Definition of unit normal vector in do Carmo

Question

Let $S\subset R^3$ be a regular surface. Suppose that $\mathbf{x}:U\subset \mathbb{R}^2\to S$ is a parametrization of $S$ at $p\in S$. Then for every $q\in \mathbf{x}(U)$, do Carmo defines a unit normal vector by $$N(q) = \frac{\mathbf{x}_u \wedge \mathbf{x}_v}{\lvert \mathbf{x}_u \wedge \mathbf{x}_v\rvert}(q).$$ What I don't understand here is that $\mathbf{x}_u$ and $\mathbf{x}_v$ are defined on $U$, so the above expression isn't defined. Shouldn't it be $$N(q) = \frac{\mathbf{x}_u \wedge \mathbf{x}_v}{\lvert \mathbf{x}_u \wedge \mathbf{x}_v\rvert}(\mathbf{x}^{-1}(q))?$$