or countable union of smooth -manifolds, each of which has the domain on only one side. He then proves that if is a standard domain in , is an -form which is defined, continuous, and bounded on , smooth on , integrable on , and such that is integrable on , then Stokes' theorem holds, that is, The study of measure-theoretic properties of rough sets leads to geometric measure theory. Even more general versions of Stokes' theorem have been proved by Federer and by Harrison. Special cases The general form of the Stokes theorem using differential forms is more powerful and easier to use than the special cases. The traditional versions can be formulated using Cartesian coordinates without the machinery of differential geometry, and thus are more accessible. Further, they are older and their names are more familiar as a result. The traditional forms are often considered more convenient by practicing scientists and engineers but the non-naturalness of the traditional formulation becomes apparent when using other coordinate systems, even familiar ones like spherical or cylindrical coordinates. There is potential for confusion in the way names are applied, and the use of dual formulations. Classical (vector calculus) case This is a (dualized) (1 + 1)-dimensional case, for a 1-form (dualized because it is a statement about vector fields). This special case is often just referred to as Stokes' theorem in many introductory university vector calculus courses and is used in physics and engineering. It is also sometimes known as the curl theorem. The classical Stokes' theorem relates the surface integral of the curl of a vector field over a surface in Euclidean three-space to the line integral of the vector field over its boundary. It is a special case of the general Stokes theorem (with ) once we identify a vector field with a 1-form using the metric on Euclidean 3-space. The curve of the line integral, , must have positive orientation, meaning that points counterclockwise when the surface normal, , points toward the viewer. One consequence of this theorem is that the field lines of a vector field with zero curl cannot be closed contours. The formula can be rewritten as: Green's theorem Green's theorem is immediately recognizable as the third integrand of both sides in the integral in terms of , , and cited above. In electromagnetism Two of the four Maxwell equations involve curls of 3-D vector fields,