Lecture 21, November 21
• Green’s theorem. If R is a simply connected region in R2 whose boundary C is a
simple, closed piecewise smooth curve oriented counterclockwise, then
)
ffi
¨ (
∂F2 ∂F1
−
dA.
F1 dx + F2 dy =
∂x
∂y
C
R
In particular, the area of the region R may be computed using any of the formulas
ffi
ffi
ffi
1
Area =
x dy = − y dx =
x dy − y dx.
2 C
C
C
.....................................................................................
Example 1. Consider the triangle C whose vertices are (0, 0), (1, 0) and (1, 2). Then
¨
ffi
2 3
xy dx + x y dy =
(2xy 3 − x) dA,
C
R
where R is the interior of the triangle. This actually gives
ffi
ˆ
1
ˆ
1
[
3
xy dx + x y dy =
C
ˆ
2x
(2xy − x) dy dx =
2 3
0
ˆ
0
0
1
(8x5 − 2x2 ) dx =
=
0
2xy 4
− xy
4
]2x
dx
y=0
8 2
2
− = .
6 3
3
Example 2. Let C be the circle of radius 2 around the origin and let
⟨
⟩
F (x, y) = ex − y 3 , cos y + x3 .
According to Green’s theorem, we then have
ffi
¨
F · dr =
(3x2 + 3y 2 ) dA,
C
R
where R is the interior of the circle. Switching to polar coordinates, we find that
ffi
ˆ 2π ˆ 2
3r3 dr dθ
F · dr =
0
C
0
ˆ 2π [ 4 ]2
ˆ 2π
3r
=
dθ =
12 dθ = 24π.
4 r=0
0
0