From the definition of a covariant vector (covariant tensor of rank 1)
the corresponding transformation matrix is
In order to calculate the transformation matrix, we need the equations relating the two coordinates systems. For cartesian to polar, we have
and for polar to cartesian
So if we designate as the bar coordinates, then the transformation components from a polar basis vector to a cartesian basis vector is calculted as
The components of cartesian basis vectors to polar basis vectors transform the same way, but now the polar coordinates have the bar
In summary, the {\mathbf components of covariant basis vectors} in cartesian coordinates and polar coordinates transform between each other according to