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Regular Patch

A regular patch is a patch x:U->R^n for which the Jacobian J(x)(u,v) has rank 2 for all (u,v) in U. A patch is said to be regular at a point (u_0,v_0) in U provided that its Jacobian has rank 2 at (u_0,v_0). For example, the points at phi=+/-pi/2 in the standard parameterization of the sphere (costhetasinphi,sinthetasinphi,cosphi) are not regular.

An example of a patch which is regular but not injective is the cylinder defined parametrically by (cosu,sinu,v) with u in (-infty,infty) and v in (-2,2). However, if x:U->R^n is an injective regular patch, then x maps U diffeomorphically onto x(U).

See also

Injective Patch, Patch, Regular Surface

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References

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 273, 1997.

Referenced on Wolfram|Alpha

Regular Patch

Cite this as:

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

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