Multivariate Calculus - Bruce E. Shapiro

LECTURE 23. GREEN’S THEOREM 193
Therefore
Hence
∮
R i
F · dr =
=
=
∫ x+∆x
x
∫ x
+
x+∆x
∫ x+∆x
x
∫ y+∆y
y
∫ x+∆x
x
∫ y+∆y
y
∫ y+∆y
M(u, y)du + N(x + ∆x, v)dv
y
M(u, y + ∆y)du +
∫ y
y+∆y
[M(u, y) − M(u, y + ∆y)] du+
[N(x + ∆x, v) − N(x, v)] dv
N(x, v)dv
M(u, y) − M(u, y + ∆y)
∆y du+
∆y
N(x + ∆x, v) − N(x, v)
∆x dv
∆x
∮
∫ x+dx
∫
∂M
y+dy
lim F · dr = −
∆x,∆y→0 R i x ∂y dydu + ∂N
y ∂x dxdv
= − ∂M ∫ x+dx
∂y dy du + ∂N ∫ y+dy
∂x dx dv
= − ∂M ∂N
dydx +
∂y ∂x dxdy
( ∂N
=
∂x − ∂M )
dxdy
∂y
Hence from equation 23.2
∮
n∑
∮
F · dr = lim F · dr
C
n→∞
i=1
R i
n∑
= lim
n→∞
=
R
i=1
x
( ∂N
∂x − ∂M
∂y
( ∂N
∂x − ∂M
∂y
)
dA
)
dxdy
Example 23.1 Let C be a circle of radius a centered at the origin. Find ∮ C F · dr
for F = (−y, x) using Green’s theorem.
Solution. We have M = −y, N = x. Hence
M y = −1, N x = −1
y
From Green’s theorem
∮
F · dr =
C
R
(N x − M y )dA = 2
R
dA = 2πa 2
Math 250, Fall 2006 Revised December 6, 2006.