Let S be the two-dimensional plane x = 0 in three-dimensional Euclidean space. Let ( 0 be a normal one-form to S.
(a) Show that if is a vector which is not tangent to S, then () ( 0.
(b) Show that if () > 0, then () > for any , which points toward the same side of S as does (i, e. any whose x components has the same sign as Vx).
(c) Show that any normal to S is a multiple of .
(d) Generalize these statements to an arbitrary three-dimensional surface in four dimensional space time?