The covariant derivative differs from partial derivatives even in flat spacetime if one uses non- Cartesian coordinates. This corresponds to going to a non inertial frame. To illustrate this we will focus on two dimensional Euclidian space with Cartesian coordinates ( , ) and polar coordinates ( , ). The coordinates are related by
For neighboring points we have
We can represent this by a transformation matrix :
Any vector components must transform in the same way.
For any scalar field , we can define a one- form:
This transformation can be represented by another matrix :
Any one- form components must transform in the same way.
The matrices and are different but related:
This is just what you would expect since in general .
The basis vectors and basis one- forms are
Note that the basis vectors change from point to point in polar coordinates and need not have unit length so they do not form an orthonormal basis:
The inverse metric tensor is:
so the components of the vector gradient of a scalar field are:
This is exactly what we would expect from our understanding of normal vector calculus.
We also have:
we can work out all the components of the Christoffel symbols:
and all other components are zero.
Alternatively, we can work out these components from the metric:
This is the best way of working out the components of , and it is the way we will adopt in General Relativity.
Finally we can check that all the components of as required. For example