DOI link for Regular Surfaces
Regular Surfaces book
This chapter shows why one requires more technical definitions to arrive at a workable definition that matches what one typically means by a "surface". A cube, for example, is not a regular surface because for whatever parametrization is used in the neighborhood of an edge where two faces meet, at least one of the partial derivatives will not exist. A particular patch of a surface can be parametrized in a variety of ways. Comparing surfaces to curves, recall that at every regular point on a curve, the unit tangent vector to a curve at a point is invariant under a positively oriented reparametrization of the curve, and the tangent line to the curve at a point is an entirely geometric object, completely unchanged under reparametrizations. The concept of an orientable surface encapsulates the notion of being able to define an inside and an outside to the surface.