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 differentiableMathworldPlanetmathPlanetmath 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 mappingMathworldPlanetmathPlanetmath f:. If you however look at the mapping g(z):=f(z¯), then that is a sense-reversing mapping. In general if f: is a smooth mapping then the Jacobian in fact is defined as J=|fz|-|fz¯|, and so a mapping is sense preserving if the modulus of the partial derivativeMathworldPlanetmath with respect to z is strictly greater then the modulus of the partial derivative with respect to z¯.

This does not that this notion is to the complex plane. For example f: defined by f(x)=2x is a sense preserving mapping, while f(x)=x2 is sense preserving only on the intervalMathworldPlanetmathPlanetmath (0,).

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