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\begin{center}
\textbf{Vector Calculus}

\textit{\textbf{Grad, Div and Curl Identities}}
\end{center}

\textbf{Question}

If the field lines of the vector field $\un{F}(x,y,z)$ are parallel
straight lines, what does this tell you about div$\un{F}$ and
curl$\un{F}$?


\textbf{Answer}

If the field lines are parallel straight lines, in the direction of the
non-zero vector $\un{a}$, where $\un{a}$ is a constant, then 
$$\un{F}(x,y,z) = \phi (x,y,z) \un{a}$$
with $\phi$ being a smooth scalar field. It is also given that
\begin{eqnarray*}
\textrm{div}\un{F} & = & \textrm{div}(\phi \un{a}) = \nabla \phi
\bullet \un{a}\\
\textrm{curl}\un{F} & = & \textrm{curl}(\phi \un{a}) = \nabla \phi
\times \un{a}.
\end{eqnarray*}
As $\nabla\phi$ is an arbitrary gradient, so div$\un{F}$ can take any
value. However, curl$\un{F}$ will be perpendicular to $\un{a}$, and so
also perpendicular to $\un{F}$.

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