There are surfaces which do not have a surface normal at each point with consistent results (for example, the Möbius strip). If such a surface is split into pieces, on each piece a parametrization and corresponding surface normal is chosen, and the pieces are put back together, the normal vectors coming from different pieces cannot be reconciled. This means that at some junction between two pieces will have normal vectors pointing in opposite directions. Such a surface is called non-orientable. Vector fields cannot be integrated on non-orientable surfaces.