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\section*{implicit function theorem}

\hypertarget{implicit_function_theorems}{}\section*{{Implicit function theorems}}\label{implicit_function_theorems}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{in_}{In $\mathbf{R}^n$}\dotfill \pageref*{in_} \linebreak
\noindent\hyperlink{local_statement_on_manifolds}{Local statement on manifolds}\dotfill \pageref*{local_statement_on_manifolds} \linebreak
\noindent\hyperlink{global_statement_on_manifolds}{Global statement on manifolds}\dotfill \pageref*{global_statement_on_manifolds} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Implicit function theorems give sufficient conditions for the existence of a [[differentiable map|differentiable]] [[inverse]] of a [[germ]] $f_p$ of a [[differentiable map]] $f\colon M \to N$ of [[smooth manifold]]s at a point $p$. The invertibility is trivially equivalent to the statement that the germ is a [[local diffeomorphism]] of some [[neighborhood]] of $p$ to some neighborhood of $f(p)$. If it is invertible, then we can consider the [[tangent map]] $T_p f\colon T_p M \to T_{f(p)}N$. If $f$ is locally invertible with differentiable inverse, then for all $y$ in some neighborhood of $y$ the [[functor]]iality of $T$ implies that $Id_{T_y} = T_y (f^{-1} \circ f) = T_{f(y)} f^{-1} \circ T_y f$ and alike for $f \circ f^{-1}$ at $f(y)$, demonstrating that $T_y f$ must then be invertible. The [[inverse function theorem]] says that the invertibility of $T_p f$ is in fact sufficient for the invertibility of the germ, which is then automatically differentiable.

\hypertarget{in_}{}\subsection*{{In $\mathbf{R}^n$}}\label{in_}

Let $U \subset \mathbf{R}^n$ be an [[open set]] in a [[cartesian space]], $a\in U$, $f\colon U \to \mathbf{R}^n$ a [[continuously differentiable map|map]] of class $C^1$ and $det\left(\frac{\partial f_i}{\partial x_j}(a)\right)\neq 0$. Then there are open sets $V \ni a$, $W \ni f(a)$, $V \subset U$ such that $f|_V\colon V \to W$ is a [[diffeomorphism]] and for all $y \in W$ and $(f^{-1})'(y) = (f'[f^{-1}(y)])^{-1}$.

\hypertarget{local_statement_on_manifolds}{}\subsection*{{Local statement on manifolds}}\label{local_statement_on_manifolds}

This is the theorem stated in the Idea section; the differentiable germ is assumed to be of class $C^1$ ([[continuously differentiable map|continuously differentiable]]). The statement is local, so one can consider it in [[chart]]s, hence the proof reduces to the case of $\mathbf{R}^n$.

\hypertarget{global_statement_on_manifolds}{}\subsection*{{Global statement on manifolds}}\label{global_statement_on_manifolds}

Let $f\colon M \to N$ be a [[smooth map]] of smooth manifolds. A point $q \in N$ is a \textbf{regular value} of $f$ if for every point $p \in f^{-1}(q)$ the [[differential]] $T_p f\colon T_p M \to T_q N$ is an [[epimorphism]]. The implicit function theorem asserts that $Q = f^{-1}(p)$ is a smooth [[submanifold]] of $M$ and the [[tangent bundle]] $T M|_N$ globally splits as $T M|_N \cong T N \oplus \mathbf{R}^n$ where $n = dim N$.

More generally, if $W \subset N$ is a submanifold, we say that the map $f$ is \textbf{transversal along} $W$ if for every point $x\in f^{-1}(W)$ there is an equality

\begin{displaymath}
T_{f(x)} N = T_{f(x)}W + (T_x f)(T_x X)
.
\end{displaymath}
In particular, $f$ is transveral along every regular value $p \in N$. The implicit function theorem asserts that the [[preimage]] $f^{-1}(W)$ is a smooth submanifold of $M$, the [[normal bundle]] $\nu(f^{-1}(W) \subset M)$ is isomorphic to $f^*(\nu(W\subset N))$, and the differential $T f$ exhibits the fiberwise isomorphism $\nu(f^{-1}(W)\subset M)\to \nu(W\subset N)$.

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item L. H. Loomis, [[S. Sternberg]], \emph{Advanced calculus}, 1968, 1990 (3.11 in 1990 edition)


\item S. Lang, \emph{Analysis I}



\end{itemize}
Various applications and related theorems can be found in chapter 5: \emph{Local and tangential properties} of

\begin{itemize}%
\item T. Br\"o{}cker, K. J\"a{}nich, C. B. Thomas, M. J. Thomas, \emph{Introduction to differentiable topology}, 1982 (translated from German 1973 edition; $\exists$ also 1990 German 2nd edition)

\end{itemize}
An invariant global statement on manifolds is at page 44 of

\begin{itemize}%
\item . . ,     , Moscow, Nauka 1984

\end{itemize}
Elementary course notes of the case in $\mathbf{R}^n$ (mainly lots of examples):

\begin{itemize}%
\item Frank Jones, \emph{Implicit function theorem}, \href{http://www.owlnet.rice.edu/~fjones/chap6.pdf}{pdf}

\end{itemize}
[[!redirects implicit function theorem]] [[!redirects implicit function theorems]]



\end{document}