PlanetMath: sense-preserving mapping

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sense-preserving mapping

(Definition)

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 $J = \vert f_z\vert - \vert f_{\bar{z}}\vert$ , and so a mapping is sense preserving if the modulus of the partial derivative with respect to is strictly greater then the modulus of the partial derivative with respect to $\bar{z}$ .

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

"sense-preserving mapping" is owned by jirka.

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See Also: orientation, Jacobian, curve

Other names:

orientation-preserving

Also defines:

sense-preserving, sense-reversing

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Cross-references: interval, complex plane, partial derivative, modulus, smooth mapping, conformal mapping, domain, point, positive, strictly, Jacobian, equivalent, differentiable, mapping, Jordan curve, orientation, preserves, continuous mapping
There are 4 references to this entry.

This is version 2 of sense-preserving mapping, born on 2004-02-05, modified 2004-02-10.
Object id is 5546, canonical name is SensePreservingMapping.
Accessed 4399 times total.

Classification:

AMS MSC:	30A99 (Functions of a complex variable :: General properties :: Miscellaneous)
	26B05 (Real functions :: Functions of several variables :: Continuity and differentiation questions)

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