The remaining part of the right-hand side can then be expressed
∫∫
R
∂N
∂x dx dy =
∫y 2
y 1
∫x 2 (y)
x 1 (y)
∂N
∂x dx dy =
∫y 2
y 1
N(x, y )
x 2 (y)
x 1 (y)
dy
=
∫y 2
y 1
[N(x 2 (y), y )−N(x 1 (y), y )] dy
=
∫y 2
y 1
N(x 2 (y), y )dy +
∫y 1
y 2
N(x 1 (y), y )dy
=
∫
C 3
N(x 1 (y), y )dy +
∫
C 4
N(x 2 (y), y )dy =
∫
C
©N(x, y )dy.
(8 .15)
Adding the results of equations (8.14) and (8.15) produces the desired result.
Example 8-5. Verify Green’s theorem in the plane in the special case
M(x, y ) = x^2 +y^2 and N(x, y ) = xy,
where Cis the wedge shaped curve illustrated in figure 8-8.
Figure 8-8. Wedge shaped path for Green’s theorem example.
Solution The boundary curve can be broken up into three parts and the left-hand
side of the Green’s theorem can be expressed
∫
C
©M dx +N dy =
∫
C 1
Mdx +N dy +
∫
C 2
Mdx +N dy +
∫
C 3
Mdx +N dy.
