SamplePublisher`GrassmannCalculus`
GrassmannLaplacian  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Set Cartesian coordinates and define scalar patterns.
In[2]:=
 | SetCoordinateVectorSpace |
| ★★S |
Laplacian of a generic function.
In[3]:=
| GrassmannLaplacian |
Out[3]=
(0,0,2)
f
(0,2,0)
f
(2,0,0)
f
Laplacian of the square of the distance from the origin.
In[4]:=
| GrassmannLaplacian |
2
x
2
y
2
z
Out[4]=
6
In cylindrical coordinates:
In[5]:=
| SetActiveSpacePreferences |
 | PublicGrassmannAtlas |
| ★★S |
The Laplacian of a generic function compared with text values.
In[6]:=
step1=
[f[ρ,φ,ζ]]
| GrassmannLaplacian |
Out[6]=
(0,0,2)
f
(0,2,0)
f
2
ρ
(1,0,0)
f
ρ
(2,0,0)
f
In[7]:=
InactivateD[ρD[f[ρ,φ,ζ],ρ],ρ]+D[f[ρ,φ,ζ],{φ,2}]+D[f[ρ,φ,ζ],{ζ,2}],DActivate[%,D]//Simplify%step1
1
ρ
1
2
ρ
Out[7]=
∂
{ζ,2}
∂
{φ,2}
2
ρ
∂
ρ
∂
ρ
ρ
Out[7]=
(0,0,2)
f
(0,2,0)
f
2
ρ
(1,0,0)
f
ρ
(2,0,0)
f
Out[7]=
True
Laplacian of the square of the distance from the origin.
In[8]:=
| GrassmannLaplacian |
2
ρ
2
ζ
Out[8]=
6
In spherical coordinates:
In[9]:=
| SetActiveSpacePreferences |
| PublicGrassmannAtlas |
| ★★S |
The Laplacian of a generic function compared with text values.
In[10]:=
step1=
[f[r,θ,φ]]
| GrassmannLaplacian |
Out[10]=
1
2
r
2
Csc[θ]
(0,0,2)
f
(0,1,0)
f
(0,2,0)
f
(1,0,0)
f
2
r
(2,0,0)
f
In[11]:=
InactivateD[D[f[r,θ,φ],r],r]+Sin[θ]D[Sin[θ]D[f[r,θ,φ],θ],θ]+D[f[r,θ,φ],{φ,2}],DActivate[%,D]//Simplify%step1
1
2
r
2
r
1
2
r
1
2
r
2
Sin[θ]
Out[11]=
2
Csc[θ]
∂
{φ,2}
2
r
∂
r
2
r
∂
r
2
r
Csc[θ](Sin[θ]f[r,θ,φ])
∂
θ
∂
θ
2
r
Out[11]=
1
2
r
2
Csc[θ]
(0,0,2)
f
(0,1,0)
f
(0,2,0)
f
(1,0,0)
f
2
r
(2,0,0)
f
Out[11]=
True
Laplacian of the square of the distance from the origin.
In[12]:=
| GrassmannLaplacian |
2
r
Out[12]=
6
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""