I'm confused about the notation used to represent the so called "local coordinates" when dealing with differential manifolds. As an example, I was given an exercise to prove that $S^{n-1}$ is a submanifold of dimension $n-1$ of $\mathbb{R^n}$, I did the following:
Let $F: \mathbb{R^n} \mapsto \mathbb{R}$ be given by $F((x^1)^2 + (x^2)^2 + ... + (x^n)^2 - 1)$. Then we have that $F^{-1}(0) = S^{n-1}$. Here the $x^i = x^i(p)$ are the coordinate functions of a point $p \in \mathbb{R^n}$. Now in order to show that $0$ is a regular value, I compute that the derivative (given by the Jacobian) of the map is surjective. The Jacobian is the row matrix: $$DF = (2x^1 \ 2x^2 ... \ 2x^n)$$.
Now from linear algebra I know that a linear a transformation $t$ is onto iff $Ker(t) = \{0\}$. In this case we have that the only possible way the following equation holds: $$(2x^1 \ 2x^2 ... \ 2x^n) \begin{bmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{n} \end{bmatrix} = 0$$ is if the $p = 0$. Thus I conclude that the linear transformation is surjective. Then by the regular value theorem we have that $S^{n-1}$ is a regular submanifold of $R^n$. Now main confusion arise when thinking about the coordinate functions in the Jacobian matrix. For example if I have the vector $\begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}$, does the Jacobian send this vector to $2x^1$ or to $2$, because if it's first case then how is that a real number? Also would it be more correct to write the Jacobian as $(2 \ 2 \ ... 2)$ or even $(2x^1(p) \ 2x^2(p) ... \ 2x^n (p))$? In other words what is the meaning of writing the coordinate functions in the Jacobian as above?