Line Integral, Double Integral, Flux and Green's Theorem

The given vector field is:
F=¿ xy ,−x2 >¿
Path of integration C is potion of unit circle in first quadrant as shown in figure below:
Figure 1 - Path of integral C and region of integration D
a) Line integral ∮
C
F · dr
In order to evaluate the line integral, we need to parametrize curve C. The curve is divided
into 3 paths: C1, C2 and C3 as shown in the figure above.
Property of line integrals:
∮
C
F · dr =∮
C 1
F · dr+∮
C 2
F · dr+∮
C 3
F · dr

1) Path C1: line from (0,0) to (1,0)
Parametric equation for C1:
r ( t ) =¿ t , 0> ,0 ≤ t ≤ 1
⇒ dr=¿ 1 , 0>dt
therefore,
∮
C 1
F · dr=∮
C 1
¿ xy ,−x2 >·<t , 0> dt
Put: x = t and y = 0
¿∮
C 1
¿ 0 ,−t2 >·<t , 0> dt
¿∮
C 1
0 ·t−t2 ·0 dt=0
2) Path C2: circle from (1,0) to (0,1)
Parametric equation for C2:
r ( t )=¿ cos t ,sin t >, 0 ≤t ≤ π
2
⇒ dr=←sint , cos t> dt
therefore,
∮
C 2
F · dr=∮
C 2
¿ xy ,−x2 >←sin t ,cos t>dt
Put: x = cos t, y = sin t
¿∮
C 2
¿ cos t·sin t ,−cos2 t >·←sin t , cos t> dt
¿∫
0
π
2
−sin2 t· cos t−cos3 t dt =∫
0
π
2
−cos t ( sin2 t+cos2 t ) dt
∫
0
π
2
−cos t · dt = [ −sin t ]0
π
2 =−1
3) Path C3: line form (1,0) to (0,1)
Parametric equation for C3:
r ( t ) =¿ 0 , 1−t >, 0 ≤t ≤ 1
⇒ dr=¿ 0 ,−1>dt
therefore,

∮
C 3
F · dr=∮
C 3
¿ xy ,−x2 >·<0 ,−1> dt
Put: x = 0, y = 1-t
∮
C 3
¿ 0 , 0>·<0 ,−1> dt=0
Therefore, the line integral over path C is:
∮
C
F · dr =0+−1+0=−1
b) Double Integral:
∂ F2
∂ x − ∂ F1
∂ y = ∂
∂ x (−x2 ) − ∂
∂ y ( xy ) =−2 x −x=−3 x
therefore,
∫∫ ( ∂ F2
∂ x −∂ F1
∂ y )dA=∫∫
D
−3 x dA
Region D is the are enclosed by path of the integral (C), as shown in the figure-1.
Limits of integration in region D are:
x : 0≤ x ≤ 1 , y :0 ≤ √ 1−x2 ≤ 1
Therefore,
∫∫
D
−3 x dA=∫
0
1
∫
0
√1−x2
−3 x dydx=∫
0
1
[ −3 xy ]0
√1− x2
dx=∫
0
1
−3 x √1−x2 dx
Substitute:
1−x2=t ⇒−2 x dx=dt
∫
0
1
3
2 √t dt=[t
3
2 ]0
1
= [ ( 1−x2 )
3
2 ]0
1
=0−1=−1
This result is equal to the result in part a).
c) Flux ∮
C
F · dr
The flux of vector field over the entire path is again calculated by computing flux over
sub-paths (C1, C2 and C3) and adding them: