Calculus 2nd Edition Rogawski

1034 C H A P T E R 18 FUNDAMENTAL THEOREMS OF VECTOR ANALYSIS
(a) Let m(r) and M(r) be the minimum and maximum values of
curl(F(P )) · e for P ∈ S r . Prove that
m(r) ≤ 1
πr 2 ∫∫
S r
curl(F) · dS ≤ M(r)
(b) Prove that
∫
1
curl(F(Q)) · e = lim r→0 πr 2 F · ds
C r
This proves that curl(F(Q)) · e is the circulation per unit area in the
plane S.
18.3 Divergence Theorem
We have studied several “Fundamental Theorems.” Each of these is a relation of the type:
Integral of a derivative
on an oriented domain
=
Integral over the oriented
boundary of the domain
Here are the examples we have seen so far:
• In single-variable calculus, the Fundamental Theorem of Calculus (FTC) relates the
integral of f ′ (x) over an interval [a,b] to the “integral” of f (x) over the boundary
of [a,b] consisting of two points a and b:
− P
C
+ Q
FIGURE 1 The oriented boundary of C is
∂C = Q − P .
D
C = ∂D
FIGURE 2 Domain D in R 2 with boundary
curve C = ∂D.
FIGURE 3 The oriented boundary of S is
C = ∂S.
C
∫ b
f ′ (x) dx = f (b) − f (a)
a
} {{ }
} {{ } “Integral” over the boundary of [a,b]
Integral of derivative over [a,b]
The boundary of [a,b] is oriented by assigning a plus sign to b and a minus sign
to a.
• The Fundamental Theorem for Line Integrals generalizes the FTC: Instead of an
interval [a,b] (a path from a to b along the x-axis), we take any path from points
P to Q in R 3 (Figure 1), and instead of f ′ (x) we use the gradient:
∫
∇V · ds = V (Q) − V (P )
C
} {{ }
} {{ }
“Integral” over the
Integral of derivative over a curve boundary ∂C = Q − P
• Green’s Theorem is a two-dimensional version of the FTC that relates the integral
of a derivative over a domain D in the plane to an integral over its boundary curve
C = ∂D (Figure 2):
∫∫ ( )
∫
∂F2
dA = F · ds
D ∂y − ∂F 1
∂x
} {{ }
Integral of derivative over domain
C
} {{ }
Integral over boundary curve
• Stokes’ Theorem extends Green’s Theorem: Instead of a domain in the plane (a flat
surface), we allow any surface in R 3 (Figure 3). The appropriate derivative is the
curl:
∫∫
∫
curl(F) · dS = F · ds
S
} {{ }
Integral of derivative over surface
C
} {{ }
Integral over boundary curve
Our last theorem—the Divergence Theorem—follows this pattern:
∫∫∫
∫∫
div(F)dW = F · dS
W
S
} {{ }
} {{ }
Integral of derivative over 3-D region Integral over boundary surface