The euclidean levi civita connection

Every ${\mathrm{\partial }}_k{g}_{ij}$ is zero, so [F1] gives ${\mathrm{\Gamma }}^k{}_{ij}=0$ for every index. Expanding $Y={\sum }_j{Y}^j{\mathrm{\partial }}_j$ by [F4] gives the claimed derivative. For example in two dimensions, $X=x{\mathrm{\partial }}_x+y{\mathrm{\partial }}_y$ and $Y=x^2{\mathrm{\partial }}_x+xy{\mathrm{\partial }}_y$ give ${\mathrm{\nabla }}_{X}Y=2x^2{\mathrm{\partial }}_x+2xy{\mathrm{\partial }}_y$.

By [F2] the parallel equation is $v^{\mathrm{\prime }}=0$. The section with constant Cartesian coefficients solves that equation on every smooth segment and is continuous across the finitely many corners. Uniqueness in [F3] identifies it as the transported section, so transport sends ${\sum }_j{v}^j{\mathrm{\partial }}_j{\mathrm{\mid }}_p$ to ${\sum }_j{v}^j{\mathrm{\partial }}_j{\mathrm{\mid }}_q$, including constant curves and singleton intervals. Zero components stay zero. For $n=0$ this is the unique map of zero tangent spaces, and $n=1$ is the ordinary scalar derivative.