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Unformatted text preview: ming the ﬁrst, apply the ﬁrst with F = f a, where
a is any constant vector.
ˆ
f a · n dS = ∇ · (f a) dV
V ∂V = (∇f ) · a + f ∇ · a dV
V = (∇f ) · a dV
V To get the second line, we used vector identity # 8. To get the third line, we just used
that a is a constant, so that it is annihilated by all derivatives. Since a is a constant, we
can factor it out of both integrals, so
ˆ
f n dS = a · a· ∇ f dV
V ∂V ˆ
f n dS − =⇒ a · ∇f dV =0 V
∂V
March 3, 2013 Divergence Theorem and Variations...
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 Fall '13
 JoelFeldman
 Calculus

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