50
CHAPTER
4V
ECTOR FIELDS IN TWO DIMENSIONS
4.2. F
LUX AND
D
IVERGENCE
From here on, we will almost always think of vector ±elds as representing ²ows of mate
rial. In this section, we seek to quantify how this ²ow is moving. The ±rst quantity we
study, outward ²ux, measures the net ²ow out of a region in the plane. The second quan
tity we study, divergence, measures ²ux per unit of area. In the next section, we study the
relationship between these two quantities (Green’s Theorem).
O
UTWARD
F
LUX
Suppose that we have a vector ±eld
F
which describes the velocity of moving air in a room
with a lit ±replace and a glass of ice water. Near the ±replace, the air is being warmed, so it
is expanding, which means that
F
near the ±replace is pointing out. Near the ice water, the
opposite is happening, so
F
near the ice water is pointing in. In terms of ²ux, this means
that the outward ²ux of this vector ±eld on a small region around the ±replace is positive,
while in a small region around the ice water, it is negative.
Flux is the rate of ²ow through a unit area during a unit time. Various sorts of ²ux
appear in physics, for example:
•
Newton’s law of viscosity describes the rate of transfer of momentum across an area,
•
Fourier’s law of conduction descibes the rate of heat ²ow across an area,
•
Fick’s law of diffusion describes the rate of movement of molecules across an area.
From our point of view, given a
2
dimensional vector ±eld
F
and a region
R
in the plane,
the outward ²ux of
F
over
R
is the amount of material ²owing out of the region
R
.Sup

pose that the boundary of
R
is parameterized by the smooth curve
r
t
.Thentocompu
te
the ²ux of
F
over
R
,wetakethelineintegralover
r
t
of the amount of material ²owing
out of the region at each point.
Consider the following region,
R
,whoseboundaryisparameterizedbythevectorfunc
tion
r
t
.
r
t
r
t
0
n
0
r
t
0
R
We want to calculate the amount of material leaving the region
R
at each point
r
t
0
on the boundary. At this point, the vector
r
t
0
is tangent to the boundary of the region.