What does it mean to "calculate in local coordinates" on a manifold? In differential geometry textbook one sometimes reads "calculating in local coordinates, we obtain..." What does this expression mean? Say, $M$ is a smooth manifold and $h$ is a function on $M$; what does it mean to calculate with $h$ in local coordinates?
 A: If $h$ is a function $h: M \longrightarrow \Bbb R$, let $p \in M$ and suppose we have a coordinate chart $(U, \varphi)$ about $p$. That means $U$ is an open set in $M$ containing $p$ and $\varphi: U \longrightarrow V$ is a homeomorphism from $U$ onto the open set $V \subset \Bbb R^m$ (where $m = \dim(M)$). Then by "$h$ in local coordinates" we mean the function
$$h \circ \varphi^{-1}: V \longrightarrow \Bbb R$$
from $V \subset \Bbb R^m$ to $\Bbb R$.
If $h$ is instead a map $h: M \longrightarrow N$, where $N$ is a smooth $n$-manifold, let $(W, \psi)$ be a coordinate chart about $h(p) \in N$, so that $\psi: W \longrightarrow Z$ is a homeomorphism where $Z \subset \Bbb R^n$ is open. Then in this case by "$h$ in local coordinates" we mean the function
$$\psi \circ h \circ \varphi^{-1}: V \longrightarrow Z$$
from $V \subset \Bbb R^m$ to $Z \subset \Bbb R^n$.
Essentially, doing something in "local coordinates" means using coordinate charts to put it in terms of subsets of $\Bbb R^n$ instead of subsets of the manifold.
