SamplePublisher`GrassmannCalculus`
LieDerivative  |
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Details and Options
Examples
(2)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
In[2]:=
SetActiveAssociation
"Grassmann Plane"
 | PublicGrassmannAtlas |
Lie derivative of a function along a field. This is the same as the directional derivative at a general position.
In[3]:=
VField=;SField=x;[VField,★+x+y][SField]
e
x
2
y
ℒ
VField
[SField]
∇
e
x
e
y
Out[3]=
2
y
Out[3]=
2
y
Lie derivative of one vector field along another field. The Lie derivative is anticommutative in the vector field arguments.
In[4]:=
VField=;WField=x;
e
x
e
y
ℒ
VField
[WField]
ℒ
WField
[VField]
Out[4]=
e
y
Out[4]=
-
e
y
From R.W.R. Darling Differential Forms and Connections p 27-28, The vector field associated with linear flow in the y direction through a gravitational field.
In[5]:=
SetEuclideanNSpace[3,{x,y,z},"Vector"]
In[6]:=
VField=-(x+y+z);WField=;
e
x
e
y
e
z
3/2
(++)
2
x
2
y
2
z
e
y
ℒ
VField
[WField]
Out[6]=
-+-
3xy
e
x
5/2
(++)
2
x
2
y
2
z
(-2+)
2
x
2
y
2
z
e
y
5/2
(++)
2
x
2
y
2
z
3yz
e
z
5/2
(++)
2
x
2
y
2
z
From John M. Lee Introduction to Smooth Manifolds Example 8.27 p 187.
In[7]:=
VField=x++x(y+1);WField=+y;
e
x
e
y
e
z
e
x
e
z
ℒ
VField
[WField]
Out[7]=
--y
e
x
e
z
Properties of the Lie Derivative
(1)
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