Page 420 - Advanced engineering mathematics

400 CHAPTER 12 Vector Integral Calculus

The conclusion of Green’s theorem can be written

∂g ∂ f
f (x, y)dx + g(x, y)dy = − dA
C D ∂x ∂y
with C the simple closed path bounding the region D of the plane.
Define the vector field
F(x, y) = g(x, y)i − f (x, y)j.

With this choice,
∂g ∂ f
∇· F = − .
∂x ∂y
Parametrize C by arc length so the coordinate functions are x = x(s), y = y(s) for 0 ≤ s ≤ L.

The unit tangent vector to C is T(s) = x (s)i + y (s)j and the unit normal vector is

n(s) = y (s)i − x (s)j. These are shown in Figure 12.19. This normal points outward away from

D, and so is called a unit outer normal. Now
dy dx
F · n = g(x, y) + f (x, y)
ds ds
so

dx dy

f (x, y)dx + g(x, y)dy = f (x, y) + g(x, y) ds
ds ds
C C

= F · nds.
C
We may therefore write the conclusion of Green’s theorem as

F · nds = ∇· FdA.
C D
In this form, Green’s theorem suggests a generalization to three dimensions. Replace the closed
curve C in the plane with a closed surface in 3-space (closed means bounding a volume, such

n
T

y

x

FIGURE 12.19 Unit tangent and normal
vectors to C.

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October 14, 2010 14:53 THM/NEIL Page-400 27410_12_ch12_p367-424