Gauss's Divergence Theorem
                                                     1
                     Gauss's Divergence Theorem 
                                 ⇀
                                F(x,y,z)                                                                                            S
                          Let               be a vector field continuously differentiable in the solid,  .
                               S   a 3-D solid
                                 S                              S
                               ∂   the boundary of   (a surface)
                               n                                                         S
                               ˆ
                                    unit outer normal to the surface ∂
                                     ⇀                         ⇀
                               div F                          F
                                        divergence of 
                          Then                         ⇀
                                   ∂S                              S
                                                                                                                                                   2
                                                                                 S 
                  The rate of flow through a boundary of   =
                                                                                       ∂S
                  If there is net flow out of the closed surface, the integral is positive. 
                  If there is net flow into the closed surface, the integral is negative. 
                                                                  ⇀                                      ⇀
                                                                 F                                S "    F
                  This integral is called "flux of   across a surface ∂                               .     can be any vector 
                  field, not necessarily a velocity field.
                                                                                                       ⇀
                                                                                                       F                S
                  Gauss's Divergence Theorem tells us that the flux of   across ∂  can be 
                                                                                ⇀
                                                                                F                                                 S
                  found by integrating the divergence of   over the region enclosed by ∂ .
                                                                                                                                                   3
                       ⇀             3^   3^   3^
                       F(x,y,z) = x i+y j+z k
              EX  1 
                       S
                         is the hemisphere
                                       ⇀⇀                                         .
                       Calculate   F·n dS.
                                    ∫∫
                                    ∂S
                                                                                                                           4