Example

Find the flux for the vector field v(x,y,z)=xi+yj+zk out of the surface
z=1-x^2-y^2, z>=0.


  Clear[v,S,g,n]
  v[x_,y_,z_]:={x,y,z}
  S[x_,y_]:=1-x^2-y^2
  g[x_,y_,z_]:=(1-x^2-y^2)-z


  vf=Plot3D[S[x,y],{x,-1,1},{y,-1,1}];


  grad3[g][x,y,z]


  n=grad3[g][x,y,z]/norm[grad3[g][x,y,z]]


  vp=vectorPlot[{{0,0,S[0,0]}+{0,0,0}},{{0,0,S[0,0]}+n/.{x->0,y->0,z->0}}];


  Show[vf,vp];

Where's the normal?


  Show[vf,vp,ViewPoint->{0,-1,-1}];

It is an inward pointing normal. We will get the flux from the outside to the inside.


  Simplify[v[x,y,S[x,y]].n]


  Dx=D[S[x,y],x]
  Dy=D[S[x,y],y]


  dS=Sqrt[1+Dx^2+Dy^2]


  Simplify[(v[x,y,S[x,y]].n)*dS]


  Integrate[1+x^2+y^2,{x,-1,1},
    {y,-Sqrt[1-x^2],Sqrt[1-x^2]}]