Patch

A patch (also called a Local Surface) is a differentiable mapping ${\bf x}:U\to\Bbb{R}^n$, where $U$ is an open subset of $\Bbb{R}^2$. More generally, if $A$ is any Subset of $\Bbb{R}^2$, then a map ${\bf x}: A\to \Bbb{R}^n$ is a patch provided that ${\bf x}$ can be extended to a differentiable map from $U$ into $\Bbb{R}^n$, where $U$ is an open set containing $A$. Here, ${\bf x}(U)$ (or more generally, ${\bf x}(A)$) is called the Trace of x.

See also Gauss Map, Injective Patch, Monge Patch, Regular Patch, Trace (Map)

References

Gray, A. ``Patches in $\Bbb{R}^3$.'' §10.2 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 183-184 and 192-193, 1993.