Coordinate geodesic equation

Along the chart segment, ${\gamma }^{\mathrm{\prime }}={\dot{x}}^j{\mathrm{\partial }}_j$. The connection product rule and [F2] give $D_t{\gamma }^{\mathrm{\prime }}={\ddot{x}}^k{\mathrm{\partial }}_k+{\dot{x}}^j{\dot{x}}^i{\mathrm{\nabla }}_{{\mathrm{\partial }}_i}{\mathrm{\partial }}_j=({\ddot{x}}^k+{\mathrm{\Gamma }}^k{}_{ij}(x){\dot{x}}^i{\dot{x}}^j){\mathrm{\partial }}_k\mathrm{.}$

The coordinate vectors are a basis at every point, so the vector in step 1.1 vanishes if and only if every displayed coefficient vanishes. By [F1], these two conditions are respectively equivalent to the intrinsic geodesic equation, proving both directions. For a constant curve all first and second derivatives vanish. In dimension zero both lists of equations are empty and both conditions hold; in dimension one the formula is $\ddot{x}+{\mathrm{\Gamma }}^1{}_{11}(x){\dot{x}}^2=0$. Included endpoints use one-sided derivatives, chart seams are handled on overlapping subintervals by the intrinsic equation, and no choices are made.