14.4 Green's Theorem
Green's Theorem gives the relationship between a line integral around a simple closed
curve C and a double integral over the plane region D bounded by C . Recall that a
curve C in the plane is a simple closed curve if its starting point is also the end point
and has no self-intersections. In stating Green's Theorem we use the convention that
the positive orientation of a simple closed curve C .
Denition. A positively oriented curve is a planar simple closed curve such that
when traveling on it one always has the curve interior to the left.
If in the above denition one interchanges left and right, one obtains a negatively
oriented curve.
Green's Theorem. Let C be a positively oriented, piecewise-smooth, simple closed
curve in the plane and let D be the region bounded by C . If P (x, y) and Q(x, y) have
continuous partial derivatives on an open region that contains D, then
ˆ
¨ P dx + Q dy =
C
∂Q ∂P
−
∂x
∂y
dA
D
Notation.
¸
(a) C P dx + Q dy is sometimes used to indicate that the line integral is calculated
using the positive orientation of the closed curve C .
(b) The positively oriented boundary curve of a region D is denoted by ∂D.
With these notation the equation in Green's Theorem can be written as
˛
¨ P dx + Q dy =
∂D
D
∂Q ∂P
−
∂x
∂y
dA
Example 1. Let C be a triangular curve consisting of the line segments from (0, 0) to
(5, 0), from (5, 0) to (0, 5) and from (0, 5) to (0, 0). Evaluate the integral
˛
1
1
(x2 y + y 2 + ex sin x ) dx + (xy + x3 + x − 4 arccos(ey )) dy
2
3
C
¸
Example 2. Evaulate the integral C F · dr, where F(x, y) = h e3x + x2 y, e3y − xy 2 i
and C is the circle x2 + y 2 = 1 oriented clockwise.
Application of Green's Theorem
The area of D is
˛
˛
1
A=
x dy = −
y dx =
2
∂D
∂D
˛
x dy − y dx
∂D
x2 y 2
Example 3. Find the area enclosed by the ellipse
+
= 1.
4
9