Circulation and Area using Green's Theorem

1.
Given vector field is:
F = 1
2 (−y ˆi + x ˆj)
Circulation of a vector field F = 〈F1, F2〉 around a closed curve C is evalu-
ated using Green’s Theorem in the following way:
∮
C
F · dr =
∫ ∫
D
( ∂F2
∂x − ∂F1
∂y
)
dA
where, C is oriented counter-clockwise and D is the area enclosed by C.
In the given problem, C is an ellipse and D is the area enclosed by ellipse,
as shown in figure-1 below.
(i)
The double integral from Green’s Theorem is evaluated in the following way:
F = 〈F1, F2〉 = 1
2 〈−y, x〉
Implies, F1 = −y
2 and F2 = x
2
Therefore,
∂F2
∂x − ∂F1
∂y = ∂
∂x
(x
2
)
− ∂
∂y
(
−y
2
)
= 1
2 −
(
−1
2
)
= 1
Therefore, ∫ ∫
D
( ∂F2
∂x − ∂F1
∂y
)
dA =
∫ ∫
D
dA
1

Thus, the double integral from Green’s Theorem, for the given F, gives area
of D, since
∫ ∫
D
dA = Area of D.
(ii)
Since the line and double integral are equal by Green’s Theorem, the area of
region D can be determined using the line integral:
∮
C
F · dr
In order to evaluate this integral, we make use of the parametric equation of
C :
C = r(t) = 〈a cos t, b sin t〉 ∀ t ∈ [0, 2π]
=⇒ dr = 〈−a sin t, b cos t〉 dt
Therefore,
Area of D =
∮
C
F · dr
=
∮
C
1
2〈−y, x〉 · 〈−a sin t, b cos t〉 dt
[Substitute: x = a cos t, y = b sin t]
= 1
2
∮
C
〈−b sin t, a cos t〉 · 〈−a sin t, b cos t〉 dt
= 1
2
∮
C
ab sin2 t + ab cos2 t dt
[Trig. Property: sin2 t + cos2 t = 2]
2

= 1
2
∫ 2π
0
ab dt
= 1
2ab
[
t
]2π
0
∴ Area of D = π ab program@epstopdf
Thus, the area of ellipse: x2
a2 + y2
b2 = 1, as determined by the line integral, is:
π ab.
2.
Given vector field is:
F = x2 ˆi + (x2 − y) ˆj
Circulation of a vector field F = 〈F1, F2〉 around a closed curve C is evalu-
ated using Green’s Theorem in the following way (Orloff, 2012):
∮
C
F · dr =
∫ ∫
D
( ∂F2
∂x − ∂F1
∂y
)
dA
where, C is oriented counter-clockwise and D is the area enclosed by C.
In the given problem, C is a composite curve of two curve: C1 = x3 and
C2 = x2, and D is the area enclosed by C, as shown in figure-2 below.
3

To verify the Green’s Theorem, we need to evaluate both, the line integral
and double integral individually, and then compare them.
a) Line integral:
∮
C
F · dr
We evaluate the line integral form Green’s Theorem using the property of
line integrals: ∮
C
F · dr =
∮
C1
F · dr +
∮
C2
F · dr
The line integrals for path C1 and C2 require parametric curve equations for
their evaluation.
Path C1: y = x3 for 0 ≤ x ≤ 1
C1 = r(t) = 〈t, t3〉 [0 ≤ t ≤ 1]
=⇒ dr = 〈1, 3t2〉 dt
therefore,
∮
C1
F · dr =
∮
C1
〈x2, x2 − y〉 · 〈1, 3t2〉 dt
[Substitute: x = t & y = t3]
4