Covariant derivative along a curve is independent of frame and extension

The connection ${\gamma }^{\ast }\mathrm{\nabla }$ is intrinsically well defined, so its evaluation on the globally specified vector field ${\mathrm{\partial }}_t$ is frame independent. Apply the ambient-section identity in [F2] with $f=\gamma$ and $X={\mathrm{\partial }}_t$; its differential is $\dot{\gamma }(t)$, giving the displayed formula.

If $s_1\circ \gamma =s_2\circ \gamma =V$ on a parameter neighbourhood, both displayed derivatives equal the same $D_{t}V$, proving independence of the extension. At an included endpoint, equality on the one-sided interval makes the derivative equality hold by the endpoint convention. For a constant curve at $p$, $V(t)=tv$ with $v\mathrm{\ne }0$ is a valid section with $D_{t}V=v$, but no ambient extension can have these varying values at $p$. Zero sections and rank zero cause no exception. No ambient extension is selected in defining $D_t$, so this statement uses no choice principle.