Multivariate Calculus - Bruce E. Shapiro

Lecture 23
Green’s Theorem
Theorem 23.1 Green’s Theorem. Let C be a closed, oriented curve enclosing a
region R ⊂ R 2 with no holes, and let F = (M(x, y), N(x, y) be a vector field in R 2 .
The
∮ ∮
( ∂N
F · dr = (Mdx + Ndy) =
∂x − ∂M )
dA (23.1)
∂y
C
C
Proof. One way to evaluate the line integral about a closed path is to divide up
the enclosed area into small rectangles. Focusing on a pair of small rectangles,
R
Figure 23.1: A closed curve may be filled with an large number of tiny rectangles.
say R 1 and R 2 , the integral over the enclosing rectangle R 3 is equal to the sum of
the line integrals over the individual rectangles because the common side cancels
out, assuming the orientation of each individual small square is the same as the
orientation of the enclosing curve.
∫
F · dr =
R 3
∫
F · dr +
R 1
∫
F · dr
R 2
The integral over the shared path cancels out because the two times the path
is traversed it is traversed in different directions - hence the two integrals differ by
a minus sign factor. This cancellation repeats itself as we add additional squares,
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