Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 6; Problem 6.2.
One example of a covariant vector is the gradient. As an example, suppose we have a 2-d scalar field given by . In rectangular coordinates
In polar coordinates
Note that because we have absorbed the factor of needed for an incremental displacement in the direction into the basis vector , there is no extra factor of in the term, as there would be if we had used unit basis vectors.
Now suppose we have a vector with components given in rectangular coordinates. Then the scalar product is
If we convert to polar coords, then
The scalar product now is
Thus the scalar product is invariant.