§ 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 dφ/ds = ([part]φ/[part]x_μ) ([part]x_μ/[part]s)
so that ψ = ([part]φ/[part]x_μ) · (dx_μ/ds) is also an invariant for all curves which go out from a point in the continuum, i.e., for any choice of the vector dx_μ. From which follows immediately that
A_μ = [part]φ/[part]x_μ
is a co-variant four-vector (gradient of φ).
According to our law, the differential-quotient χ = [part]ψ/[part]s taken along any curve is likewise an invariant.
Substituting the value of ψ, we get
χ = ([part]^2φ/([part]x_μ [part]x_ν)) · (dx_μ/ds) · (dx_ν/ds) + ([part]φ/[part]x_μ) · (d^2x_μ/ds^2).
