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 $f:ℂ\to ℂ$. If you however look at the mapping $g(z):=f(\overline{z})$, then that is a sense-reversing mapping. In general if $f:ℂ\to ℂ$ is a smooth mapping then the Jacobian in fact is defined as $J=|f_z|-|f_{\overline{z}}|$, 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 $\overline{z}$.

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

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