Here we have put, following Christoffel,
§ 10. Formation of Tensors through Differentiation.
Relying on the equation of the geodetic line, we can now easily deduce laws according to which new tensors can be formed from given tensors by differentiation. For this purpose, we would first establish the general co-variant differential equations. We achieve this through a repeated application of the following simple law. If a certain curve be given in our continuum whose points are characterised by the arc-distances s, measured from a fixed point on the curve, and if further φ, be an invariant space function, then dφ/ds is also an invariant. The proof follows from the fact that dφ as well as ds, are both invariants
Since
so that
is also an invariant for all curves which go out from a point in the continuum, i.e., for any choice of the vector dxμ. From which follows immediately that
Aμ = ∂φ/∂xμ