Let $D$ be a plane region (connected open set). A surface is a continuously-differentiable mapping $\mathbf{x}$ from $D$ into Euclidean three-space. We write a surface generically as \[ \mathbf{x}(u, v) = (x(u, v), y(u, v), z(u, v)). \] The $u$-coordinate curves (purple) are obtained by holding $v$ constant. The $v$-coordinate curves (gold) are obtained by holding $u$ constant.
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