12:37 PM
I am reading the proof of the Implicit Function Theorem in Geometry and Topology by Bredon; I am having trouble with a few parts.
Here's the statement of the theorem: Let $g : \Bbb{R}^n \times \Bbb{R}^m \to \Bbb{R}^m$ be $C^1$ and let $\xi \in \Bbb{R}^n$, $\eta \in \Bbb{R}^m$ be given with $g(\xi, \eta) = 0$.
Assume that the differential of the composition $y \mapsto (\xi, y) \mapsto g(\xi, y)$ is onto at $\eta$. [This is equivalent to the statement that the Jacobian determinant $J(g_i,y_j) \neq 0$ at $(\xi, \eta)$]. Then there are numbers $a > 0$ and $b > 0$ such that there exists a unique function $\phi : A \to B$ (with $A,B$ as in Lemma 1.4), with $\phi (\xi ) = \eta$, such that $g(x,\phi (x)) = 0$ for all $x \in A$.
First, how is the differential of $y \mapsto (\xi, y) \mapsto g(\xi, y)$ defined? I ask for the general definition because later in the proof that want to compute the differential of $y \mapsto f(\xi , y)$ at $\eta$, where $f(x,y) = y - L^{-1}(g(x,y))$ and $L : \Bbb{R}^m \to \Bbb{R}^m$ is the linear map represented by the Jacobian(?) And what does it mean to say that it is onto at $\eta$?
