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

1138 Chapter 16 Vector Calculus
and recall that v T is the component of v in the direction of the unit tangent vector T.
This means that the closer the direction of v is to the direction of T, the larger the value
of v T. Thus y C
v dr is a measure of the tendency of the fluid to move around C and
is called the circulation of v around C. (See Figure 5.)
T
T
C
FIGURE 5
C
(a) j C
v dr>0, positive circulation
v
(b) j C
v dr<0, negative circulation
v
Now let P 0 sx 0 , y 0 , z 0 d be a point in the fluid and let S a be a small disk with radius a and
center P 0 . Then (curl FdsPd < scurl FdsP 0 d for all points P on S a because curl F is continuous.
Thus, by Stokes’ Theorem, we get the following approximation to the circulation
around the boundary circle C a :
y C a
v dr − y
S a
y curl v dS − y
S a
y curl v n dS
< y
S a
y curl vsP 0 d nsP 0 d dS − curl vsP 0 d nsP 0 da 2
Imagine a tiny paddle wheel placed in
the fluid at a point P, as in Figure 6; the
paddle wheel rotates fastest when its
axis is parallel to curl v.
This approximation becomes better as a l 0 and we have
4 curl vsP 0 d nsP 0 d − lim
a l 0
1
y v dr
a 2 C a
FIGURE 6
curl v
Equation 4 gives the relationship between the curl and the circulation. It shows that
curl v n is a measure of the rotating effect of the fluid about the axis n. The curling
effect is greatest about the axis parallel to curl v.
Finally, we mention that Stokes’ Theorem can be used to prove Theorem 16.5.4
(which states that if curl F − 0 on all of R 3 , then F is conservative). From our pre vious
work (Theorems 16.3.3 and 16.3.4), we know that F is conservative if y C
F dr − 0 for
every closed path C. Given C, suppose we can find an orientable surface S whose boundary
is C. (This can be done, but the proof requires advanced techniques.) Then Stokes’
Theorem gives
y F dr −
C
yy curl F dS − y
S
S
y 0 dS − 0
A curve that is not simple can be broken into a number of simple curves, and the integrals
around these simple curves are all 0. Adding these integrals, we obtain y C
F dr − 0 for
any closed curve C.
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