Green’s Theorem
Reading Assignment: 6.2
Suggested Problems: 6.2: 1,3, 5,6, 13-19 odd, 16, 25
1) Verify Green’s theorem for

C
y dx  x dy where C is the boundary of the unit square
[0, 1]  [0, 1] by calculating the line integral, and then evaluating the double integral into
which it transforms.
2) Use Green’s theorem to find the area bounded by the x-axis and one arc of the cycloid
x  a(  sin ), y  a(1  cos ), a  0, and 0    2 .
3) Verify Green’s Theorem for the functions M(x, y) = 2x3 + y3, N(x, y) = 3xy2, and the
region D which is the annulus between the circles of radii a and b (a < b), centered at the
origin. Be careful with the orientations!
4) Let D be a region to which Green’s Theorem applies and assume that f(x, y) is a
 f
f
2 f 2 f
dx  dy  0.
harmonic function (i.e., 2 f  2  2  0 ). Prove that 
x
x
y
D y
5) Let F(x, y) be the vector field 2xyi  y 2 j . What is the total outward flow of this field
across the ellipse
x2 y 2

 1?
a 2 b2
y
x
and N ( x, y )  2
.
2
(x  y )
(x  y2 )
a) Verify that the curl of Mi + Nj is zero.
b) Verify that Green’s Theorem fails for this function on the unit disk.
c) Explain.
6) Let M ( x, y ) 
2
7) Determine whether Green’s theorem can be used to evaluate the integral
y
x
C x2  y 2 dx  x2  y 2 dy
where C is the given curve: Explain.
a.
the line segment from (0,1/2) to (2,1)
b.
the circle (x – 1)2 + (y – 1)2 = 1
c.
the circle x2 + y2 = 1
d.
Evaluate the integral in c.