Multivariate Calculus - Bruce E. Shapiro

194 LECTURE 23. GREEN’S THEOREM
Example 23.2 Let C be a circle of radius a centered at the origin. Find ∮ C F · dr
for F = (−x 2 y, xy 2 ) using Green’s theorem.
Solution. Since M = −x 2 y and N = xy 2 , we have
M y = −x 2 , N x = y 2
From Green’s theorem
∮
F · dr = (N x − M y )dA = (x 2 + y 2 )dA
Converting to polar coordinates
∮
F · dr =
C
C
R
∫ 2π ∫ a
0
0
r 3 drdθ =
∫ 2π
0
R
a 4
4 dθ = a4 π
2
Theorem 23.2 Gauss’ Theorem on a Plane Let C be a simple closed curve
enclosing a region S ⊂ R 2 , and let
F = (M(x, y), N(x, y))
be a smoothly differentiable vector field on S. Then
∮
F · nds = ∇ · FdA (23.9)
where n(x, y) is a nunit tangent vector at (x, y).
C
Proof. Suppose that C is described by a parameter t as Recall that we derived
earlier that if s is the arclength,
T = dr ( dx
ds = ds , dy )
(23.10)
ds
is a unit tangent vector, and an outward pointing unit normal vector is
( ) dy
n =
ds , −dx ds
Hence
∮
C
∮
F · nds =
∮
=
C
C
=
R
∫
=
R
S
( ) dy
(M(x, y), N(x, y)) ·
ds , −dx ds
ds
(M(x, y)dy − N(x, y)dx)
( ∂M
∂x + ∂N )
dA
∂y
∇ · FdA
(23.11)
Revised December 6, 2006. Math 250, Fall 2006