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

1106 Chapter 16 Vector Calculus
Differentiating (8) with respect to y, we get f y sx, y, zd − 2xyz 3 1 t y sy, zd, so comparison
with (6) gives t y sy, zd − 0. Thus tsy, zd − hszd and
Then (7) gives h9szd − 0. Therefore
f z sx, y, zd − 3xy 2 z 2 1 h9szd
f sx, y, zd − xy 2 z 3 1 K
■
(x, y, z)
FIGURE 1
curl F(x, y, z)
The reason for the name curl is that the curl vector is associated with rotations. One
connection is explained in Exercise 37. Another occurs when F represents the velocity
field in fluid flow (see Example 16.1.3). Particles near sx, y, zd in the fluid tend to rotate
about the axis that points in the direction of curl Fsx, y, zd, and the length of this curl
vector is a measure of how quickly the particles move around the axis (see Figure 1). If
curl F − 0 at a point P, then the fluid is free from rotations at P and F is called irrotational
at P. In other words, there is no whirlpool or eddy at P. If curl F − 0, then a
tiny paddle wheel moves with the fluid but doesn’t rotate about its axis. If curl F ± 0, the
paddle wheel rotates about its axis. We give a more detailed explanation in Section 16.8
as a consequence of Stokes’ Theorem.
Divergence
If F − P i 1 Q j 1 R k is a vector field on R 3 and −Py−x, −Qy−y, and −Ry−z exist, then
the divergence of F is the function of three variables defined by
9 div F − −P
−x 1 −Q
−y 1 −R
−z
Observe that curl F is a vector field but div F is a scalar field. In terms of the gradient
operator = − s−y−xd i 1 s−y−yd j 1 s−y−zd k, the divergence of F can be written
symbolically as the dot product of = and F:
10 div F − = F
ExamplE 4 If Fsx, y, zd − xz i 1 xyz j 2 y 2 k, find div F.
SOLUTION By the definition of divergence (Equation 9 or 10) we have
div F − = F − − −x sxzd 1 − −y sxyzd 1 − −z s2y 2 d − z 1 xz
■
If F is a vector field on R 3 , then curl F is also a vector field on R 3 . As such, we can
compute its divergence. The next theorem shows that the result is 0.
11 Theorem If F − P i 1 Q j 1 R k is a vector field on R 3 and P, Q, and R
have continuous second-order partial derivatives, then
div curl F − 0
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.