What is mean of $\dot{\gamma}^j(t)$?

Question

In picture below ,I know $\dot{\gamma}(t)=\frac{\partial \gamma}{\partial t}$, but , what is mean of $\dot{\gamma}^j(t)$? $\gamma(t)$ is a path on Riemannian manifold.

Answer 1

That formula is written in local coordinates, so if the coordinates are $(x^1, \ldots, x^n):M\rightarrow U \subseteq \mathbb R^n$ then $$ \gamma^j(t) := x^j(\gamma(t))$$ and $$ \dot \gamma^j(t) := \frac{d}{dt}\gamma^j(t)$$

Answer 2

I read $\dot{\gamma}^j$ as the $j$-th contravariant component of the time derivative of $\gamma$.

Answer 3

Do you mean $\dot{\gamma}^j(t)$ ? Here $\gamma^j$ means the $j^\textrm{th}$ component of $\gamma$ you take the derivative of that and it is a function of time.

Answer 4

Generally you will have a chart $\psi:U\to\mathbb{R}^n$ such that $\gamma(t)\in U$. Then $\dot\gamma^j(t)=(\partial/\partial t)\psi^j(\gamma(t))$.