and
a double integral over the plane region
bounded
by
It
is the two-dimensional special case of the more general Stokes'
theorem, and is named after British mathematician George Green.
University Maths Notes: Calculus – Green's Theorem
Green's theorem gives the relationship between a line integral
around a simple closed curveand
a double integral over the plane regionbounded
byIt
is the two-dimensional special case of the more general Stokes'
theorem, and is named after British mathematician George Green.
Letbe
a positively oriented (counterclockwise), piecewise smooth, simple
closed curve in the plane
and
let
be the region bounded byIf
and
are
functions of
defined
on an open region containingand
have continuous partial derivatives there, then
For example letbe
the triangular region illustrated below and let
We need to find the limits. If we integrate with respect to
first
then we must findas
a function of
on
the line BC:
Our
integral becomes
Expanding the integrand and simplifying gives
