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

1104 chapter 16 Vector Calculus
ExamplE 1 If Fsx, y, zd − xz i 1 xyz j 2 y 2 k, find curl F.
SOLUTION Using Equation 2, we have
CAS Most computer algebra systems
have commands that compute the curl
and divergence of vector fields. If
you have access to a CAS, use these
commands to check the answers to the
examples and exercises in this section.
curl F − = 3 F −
i j k
− − −
−x −y −z
xz xyz 2y 2
−F − −y s2y 2 d 2 − −z sxyzd G i 2F − −x s2y 2 d 2 − −z sxzd G j
1F − −x sxyzd 2 − −y sxzd G k
− s22y 2 xyd i 2 s0 2 xd j 1 syz 2 0d k
− 2ys2 1 xd i 1 x j 1 yz k
■
Recall that the gradient of a function f of three variables is a vector field on R 3 and
so we can compute its curl. The following theorem says that the curl of a gradient vector
field is 0.
3 Theorem If f is a function of three variables that has continuous secondorder
partial derivatives, then
curls=f d − 0
Notice the similarity to what we know
from Section 12.4: a 3 a − 0 for every
three-dimensional vector a.
Proof We have
curls=f d − = 3 s=f d −
−S −2
f
−y −z 2
i j k
−
−x
−f
−x
−
−y
−f
−y
−2 f
i
−z −yD 1S −2
− 0 i 1 0 j 1 0 k − 0
−
−z
−f
−z
f
−z −x 2
−2 f
j
−x −zD 1S −2
f
−x −y 2
−2 f
−y −xD k
by Clairaut’s Theorem.
■
Since a conservative vector field is one for which F − =f , Theorem 3 can be re phrased
as follows:
Compare this with Exercise 16.3.29.
If F is conservative, then curl F − 0.
This gives us a way of verifying that a vector field is not conservative.
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