I am trying to understand Theorem 9.1 of 1991 copy of Munkres' Analysis on Manifolds. I have stated what I don't understand below; there is a heading in bold. This theorem is a precursor to the implicit function theorem and on my copy of the book is on page 73.
Now on page 72 he states the following definition:
Let $A$ be open in $\Bbb{R}^m$; let $f : A \rightarrow \Bbb{R}^n$ be differentiable. Let $f_1,\ldots,f_n$ be the component functions of $f$. We sometimes use the notation $Df = \frac{\partial(f_1,\ldots,f_n)}{\partial(x_1,\ldots,x_m)}$ for the derivative of $f$. On occasion we shorten this to the notation $Df = \partial f /\partial \Bbb{x}$.
This is all good, so now on to theorem 9.1 (which is where my confusion lies).
Theorem 9.1: Let $A$ be open in $\Bbb{R}^{k+n}$; let $f : A \rightarrow \Bbb{R}^n $ be differentiable. Write $f$ in the form $f(\Bbb{x},\Bbb{y})$, for $\Bbb{x} \in \Bbb{R}^k$ and $\Bbb{y} \in \Bbb{R}^n$; then $Df$ has the form $Df = \Big[ \partial f/\partial \Bbb{x} \hspace{5mm} \partial f / \partial \Bbb{y}\Big].$ Suppose there is a differentiable function $g : B \rightarrow \Bbb{R}^n$ defined on an open set $B$ in $\Bbb{R}^k$, such that $f(\Bbb{x},g(\Bbb{x})) = 0$ for all $\Bbb{x} \in B$. Then for $\Bbb{x} \in B$, $ \frac{\partial f}{\partial \Bbb{x}}(\Bbb{x},g(\Bbb{x})) + \frac{\partial f}{\partial \Bbb{y}}(\Bbb{x},g(\Bbb{x}))\cdot Dg(\Bbb{x}) = 0.$
The dot just before $Dg(\Bbb{x})$ means matrix multiplication.
Now the proof of this goes as follows, given $g$, we can define $h : B \rightarrow \Bbb{R}^{k+n}$ by the equation
$h(\Bbb{x}) = (\Bbb{x},g(\Bbb{x})).$
The hypotheses of the theorem then imply that the composite function $f(h(\Bbb{X})) = f(\Bbb{x},g(\Bbb{x}))$ is defined and equals zero for all $\Bbb{x} \in B$. The chain rule then implies that
$\begin{eqnarray*} 0 &=& Df(h(\Bbb{x}))\cdot Dh(\Bbb{x})\\ &=& \Big[\frac{\partial f}{\partial \Bbb{x}}(h(\Bbb{x})) \hspace{4mm} \frac{\partial f}{\partial \Bbb{y}}(h(\Bbb{x})) \Big] \cdot \left[\begin{array}{c} I_k \\ Dg(\Bbb{x}) \end{array}\right] \end{eqnarray*}.$
What I don't understand: In the last row above, I get the second matrix on the right hand side, the one involving the identity matrix. However for the first matrix, I can see the notation means that it is formed by concatenating two matrices together, one from $\frac{\partial f}{\partial \Bbb{x}}(h(\Bbb{x}))$ and the other from $\frac{\partial f}{\partial \Bbb{y}}(h(\Bbb{x}))$. My problem now is I don't even no what these matrices look like.
I have tried several ways to interpret them, but keep getting tied up. Also, for the second matrix on the right it is of dimensions
$(n + k) \times k$
yes? But if this were so, then how can $Df(h(\Bbb{x}))$ be a map from $\Bbb{R}^{k+n}$ to $\Bbb{R}^n$?
Thanks.