Lie Derivative in Coordinates¶
Here we will study the coordinate dependent realization of the Lie derivative. Let be a scalar (zero-form), a vector field, and be a one-form, where
form the basis that span the space of vector fields, and are the basis elements for the space of one-forms.
Using Cartan’s magic formula the Lie derivative is given algebraically by
In coordinate form the Lie derivative of a scalar, a one-form and a vector field become
To compute the Lie derivative of a one form we use
Combining the terms above we obtain the results for the Lie derivative of a one-form
The Lie derivative is a derivation and so it must satisfy a product rule over the pairing of forms and vector fields
To see that the above expressions for the Lie derivative do indeed satisfy Eq. (2)
After a simple relabeling of the dummy indices it can be shown that Eq. (2) is indeed satisfied.
Thus, in two dimensions it can be shown that
Given a two form
it can be shown that