A Riemmanian manifold $\mathcal{X}$ contains, for certain coordinate chart $(U, \varphi)$, the (local) coordinate map $\varphi$ that maps a neighborhood of coordinate $p$ to an Euclidean space $\tilde{U}$, which represents its tangential space $T_p \mathcal{X}$. My question regards the derivative of such a map.

In case of a submerged surface $\mathcal{X}$ in $\mathbb{R}^3$, the map $\varphi$ may be the spherical coordinates, for example. I know that, for certain curve $c: \mathbb{R} \to \mathcal{X}$, the first derivative is related by the jacobian matrix, i.e., $\dot{c} = J_\varphi \dot{u}$.

Is there some concise manner to represent $\ddot{c}$ rather than expression $\dot{J}_\varphi \dot{u} + J_\varphi \ddot{u}$? In a more direct way, how can we dry $\frac{d}{dt}J_\varphi(u(t))$ out?



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