James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

1108 Chapter 16 Vector Calculus
Vector Forms of Green’s Theorem
The curl and divergence operators allow us to rewrite Green’s Theorem in versions that
will be useful in our later work. We suppose that the plane region D, its boundary curve
C, and the functions P and Q satisfy the hypotheses of Green’s Theorem. Then we consider
the vector field F − P i 1 Q j. Its line integral is
y C
F dr − y C
P dx 1 Q dy
and, regarding F as a vector field on R 3 with third component 0, we have
i j k
− − −
curl F −
−S −Q
−x −y −z −x 2 −P k
−yD
Psx, yd Qsx, yd 0
Therefore
scurl Fd k −S −Q
−x −yD 2 −P k k − −Q
−x 2 −P
−y
and we can now rewrite the equation in Green’s Theorem in the vector form
12 y F dr −
C
yy scurl Fd k dA
D
Equation 12 expresses the line integral of the tangential component of F along C as
the double integral of the vertical component of curl F over the region D enclosed by C.
We now derive a similar formula involving the normal component of F.
If C is given by the vector equation
rstd − xstd i 1 ystd j
a < t < b
y
then the unit tangent vector (see Section 13.2) is
0
FIGURE 2
D
T(t)
r(t) n(t)
C
x
Tstd −
x9std
| r9std | i 1 y9std
| r9std | j
You can verify that the outward unit normal vector to C is given by
nstd −
y9std
| r9std | i 2 x9std
(See Figure 2.) Then, from Equation 16.2.3, we have
| r9std | j
y F n ds −
C
y b
sF ndstd | r9std | dt
a
b
− y
Psxstd, ystdd y9std Qsxstd, ystdd x9std
F
a | r9std 2
|
| r9std |
G | r9std | dt
− y b
Psxstd, ystdd y9std dt 2 Qsxstd, ystdd x9std dt
a
− y P dy 2 Q dx −
C
y S
Dy −P
−x −yD 1 −Q dA
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