Page 421 - Advanced engineering mathematics

12.7 Lifting Green’s Theorem to R 3 401

as a sphere). Replace the line integral over C with a surface integral over and allow the vector
field to be a function of three variables. We conjecture that Green’s theorem generalizes to

F · ndσ = ∇· FdV,

M
where M is the solid region bounded by and n is a unit normal to pointing out of the surface
and away from M. This conclusion is Gauss’s divergence theorem.
Now begin again with Green’s theorem and pursue a different generalization to three
dimensions. This time let

F(x, y, z) = f (x, y)i + g(x, y)j + 0k.
Including a third component allows us to take the curl:

i j k
∂g ∂ f
∇× F = ∂/∂x ∂/∂y ∂/∂z = − k.

∂x ∂y
f g 0

Then
∂g ∂ f
(∇× F) = − .
∂x ∂y

Further, with unit tangent T(s) = x (s)i + y (s)j to C, we can write

dx dy
F · Tds =[ f (x, y)i + g(x, y)j]· i + k
ds ds
= f (x, y)dx + g(x, y)dy.
Now the conclusion of Green’s theorem can be written

F · Tds = (∇× F) · kdA.
C D
Think of D as a flat surface in the x, y-plane, with unit normal k, and bounded by the closed path
C. To generalize this, allow C to be a path in 3-space, bounding a surface having unit outer
normal N, as in Figure 12.20. With these changes, the last equation suggests that

n

Σ
z

C
y

x

FIGURE 12.20 C
bounding a surface
having outer normal n.

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October 14, 2010 14:53 THM/NEIL Page-401 27410_12_ch12_p367-424