From the definition of a contravariant vector (contravariant tensor of rank 1)
we get the transformation matrix from the partial derivatives
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 polar coordinates to cartesian coordinates is calculated as
The components from cartesian coordinates to polar coordinates transform the same way, but now the polar coordinates have the bar
In summary, the {\mathbf components of contravariant vectors} in cartesian coordinates and polar coordinates transform between each other according to