文档介绍：Chapter Sixteen
Integrating Vector Functions
Introduction
Suppose water (or some other pressible fluid ) flows at a constant velocity v
in space (through a pipe, for instance), and we wish to know the rate at which the water
flows across a rectangular surface S that is normal to the stream lines:
What is the rate at which the fluid flows through S? Let M (t) denote the total volume of
fluid that has passed through the surface at time t. The amount of fluid that flows through
during the time between t and t +Dt is simply
M (t + Dt) - M(t) = |v|aDt ,
dM
where a is the area of S. Thus, the rate of flow through S is =|v|a .
dt
The result is slightly plicated when various exciting changes are made.
Clearly there is nothing special about the surface's being a rectangle. But suppose that S
is placed at an angle to the stream lines instead of being placed normal to the them. Then
dM
we have = v ×n a , where n is a unit normal to the surface S.
dt
16. 1
Observe that matters which unit normal to the plane surface we choose. If we
choose the other normal (- n ), then our rate will be the negative of this one. We must
thus specify an orientation of the surface. We puting the rate of flow from one
side of the surface to the other, and so we have to specify the "sides", so to speak.
Flux
Now, let's look at the general situation. The surface is n