Chapter 15
Section 15.6: Surface Integrals

• Start with `S` as a surface given by `z=g(x,y)`. Let `R` be the projection of `S` onto the `xy`-plane. We assume `g, g_x`, and `g_y` are continuous over `R`. Then, we consider a function `f` that is defined on all points of `S`. Then we can define the surface integral of `f` over `S` to be
  `int_Sint f(x,y,z) dS = lim_(||Delta||rarr0) sum_(i=1)^n f(x_i,y_i,z_i) DeltaS_i` where `Delta S_i=sqrt( 1 +[g_x(x_i,y_i)]^2 + [g_y(x_i,y_i)]^2) Delta A`
• Let `S` be a surface with equation `z=g(x,y)` and `R` be its projection onto the `xy-`plane. If `g, g_x`, and `g_y` are continuous on `R` and `f` is continuous on `S`, then the surface integral of `f` over `S` is
  `int_Sint f(x,y,z) dS = int_Rint f(x,y,g(x,y))sqrt( 1 +[g_x(x_i,y_i)]^2 + [g_y(x_i,y_i)]^2) dA`
• For a surface `S` defined by `bbr(u,v) =x(u,v)bbi +y(u,v)bbj + z(u,v)bbk` over a region `D` in the `uv`-plane, the surface integral of `f(x,y,z)` is given by
  `int_Sint f(x,y,z)dS = int_Dint f( x(u,v),y(u,v),z(u,v))||bbr_u(u,v) x bbr_v(u,v)|| dA`
• Orientation of surface
  For `z=g(x,y)` let `G(x,y,z)= z - g(x,y)`. [When `G(x,y,z)=0` we have the surface defined as `z=g(x,y)`.] Then we can orient `S` with either the unit normal vector given by

  `bbN = (grad G(x,y,z))/(||grad G(x,y,z)||)`

  `bbN=(-g_x(x,y)bbi - g_y(x,y)bbj + bbk)/(sqrt(1+[g_x(x,y)]^2 + [g_y(x,y)]^2))`

  or

  `bbN = (-grad G(x,y,z))/(||grad G(x,y,z)||)`

  `bbN=(g_x(x,y)bbi + g_y(x,y)bbj - bbk)/(sqrt(1+[g_x(x,y)]^2 + [g_y(x,y)]^2))`

• For a surface given by `bbr(u,v) =x(u,v)bbi +y(u,v)bbj + z(u,v)bbk` we get

  `bbN = (bbr_u x bbr_v)/(||bbr_u x bbr_v||)`

  or

  `bbN = (bbr_v x bbr_u)/(||bbr_v x bbr_u||)` [incorporating the negative in the change in order].

• Flux Integrals
  1. The flux integral of `bbF` across `S`, where `bbF(x,y,z)=Mbbi+Nbbj+Pbbk` and `bbN` is an oriented unit normal vector, [noticing the confusing use of `N` for the function in the y direction in the definition of the function `bbF` and `bbN` as the unit normal vector], is denoted

     `int_SintbbF•bbN\quad dS`

  2. If we have `S` defined by `z=g(x,y)` and we have that `R` is the projection of `S` onto the `xy`-plane,

     `int_SintbbF•bbN\quad dS = int_Sint bbF•[-g_x(x,y)bbi - g_y(x,y)bbj + bbk]\quad dA` upward

     `int_SintbbF•bbN\quad dS = int_Sint bbF•[g_x(x,y)bbi + g_y(x,y)bbj - bbk]\quad dA` downward