★★S |
Vector Laplacian of generic vector.
In[3]:=
| GrassmannVectorLaplacian |
e
x
e
y
e
z
Out[3]=
e
x
e
z
(0,0,2)
g
(0,1,1)
h
e
x
e
y
(0,1,1)
g
(0,2,0)
h
e
z
(0,0,2)
h
(0,1,1)
g
(1,0,1)
f
2
e
z
(0,1,1)
f
(1,0,1)
g
e
y
e
z
(0,0,2)
f
(1,0,1)
h
e
y
(0,1,1)
h
(0,2,0)
g
(1,1,0)
f
e
y
e
z
(0,2,0)
f
(1,1,0)
g
2
e
y
(0,1,1)
f
(1,1,0)
h
2
e
x
(1,0,1)
g
(1,1,0)
h
e
x
(1,0,1)
h
(1,1,0)
g
(2,0,0)
f
e
x
e
z
(1,1,0)
f
(2,0,0)
g
e
x
e
y
(1,0,1)
f
(2,0,0)
h
Vector Laplacian of various vector fields.
In[4]:=
| GrassmannVectorLaplacian |
2
x
e
x
2
y
e
y
2
z
e
z
 | GrassmannVectorLaplacian |
2
(xyz)
e
x
e
y
e
z
 | OrthonormalBasis |
| GrassmannVectorLaplacian |
e
x
e
y
e
z
2
x
2
y
2
z
| OrthonormalBasis |
Out[4]=
2+2+2
e
x
e
y
e
z
Out[4]=
(2+2+6x)+(2+6y+2)+(6z+2+2)
3
x
2
y
3
x
2
z
2
y
2
z
e
x
2
x
3
y
2
x
2
z
3
y
2
z
e
y
2
x
2
y
2
x
3
z
2
y
3
z
e
z
Out[4]=
---
2x
e
x
2
(++)
2
x
2
y
2
z
2y
e
y
2
(++)
2
x
2
y
2
z
2z
e
z
2
(++)
2
x
2
y
2
z
In cylindrical coordinates.
In[5]:=
| SetActiveSpacePreferences |
| PublicGrassmannAtlas |
| ★★S |
In[6]:=
| GrassmannVectorLaplacian |
e
ρ
e
φ
e
ζ
Out[6]=
1
2
ρ
e
ρ
2
ρ
(0,0,2)
f
(0,1,0)
g
(0,2,0)
f
(1,0,0)
f
2
ρ
(2,0,0)
f
1
2
ρ
e
φ
2
ρ
(0,0,2)
g
(0,1,0)
f
(0,2,0)
g
(1,0,0)
g
2
ρ
(2,0,0)
g
e
ζ
(0,0,2)
h
(0,2,0)
h
2
ρ
(1,0,0)
h
ρ
(2,0,0)
h
In[7]:=
| GrassmannVectorLaplacian |
e
ρ
e
ζ
2
ρ
2
ζ
| OrthonormalBasis |
Out[7]=
--
2ζ
e
ζ
2
(+)
2
ζ
2
ρ
2ρ
e
ρ
2
(+)
2
ζ
2
ρ
In spherical coordinates.
In[8]:=
| SetActiveSpacePreferences |
| PublicGrassmannAtlas |
| ★★S |
In[9]:=
| GrassmannVectorLaplacian |
e
r
e
θ
e
φ
Out[9]=
1
2
r
e
r
(0,0,1)
h
2
Csc[θ]
(0,0,2)
f
(0,1,0)
f
(0,1,0)
g
(0,2,0)
f
(1,0,0)
f
2
r
(2,0,0)
f
1
2
r
e
θ
2
Csc[θ]
(0,0,1)
h
2
Csc[θ]
(0,0,2)
g
(0,1,0)
f
(0,1,0)
g
(0,2,0)
g
(1,0,0)
g
2
r
(2,0,0)
g
1
2
r
e
φ
2
Csc[θ]
(0,0,1)
f
(0,0,1)
g
2
Csc[θ]
(0,0,2)
h
(0,1,0)
h
(0,2,0)
h
(1,0,0)
h
2
r
(2,0,0)
h
In[10]:=
| GrassmannVectorLaplacian |
e
r
2
r
| OrthonormalBasis |
Out[10]=
-
2
e
r
3
r
 |
|
""