Injective Patch

An injective patch is a patch such that x(u_1,v_1)=x(u_2,v_2) implies that u_1=u_2 and v_1=v_2. An example of a patch which is injective but not regular is the function defined by (u^3,v^3,uv) for u,v in (-1,1). However, if x:U->R^n is an injective regular patch, then x maps U diffeomorphically onto x(U).

See also

Patch, Regular Patch

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

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Injective Patch

Cite this as:

Weisstein, Eric W. "Injective Patch." From MathWorld--A Wolfram Web Resource.

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