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Regular Patch -- from Wolfram MathWorld

TOPICS Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index New in MathWorld Regular Patch

A regular patch is a patch for which the Jacobian has rank 2 for all . A patch is said to be regular at a point provided that its Jacobian has rank 2 at . For example, the points at in the standard parameterization of the sphere are not regular.

An example of a patch which is regular but not injective is the cylinder defined parametrically by with and . However, if is an injective regular patch, then maps diffeomorphically onto .

See alsoInjective Patch

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Patch

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Regular Surface Explore with Wolfram|Alpha

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ReferencesGray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 273, 1997. Referenced on Wolfram|AlphaRegular Patch Cite this as:

Weisstein, Eric W. "Regular Patch." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RegularPatch.html

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