# 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^{} $f:\u2102\to \u2102$. If you however look at the mapping $g(z):=f(\overline{z})$, then that is a sense-reversing mapping. In general if $f:\u2102\to \u2102$ 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 $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)=2x$ is a sense preserving mapping, while $f(x)={x}^{2}$ is sense preserving only on the
interval^{} $(0,\mathrm{\infty})$.

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 |