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
and
We can represent this by a transformation matrix
:
where
Any vector components must transform in the same way.
For any scalar field 
, we can define a one- form:
We have
and
This transformation can be represented by another matrix
:
where
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
and
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:
and
Since
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
