Atlas Paclet Symbol

cov - covariant derivative

cov[v, expr]

cov_v[expr]
allows one to calculate the covariant derivative on an expression along given vector field.

MORE INFORMATION

expr - any expression. v - vector field.

The derivative has the following properties.

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

- For vector fields X, Y and Z we have: _X(Y+Z)=_X(Y)+_X(Z) and _X+Y for any functions f and h.

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

Declare functions:

Declare p-forms:

Declare vectors:

Declare tensors:

Using cov- procedure: As h is 0-form by defaults then:

In[6]:=

Out[6]=

f - declared as function f=f[x,y] thus:

Obviously that:

As declared as constant thus:

In[9]:=

Out[9]=

Lie derivative is linear with respect to any argument:

And:

More complex examples:

And:

Verify that Leibniz rule takes place for tensor product:

Declare constants:

Declare p-forms:

Declare vectors:

Declare coframe:

Declare frame:

Declare metric:

Calculate connection:

In[8]:=

Out[8]=

For frame vectors

_j:

In[9]:=

Out[9]=

As g is a metric then for any vector field X:

By definition:

For coframe 1-forms:

For exterior product:

For tensor product:

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cov - covariant derivative

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