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 <!-- --- begin quiz question --- -->
 The equation

 $$
 \begin{equation}
 \nabla\cdot\boldsymbol{u} = 0
 \tag{3}
 \end{equation}
 $$

 is famous in physics. Select the wrong assertion(s):
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 <!-- --- keywords:['gradient', 'divergence', 'curl', 'vector calculus'] -->
 <!-- --- label:div:assert -->

 <!-- --- begin quiz choice 1 (wrong) --- -->
 The equation tells that the net outflow of something with
 velocity \( \boldsymbol{u} \) in region is zero.
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 <!-- --- begin explanation of choice 1 --- -->
 This is right: integrating (3) over an arbitrary
 domain \( \Omega \) and using Gauss' divergence theorem, we
 get the surface integral

 $$ \int_{\partial\Omega}\boldsymbol{u}\cdot\boldsymbol{n}dS=0,$$

 where \( \boldsymbol{n} \) is an outward unit normal on the
 boundary \( \partial\Omega \).
 The quantity \( \boldsymbol{u}\cdot\boldsymbol{n}dS \) is the
 outflow of volume per time unit if \( \boldsymbol{u} \) is
 velocity.
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 <!-- --- begin quiz choice 2 (wrong) --- -->
 The equation tells that the vector field \( \boldsymbol{u} \)
 is divergence free.
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 <!-- --- begin explanation of choice 2 --- -->
 Yes, <em>divergence free</em> is often used as synonym for
 <em>zero divergence</em>, and \( \nabla\cdot\boldsymbol{u} \)
 is the divergence of a vector field \( \boldsymbol{u} \).
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 <!-- --- begin quiz choice 3 (wrong) --- -->
 The equation implies that there exists a vector potential
 \( \boldsymbol{A} \) such that
 \( \boldsymbol{u}=\nabla\times\boldsymbol{A} \).
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 <!-- --- begin explanation of choice 3 --- -->
 Yes, this is an important result in vector calculus that is much
 used in electromagnetics.
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 <!-- --- begin quiz choice 4 (right) --- -->
 The equation implies \( \nabla\times\boldsymbol{u}=0 \).
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 <!-- --- begin explanation of choice 4 --- -->
 No, only if \( \boldsymbol{u}=\nabla\phi \), for some scalar
 potential \( \phi \), we have \( \nabla\times\boldsymbol{u}=0 \).
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 The equation implies that \( \boldsymbol{u} \)
 must be a constant vector field.
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 <!-- --- begin explanation of choice 5 --- -->
 No, it is the <em>sum</em> of derivatives of different components
 of \( \boldsymbol{u} \) that is zero. Only in one dimension,
 where \( \boldsymbol{u}=u_x\boldsymbol{i} \) and consequently
 \( \nabla\cdot\boldsymbol{u}=du/dx \), the vector field must be
 constant.
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