I am trying to make sense of the following statement:
Let $P: \Omega \subseteq \mathbb{R^2} \to \mathbb{R^3} $ be a parameterized surface. Let $S=Im(P)$ Suppose $P$ is injective. If $X$ is a vector field over $\Omega$,
$P_{*}(X)\circ P^{-1}: s \in S \to(P_*X)_{P^{-1}(s)} \in T_s S$ is a vector field over $S$
where $P_{*}$ is the tangent mapping of P, $T_s S$ is the tangent space of $S$ at $s$
I think the objects are not well-defined:
The composition $P_{*}(X)\circ P^{-1}$ makes no sense. $P_*(X)$ is a vector space over $\mathbb{R^3} $, so it is a mapping $ \mathbb{R^3} \to \mathbb{R^3} $ , while $P^{-1}: \mathbb{R^3} \to \mathbb{R^2} $, therefore I can't compose them
$(P_*X)_{P^{-1}(s)} \in T_s S$ makes no sense. It is supposed to mean the vector field $P_{*}(X)$ at point $P^{-1}(s)$ , but $P_{*}(X)$ is a vector field over $\mathbb{R^3} $, while $P^{-1}(s)\in \mathbb{R^2}$. I need a point in $\mathbb{R^3}$ where the field $P_{*}(X)$lives, not in $\mathbb{R^2}$ where the field $X$ lives.
What is going on here?