Atlas Paclet Symbol

L - Lie derivative

L_{v1, v2, v3..., vn}[expr]

L[v1, v2, v3..., vn, expr]
allows one to calculate the Lie derivative on an expression along given vector field

MORE INFORMATION

expr - any expression (on which Lie derivative is defined). v1, v2, v3..., vn - vector fields.

The derivative has the following properties:

- For any vector fields X₁, X₂, .., X_n and expression a we have: L_{X₁, X₂, .., X_n}(a)=L_X₁(L_X₂(..L_{X_n}(a)))

- For any vector field X and 0-form f we have: L_x(f)=_X(d(f))

- For vector fields X and Y we have: L_x(Y)=[X, Y]

- For any vector field X and tensor fields and T the Leibniz rule for the Lie derivative takes place: L_X(T)=(L_X())T+(L_X(T))

- For any vector field X and p-form we have: L_X()=_X(d())+d(_X())

Declare constants:

Declare functions:

Declare p-forms:

Declare vectors:

Declare tensors:

Using L- procedure:

Just definition for "long" Lie operator:

As h is 0-form by defaults then:

In[10]:=

Out[10]=

F- declared as function F=F(x,y) thus:

In[11]:=

Out[11]=

As declared as constant thus:

In[12]:=

Out[12]=

Lie derivative is linear with respect to any argument:

And:

More complex examples:

And:

- declared as p-form thus:

In[17]:=

Out[17]=

Verify that Leibniz rule takes place for tensor product:

In[18]:=

Out[18]=

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L - Lie derivative

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