In do Carmo's Riemannian geometry, in ch. 0, sec. 4, ex. 4.2, he discusses the generalization of a 2-surface in $\mathbb{R}^3$ to a $k$-surface in $\mathbb{R}^n$, $k\leq n$ by defining a subset $M^k\subset \mathbb{R}^n$ as a regular surface of dimension $k$ if, for every $p \in M^k$ there exists a neighborhood $V$ of $p \in \mathbb{R}^n$ and a mapping $\textbf{x}:U\subset \mathbb{R}^k \rightarrow M\cap V$ of an open set $U\subset \mathbb{R}^k$ onto $M\cap V$ such that:
- $\textbf{x}$ is a diff'able homeomorphism
- $(d\textbf{x})_q:\mathbb{R}^k \rightarrow \mathbb{R}^n$ is injective for all $q\in U$.
Okay, all is fine and dandy, I can see exactly how this makes sense as a generalization of a regular surface in $\mathbb{R}^3$.
He goes firther and shows that if $\textbf{x}$ and $\textbf{y}$ are two parameterizations of $M$ from open sets $U,V \subset \mathbb{R}^k$, respectively, such that $\textbf{x}(U) \cap \textbf{y}(V) = W \neq \emptyset$ then the mapping $h=\textbf{x}^{-1}\circ\textbf{y}:\textbf{y}^{-1}(W)\rightarrow\textbf{x}^{-1}(W)$ is a diffeomorphism. During his little "proof" that $h$ is a diffeomorphism, he states that from condition (2) of $M$ being a regular surface, we can suppose that
$$\frac{\partial(v_1, ..., v_k)}{\partial(u_1,...,u_k)}(q) \neq 0.$$ where $(u_i, ..., u_k)\in U$, $(v_i, ..., v_k)\in \mathbb{R}^n$, and $q$ is the image of some point in the image $h$.
So my question is: what exactly is meant by $\frac{\partial(v_1, ..., v_k)}{\partial(u_1,...,u_k)}(q) \neq 0$? I have never seen this before, and am not quite sure what it means.