A mapping f of a surface into space is smooth provided it has sufficiently many partial derivatives, and that the mixed partials do not depend on the order of differentiation. Here "sufficiently many" depends on the context. Usually 2 is sufficient, but often smooth means infinitely many.

The basic idea is that the surface admits a differentiable structure, so that it is possible to do calculus on the surface. For most points, there will be a well-defined tangent plane, though this does not rule out self-intersection and some other degenerate behavior.

See also:

[More] Polyhedral surfaces

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8/13/94 -- The Geometry Center