I'm confused on what I think might be a definition. From "The wave equation on a curved space-time" by Friedlander:
Let $(\Omega,\pi)$ be the coordinate chart at $p$. The coordinate curves are the images, under $\pi^{-1}$,, of the coordinate lines in $\mathbb{R}$ at $\pi p$, $$x^i = (\pi p)^{i} +\delta^{i}_j t \quad (i,j =1,\ldots, n)$$ I'm not sure where the $t$ term is coming from.
Also if I can squeeze in another question (I know this is "wrong").
My same book states:
$$du(p)=\frac{d}{dt}(u\circ f(t))_{t=0}$$ then says $du(p)=\langle \nabla u(p),v\rangle$ and that because $du(p)=\frac{d}{dt}(u\circ \pi^{-1})\circ (\pi \circ f)_{t=0}$ that it follows that $du=\langle u(p),v\rangle =\xi^j \frac{\partial }{\partial x^j} u\circ \pi^{-1}(x)$. I don't see why they write $du$ rather than $du(p)$ and I'm not seeing where the final sum comes from. In both questions, I'm lost in the notation. Thank you for any help.