30A99-SensepreservingMapping.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{SensepreservingMapping}
\pmcreated{2013-03-22 14:08:01}
\pmmodified{2013-03-22 14:08:01}
\pmowner{jirka}{4157}
\pmmodifier{jirka}{4157}
\pmtitle{sense-preserving mapping}
\pmrecord{5}{35546}
\pmprivacy{1}
\pmauthor{jirka}{4157}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{30A99}
\pmclassification{msc}{26B05}
\pmsynonym{orientation-preserving}{SensepreservingMapping}
\pmrelated{Orientation}
\pmrelated{Jacobian}
\pmrelated{Curve}
\pmdefines{sense-preserving}
\pmdefines{sense-reversing}

\endmetadata

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\begin{document}
A continuous mapping which preserves the orientation of a Jordan curve is called sense-preserving or orientation-preserving.  If on the other hand a mapping reverses the orientation, it is called sense-reversing.

If the mapping is furthermore differentiable then the above statement is equivalent to saying that the Jacobian is strictly positive at every point of the domain.

An example of sense-preserving mapping is any conformal mapping $f : {\mathbb{C}} \rightarrow {\mathbb{C}}$.  If you however look at the mapping $g(z) := f(\bar{z})$, then that is a sense-reversing mapping.  In general if $f : {\mathbb{C}} \rightarrow {\mathbb{C}}$ is a smooth mapping then the Jacobian in fact is defined as $J = |f_z| - |f_{\bar{z}}|$, and so a mapping is sense preserving if the modulus of the partial derivative with respect to $z$ is strictly greater then the modulus of the partial derivative with respect to $\bar{z}$.

This does not \PMlinkescapetext{mean} that this notion is \PMlinkescapetext{restricted} to the complex plane.  For example $f : {\mathbb{R}} \rightarrow {\mathbb{R}}$ defined by $f(x) = 2x$ is a sense preserving mapping, while $f(x) = x^2$ is sense preserving only on the
interval $(0,\infty)$.
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