"Flip Tomato" ,

As I have understood this, I think that the story is
something like this: A connection "A" defines a "parallel
transport"(In some sense a parallel transport can be seen
as a prescription that given a parametrized curve and a
vector w on one end of the curve, assigns to this a unique
vector field along the curve (where of course, the vector
that the vector field assigns to the endpoint is w.). Now,
once we have this, we can parallel transport n basis basis
vectors along any curve , and express any vector field along
the curve in terms of this basis i.e., with components C_i,
then take the derivative of the components C_i(t) and that
vector field is the covariant derivative. Ugh... I think