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%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\section{Curl and Divergence\label{Curl}}

Two operations on vector fields---curl and divergence---play a basic role in
the applications of vector calculus. Both operations resemble
differentiation, but in one case the result is another vector field, and in
the other case the result is a scalar function.

\subsubsection{Curl}

\begin{itemize}
\item  Let $\mathbf{F}=P\mathbf{i}+Q\mathbf{j}+R\mathbf{k}$ denote a vector
field for which the partial derivatives of \ $P$, $Q$, and $R$ all exist. \
The \textbf{curl} of $\mathbf{F}$ is the vector field defined by 
\[
\limfunc{curl}\mathbf{F}=\left( \frac{\partial R}{\partial y}-\frac{\partial
Q}{\partial z}\right) \mathbf{i}+\left( \frac{\partial P}{\partial z}-\frac{%
\partial R}{\partial x}\right) \mathbf{j}+\left( \frac{\partial Q}{\partial x%
}-\frac{\partial P}{\partial y}\right) \mathbf{k}
\]
The easiest way to remember this formula is by means of the symbolic
expression 
\[
\limfunc{curl}\mathbf{F}=\nabla \times \mathbf{F}=\left| 
\begin{array}{lll}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\ 
\frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial 
}{\partial z} \\ 
P & Q & R
\end{array}
\right| 
\]
Here we have made use of the operator notation 
\[
\nabla =\frac{\partial }{\partial x}\mathbf{i+}\frac{\partial }{\partial y}%
\mathbf{j+}\frac{\partial }{\partial z}\mathbf{k}\text{.}
\]
The main theorem about curl is the following:
\end{itemize}

\begin{theorem}
Let $\mathbf{F}$ be a vector field defined on all of three dimensional
space, which has continuous partial derivatives. Then $\limfunc{curl}\mathbf{%
F}=0$ if and only if $\mathbf{F}$ is a conservative vector field. \ In
particular $\limfunc{curl}\left( \nabla f\right) =0.$
\end{theorem}

\subsubsection{Divergence}

\begin{itemize}
\item  Let $\mathbf{F}=P\mathbf{i}+Q\mathbf{j}+R\mathbf{k}$ denote a vector
field for which the partial derivatives of \ $P$, $Q$, and $R$ all exist. \
The \textbf{divergence} of $\mathbf{F}$ is the vector field defined by 
\[
\limfunc{div}\mathbf{F}=\nabla \cdot \mathbf{F=}\frac{\partial P}{\partial x}%
+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z} 
\]

The main theorem about divergence is the following:
\end{itemize}

\begin{theorem}
Let $\mathbf{F}$ be a vector field defined on all of three dimensional
space, which has continuous partial derivatives. Then $\limfunc{div}\mathbf{F%
}=0$ if and only if $\mathbf{F}$ is a curl of another vector field $\mathbf{G%
}$ (called the vector potential of $\mathbf{F}$\textbf{)}. \ In particular $%
\limfunc{div}\limfunc{curl}\mathbf{F}=0.$
\end{theorem}

\strut

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Gradient, Divergence, and Curl}{}{}{%
../../Help/DM9-2.tex#Gradient}}\hfill

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