58
CHAPTER
4V
ECTOR FIELDS IN TWO DIMENSIONS
4.3. G
REEN
’
S
T
HEOREM FOR
F
LUX AND
D
IVERGENCE
In terms of our interpretation of vector ±elds as ²ows of material, outward ²ux measures
the amount of material leaving a region, while divergence measures the pressure at points
within the region. Because matter can be neither created nor destroyed, the only way to
have an overall decrease in pressure (increase in divergence) within a region is for mate
rial to leave the region (positive ²ux). Therefore, we should expect that the ²ux of a ±eld
over a region is equal to the double integral of divergence over the region. The following
theorem, named for George Green
1
shows that this intuition is correct, not just for vector
±elds with physical interpretations, but in general for all wellbehaved vector ±elds over
wellbehaved regions. The de±nition of “wellbehaved region” is that it should have a
piecewise smooth boundary, a term we’ve already encountered, and also be
simply con
nected
,whichmeanstwothings
:
•
between every two points in the region, there is a path (in other words, the region
isn’t split into two or more disconnected pieces), and
•
every closed curve in the region encloses only other points inthereg
ion(
ino
ther
words, the region can’t contain a hole).
For example, the region on the left, below, is simply connected, but the two other regions
are not simply connected.
With the hypotheses explained, we can now state Green’s Theorem.
Green’s Theorem for Flux and Divergence.
Let
C
be a piecewise
smooth curve enclosing a simplyconnected region
R
in the
xy

plane. Let
F
P
i
Q
j
be a vector ±eld, where
P
and
Q
have
continuous ±rst partial derivatives in an open region containing
R
.
Then
C
F
n
ds
C
Pdy
Qdx
outward ²ux
R
div
F
dA
R
∇
F
dA
integral of divergence
.
1
George Green (1793–1841) published Green’s Theorem in his 1828 article
An essay on the application of
mathematical analysis to the theories of electricity and magnetism
,o
fwh
ichheso
ld51cop
ies
. Uptoth
ispo
in
t
,
Green had attended school for precisely one year, when he was 8, and worked maintaining his father’s mill.
When his father died in 1829, Green sold the mill and used the proceeds to attend Cambridge, in 1833, at the
age of 40.