I am reading Analysis on Manifolds by Munkres, and have a question about the proof about the Implicit Function Theorem (both the statement and proof included below):
- Note (3rd paragraph of the proof) how Munkres chooses $U \times V$ as a neighborhood of $(a,b) \in \mathbb{R}^{k+n}$. I know this can be done by restricting the open set guaranteed to exist by the Inverse Function Theorem, but I don't see why we want it to be a Cartesian product.
- Regarding uniqueness (last paragraph of the proof), why is the argument provided necessary? It seems unnecessarily complicated. Here is how I reasoned it: say $(x,g(x)) \in U \times V$ s.t. $f(x,g(x))= \textbf{0}_n$. Then $F(x,g(x))=(x,\textbf{0}_n)$, so $$(x,g(x))=G(x,\textbf{0}_n)=(x,h(x,\textbf{0}_n)).$$ (Here $G=F^{-1}$ and $h$ is the last $n$ coordinate functions of $G$, following Munkres' notation). By inspection, $g(x)=h(x,\textbf{0}_n)$, hence uniqueness is shown because we just derived what $g(x)$ has to be.
Many thanks in advance.