ECTOR FIELDS IN TWO DIMENSIONS
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
shows that this intuition is correct, not just for vector
±elds with physical interpretations, but in general for all well-behaved vector ±elds over
well-behaved regions. The de±nition of “well-behaved region” is that it should have a
piecewise smooth boundary, a term we’ve already encountered, and also be
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
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.
be a piecewise
smooth curve enclosing a simply-connected region
be a vector ±eld, where
continuous ±rst partial derivatives in an open region containing
integral of divergence
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
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.