Surface Integrals

 1 In a Nut Shell:  Two questions are relevant.  What is a surface integral and how do you evaluate a surface integral?   Recall the line integral, I, provides the value of a function, f(x,y,z), evaluated along a curve, C, in space.  Here the line integral is       I  =  ∫ f(x, y, z) ds                                                                                        C where   ds  is the arc length along the curve, C.   The surface integral, Is, is analogous to the line integral in that it provides the value of a function, f(x,y,z), evaluated over a “smooth” surface, S, in space.  Here the surface integral is  Is   =     ∫  ∫ f(x, y, z) dS                                                                                                               S where   dS  is the element of surface area on the spatial surface. 2 How do you evaluate a surface integral?   Usually the surface, S, in space is somewhat complicated.  So it is usually convenient to transform the element of surface, dS, from the x-y-z space into a parallelogram dA = du dv  in the u-v plane.            where  r(u,v) is the parametric representation of the surface, S.   r(u,v)  =  <  x(u,v), y(u,v), z(u,v) >   is the position vector to point on surface, S   Click here to continue with discussion of surface integrals.