Multivariate Calculus - Bruce E. Shapiro

192 LECTURE 23. GREEN’S THEOREM
Figure 23.2: The integral over two adjacent rectangles is the sum over the individual
rectangles because the integrals over the common boundary cancels out.
eventually filling out the entire original closed curve. The only edges that do not
cancel are the ones belonging to the original curve.
∮
n∑
∮
F · dr = F · dr (23.2)
C
R i
i=1
Let us consider the path integral over a single rectangle as illustrated in figure 23.3
Figure 23.3: The path around a single tiny rectangle.
Figure 23.4: 17-4,5,6,7Fig6.pict about here.
∮ ∫
F · dr =
R i
Writing the components of F as
Then
∫
∫
∫
∫
A
B
C
D
A
∫ ∫ ∫
F · dr + F · dr + F · dr + F · dr (23.3)
B
C
D
F = (M(x, y), N(x, y)) (23.4)
F · dr =
F · dr =
F · dr =
F · dr =
∫ x+∆x
x
∫ y+∆y
y
∫ x
x+∆x
∫ y
y+∆y
M(u, y)du (23.5)
N(x + ∆x, v)dv (23.6)
M(u, y + ∆y)du (23.7)
N(x, v)dv (23.8)
Revised December 6, 2006. Math 250, Fall 2006