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.

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