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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Vector Fields">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sun Mar 30 14:20:49 1997}
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\section{Vector Fields\label{Vector Field}}

\begin{itemize}
\begin{definition}
Let $D$ denote a subset of the plane. \ A \textbf{vector field} on $D$ is a
function $\mathbf{F}$ that assigns to each point $(x,y)$ in $D$ a
two-dimensional vector $\mathbf{F}(x,y)$. \ In terms of its component
functions, the vector field $\mathbf{F}$ is given by 
\[
\mathbf{F}(x,y)=P(x,y)\mathbf{i}+Q(x,y)\mathbf{j}=\left\langle P\left(
x,y\right) ,Q\left( x,y\right) \right\rangle 
\]
or, for short, 
\[
\mathbf{F}=P\mathbf{i}+Q\mathbf{j}
\]
\end{definition}

\begin{definition}
Let $D$ denote a subset of space. \ A \textbf{vector field} on $D$ is a
function $\mathbf{F}$ that assigns to each point $(x,y,z)$ in $D$ a
three-dimensional vector $\mathbf{F}(x,y,z)$. \ In terms of its component
functions, the vector field $\mathbf{F}$ is given by 
\[
\mathbf{F}(x,y,z)=P(x,y,z)\mathbf{i}+Q(x,y,z)\mathbf{j}+R(x,y,z)\mathbf{k}%
=\left\langle P\left( x,y,z\right) ,Q\left( x,y,z\right)
,R(x,y,z)\right\rangle 
\]
or, for short, 
\[
\mathbf{F}=P\mathbf{i}+Q\mathbf{j}+R\mathbf{k}
\]
\end{definition}

\begin{definition}
If $f$ is a scalar function of two variables, its \textbf{gradient vector
field} $\nabla f$ is defined by 
\[
\nabla f=\frac{\partial f}{\partial x}\mathbf{i}+\frac{\partial f}{\partial y%
}\mathbf{j}
\]
If $f$ is a scalar function of three variables, its \textbf{gradient vector
field} $\nabla f$ is defined by 
\[
\nabla f=\frac{\partial f}{\partial x}\mathbf{i}+\frac{\partial f}{\partial y%
}\mathbf{j}+\frac{\partial f}{\partial z}\mathbf{k}
\]
\end{definition}

\begin{definition}
$\mathbf{F}$ is a \textbf{conservative vector field} if there exists a
scalar function $f$ such that $\nabla f=\mathbf{F}$. \ In this case $f$ is
called a \textbf{potential function} for $\mathbf{F}$\textbf{.}
\end{definition}
\end{itemize}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Vectors}{}{}{../../Help/DM9-1.tex#Vectors}}

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

{\small \hyperref{Plots of Vector Fields and Gradients}{}{}{%
../../Help/DM9-3.tex#Vector field}}

{\small \hyperref{Scalar and Vector Potentials}{}{}{%
../../Help/DM9-4.tex#Scalar potential}}

\hrulefill

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