Page 422 - Advanced engineering mathematics

402 CHAPTER 12 Vector Integral Calculus

F · Tds = (∇× F) · ndσ.
C
We will see this in Section 12.9 as Stokes’s theorem.

SECTION 12.7 PROBLEMS

1. Let C beasimpleclosedpathinthe x, y-plane with 2. Under the conditions of Problem 1, show that
interior D.Let ϕ(x, y) and ψ(x, y) be continuous with
2
2
continuous first and second partial derivatives on C and (ϕ∇ ψ − ψ∇ ϕ)dA
throughout D.Let D
∂ϕ ∂ψ ∂ψ ∂ϕ

= ψ − ϕ dx + ϕ − ψ dy.
2
2
∂ ψ ∂ ψ C ∂y ∂y ∂x ∂x
2
∇ ψ = + .
∂x 2 ∂y 2 3. Let C be a simple closed path in the x, y-plane, with
interior D.Let ϕ be continuous with continuous first
Prove that
and second partial derivatives on C and throughout D.
Let N(x, y) be the unit outer normal to C (outer mean-

2
ϕ∇ ψ dA ing pointing away from D from points on C). Prove that
D
2
∂ψ ∂ψ ϕ N (x, y)ds = ∇ ϕ(x, y)dA.
= −ϕ dx + ϕ dy − ∇ϕ ·∇ψ dA. C D
C ∂y ∂x D
(Recall that ϕ N is the directional derivative of ϕ in the
direction of N.)
12.8 The Divergence Theorem of Gauss
The discussion of the preceding section suggested a possible extension of Green’s theorem to
three dimensions, yielding what is known as the divergence theorem.

THEOREM 12.8 The Divergence Theorem of Gauss

Let be a piecewise smooth closed surface bounding a region M of 3-space. Let have unit
outer normal n.Let F be a vector field with continuous first and second partial derivatives on
and throughout M. Then

F · ndσ = ∇· FdV. (12.10)

M
The theorem is named for the great nineteenth-century German mathematician and scientist
Carl Friedrich Gauss, and is actually a conservation of mass equation. Recall that the diver-
gence of a vector field at a point is a measure of the flow of the field away from that point.
Equation (12.10) states that the flux of the vector field outward from M across exactly bal-
ances the flow of the field from each point in M. Whatever crosses the surface and leaves M must
be accounted for by flow out of M (in the absence of sources or sinks in M).
We will look at two computational examples to get some feeling for equation (12.10), and
then consider applications.

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October 14, 2010 14:53 THM/NEIL Page-402 27410_12_ch12_p367-424