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

section 16.5 Curl and Divergence 1107
Proof Using the definitions of divergence and curl, we have
Note the analogy with the scalar triple
product: a sa 3 bd − 0.
div curl F − = s= 3 Fd
− S S − −R
−x −y 2 −Q
−zD1 − −P
−y −z −xD 2 −R 1 S − −Q
−z −x −yD 2 −P
−
− 0
−2 R
−x −y 2 −2 Q
−x −z 1
−2 P
−y −z 2
−2 R
−y −x 1 −2 Q
−z −x 2
−2 P
−z −y
because the terms cancel in pairs by Clairaut’s Theorem.
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ExamplE 5 Show that the vector field Fsx, y, zd − xz i 1 xyz j 2 y 2 k can’t be
written as the curl of another vector field, that is, F ± curl G.
SOLUTION In Example 4 we showed that
div F − z 1 xz
and therefore div F ± 0. If it were true that F − curl G, then Theorem 11 would give
div F − div curl G − 0
which contradicts div F ± 0. Therefore F is not the curl of another vector field.
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The reason for this interpretation of
div F will be explained at the end of
Section 16.9 as a consequence of the
Divergence Theorem.
Again, the reason for the name divergence can be understood in the context of fluid
flow. If Fsx, y, zd is the velocity of a fluid (or gas), then div Fsx, y, zd represents the net
rate of change (with respect to time) of the mass of fluid (or gas) flowing from the point
sx, y, zd per unit volume. In other words, div Fsx, y, zd measures the tendency of the fluid
to diverge from the point sx, y, zd. If div F − 0, then F is said to be incompressible.
Another differential operator occurs when we compute the divergence of a gradient
vector field =f . If f is a function of three variables, we have
divs=f d − = s=f d − −2 f
−x 2
1 −2 f
−y 2
1 −2 f
−z 2
and this expression occurs so often that we abbreviate it as = 2 f . The operator
= 2 − = =
is called the Laplace operator because of its relation to Laplace’s equation
= 2 f − −2 f
−x 2
1 −2 f
−y 2
1 −2 f
−z 2 − 0
We can also apply the Laplace operator = 2 to a vector field
in terms of its components:
F − P i 1 Q j 1 R k
= 2 F − = 2 P i 1 = 2 Q j 1 = 2 R k
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