Green's Theorem and Circulation of Vector Field

Q 3.
a)
The given vector field is:
F = 〈−y, x〉
Green’s Theorem relates circulation of a vector field F around a closed curve
C to curl of the field over the area D enclosed by C:
Circulation of F =
∮
C
F · dr =
∫ ∫
D
curl(F) dA
Curve C is oriented counter-clockwise. A general closed curve C oriented
counter-clockwise, enclosing area D is shown in figure-1 below.
Figure 1: Positively oriented curve C and the region inside it D
Curl of a vector field F = 〈M, N 〉 is given by:
curl(F) = ∇ × F
=
〈 ∂
∂x , ∂
∂y
〉
× 〈M, N 〉
= ∂N
∂x − ∂M
∂y
1

Therefore, for F = 〈−y, x〉:
curl(F) = ∂
∂x (x) − ∂
∂y (−y)
= 1 − (−1) = 2
The Green’s Theorem applied to the field 〈−y, x〉 over a closed curve C
gives: ∮
C
〈−y, x〉 · dr =
∫ ∫
D
curl(F) dA
r = 〈x, y〉 is the position vector of the curve and it’s differential is given by:
dr = 〈dx, dy〉.
Therefore,
∮
C
〈−y, x〉 · 〈dx, dy〉 =
∫ ∫
D
curl(F) dA
Expanding the vector dot product in the line integral and substituting the
value of curl(F) in the double integral we get:
∮
C
−y dx + x dy =
∫ ∫
D
2 dA
But, ∫ ∫
D
dA = Area of D
It is the double integral which gives the area of region of integration.
Therefore, ∮
C
−y dx + x dy = 2Area(D)
This whole equation can be divided by 2 and expressed as:
Area(D) = 1
2
∮
C
−y dx + x dy
b)
In order to use the area formula from Green’s Theorem, let us circulate the
vector field 〈−y, x〉 around the circle C.
2