Flow Across a Surface (Flux)
Flow Across a Surface (Flux)
Consider a vector field v(x,y,z)=v1(x,y,z) i + v2(x,y,z) j + v3(x,y,z) k.
Think of a small (infinitesimal) piece of surface. The flow of v across the
small piece of surface is in the direction of the unit normal to the surface and
is v times the small piece of surface area (the bigger the piece of surface, the
more flow!). Adding these infinitesimal pieces up (surface integrating!) yields
flow across the surface:
Surface Integral of v.n dsigma
Recall that the normal vector at a point on a level curve was the gradient
vector at that point.
The gradient of a "level surface" will give us the normal now.
Consider g(x,y,z)=S(x,y)-z. If g=c we have "level surfaces". All we
care about is g=0.
The gradient will still give a normal:
The gradient of g(x,y,z) is
DxS i + DyS j - 1 k
or
{DxS,DyS,-1}.
A unit normal is given by
{DxS,DyS,-1}/|| {DxS,DyS,-1} ||
The surface integral v.n dS = double integral (v1Dxf+v2Dyf-v3) dydx
Something you might have noticed while doing the example was the cancellation
of the dS with ||normal vector||.
Example
The unit normal is given by
{DxS,DyS,-1}/|| {DxS,DyS,-1} ||,
while dS is
dS=Sqrt[1+DxS^2+DyS^2] dydx
Notice that dS=|| {DxS,DyS,-1} || dydx
So, surface integral v.n dS=double integral v.{DxS,DyS,-1} dydx
or equivalently: