Surface Integrals  with “oriented surfaces”

 1 In a Nut Shell:  Surface integrals also appear in vector form           ∫  ∫  F  ·  dS        =    ∫  ∫  F  ·  n  dS                          Note:    dS  =  n dS          S                               S They involve the “dot product”  of a vector function,  F =  F(x,y, z) with  n   the unit normal to the surface, S.  In such cases the unit normal to the surface may point out (say the top of the surface) or may point in (say the bottom of the surface).  Such a surface, S, is said to be “orientable”.  The direction of the unit vector, n, establishes the orientation of the surface.   This situation is especially true for vector fields distributed over surfaces such as flux passing through a surface, S, bounded by a spatial curve, C.   Definitions:   F  =  vector field  (one example is fluid velocity giving rise to flux across a surface) dS   element of oriented surface S       where  dS  =  n dS n  =  unit vector normal to oriented surface   as before the unit vector comes from the cross product   n  =    (ru x rv) / | ru x rv |  where  r  is the position vector to the surface, S,  given by                           r  =  < x, y, z >  =  <  x(u,v), y(u,v), z(u,v) >  =  r(u,v) so               ∫  ∫  F  ·  dS   =  ∫  ∫  F  ·  n dS    =    ∫  ∫  F  ·  (ru x rv) / | ru x rv | dS                S                        S                                     S   and recall   dS  =  | ru x rv |dA     where | ru x rv | transforms the element of area, dS, on the surface, S, to the element of area, dA,  on the uv-surface in the domain, D.                                                                                So the surface integral becomes                                                       ∫  ∫  F  ·  dS        =    ∫  ∫  F  · (ru x rv) dA                             S                               D Note that the dot product,    F  ·  (ru x rv) ,  is a scalar function and therefore it is a scalar field.  So the same two approaches, the direct approach and the transformation approach, still apply to evaluate the scalar form of surface integrals starting with this vector form of surface integral.