# sense-preserving mapping

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 that this notion is 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 $(0,\infty)$.

 Title sense-preserving mapping Canonical name SensepreservingMapping Date of creation 2013-03-22 14:08:01 Last modified on 2013-03-22 14:08:01 Owner jirka (4157) Last modified by jirka (4157) Numerical id 5 Author jirka (4157) Entry type Definition Classification msc 30A99 Classification msc 26B05 Synonym orientation-preserving Related topic Orientation Related topic Jacobian Related topic Curve Defines sense-preserving Defines sense-reversing