where the line integral is taken in a counterclockwise direction around the simple
closed curve Cwhich encloses the region R.
To prove this theorem, let y =y 2 (x) and y =y 1 (x) be single-valued continuous
functions which describe the upper and lower portions C 2 and C 1 of the simple closed
curve Cin the interval x 1 ≤x≤x 2 as illustrated in figure 8-7(a).
Figure 8-7. Simple closed curve for Green’s theorem.
The right-hand side of equation (8.13) can be expressed
−
∫∫
R
∂M
∂y
dx dy =−
∫x 2
x 1
∫y 2 (x)
y 1 (x)
∂M
∂y
dy dx =−
∫x 2
x 1
M(x, y )
y 2 (x)
y 1 (x)
dx
=
∫x 2
x 1
[M(x, y 1 (x)) −M(x, y 2 (x))] dx
=
∫x 2
x 1
M(x, y 1 (x)) dx +
∫x 1
x 2
M(x, y 2 (x)) dx
=
∫
C 1
M(x, y 1 (x)) dx +
∫
C 2
M(x, y 2 (x)) dx =
∫
C
©M(x, y )dx.
(8 .14)
Now let x =x 1 (y) and x =x 2 (y) be single-valued continuous functions which
describe the left and right sections C 3 and C 4 of the curve Cin the interval y 1 ≤y≤y 2.
