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%TCIDATA{<META NAME="Title" CONTENT="Plots of Vector Fields and Gradients">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Wed Mar 12 12:10:51 1997}
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\section{\label{Vector field}\label{Plot 2D, vector field}Plots of Vector
Fields and Gradients}

A function that assigns a vector to each point of a region in two- or
three-dimensional space is called a \textsl{vector field}. The gradient of a
scalar-valued function of two variables is a vector field.

\subsection{\ Plots of Vector Fields}

\index{Plot 2D@Plot 2D!vector field@vector field}The operation 
\index{Vector@Vector!field@field}\textsf{Plot 2D + Vector Field} requires a
pair of expressions in two variables representing the horizontal and
vertical components of the vector field.\medskip

\begin{quote}
$\blacktriangleright $ To plot a two-dimensional vector field
\end{quote}

\begin{enumerate}
\item[1.]  Type a pair of two-variable expressions, representing the
horizontal and vertical components of a vector field, into a vector.

\item[2.]  Leave the insertion point in the vector, and from the \textsf{%
Plot 2D} submenu, choose \textsf{Vector Field}.\medskip 
\end{enumerate}

\strut

\begin{example}
To visualize the vector field $F(x,y)=\left[ x+y,x-y\right] $

Place the insertion point in the vector $\left[ x+y,x-y\right] $, and from
the \textsf{Plot 2D} submenu, choose \textsf{Vector Field.\medskip }
\end{example}

\begin{quote}
$\blacktriangleright $ \textsf{Plot 2D + Vector Field}
\end{quote}

\begin{quotation}
$\left[ x+y,x-y\right] $
\end{quotation}

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\label{Direction field}At a point $\left( x,y\right) $ on a solution curve
of a differential equation of the form $\dfrac{dy}{dx}=f(x,y)$ the curve has
slope $f(x,y)$. You can get an idea of the appearance of the graphs of the
solution of a differential equation from the \emph{direction field}--that
is, a plot depicting short line segments with slope $f(x,y)$ at points $%
(x,y) $. This can be done using \textsf{Plot 2D + Vector Field}.

\begin{example}
The direction field for the differential equation $\frac{dy}{dt}=y^{2}-t^{2}$
is the two-dimensional vector field plot of the vector valued function 
\[
F\left( t,y\right) =\frac{\left( 1,y^{2}-t^{2}\right) }{\left\| \left(
1,y^{2}-t^{2}\right) \right\| }
\]
which assigns to each point $\left( t,y\right) $ a vector of length one in
the direction of the derivative at the point $(t,y)$..

$\blacktriangleright $ \textsf{Plot 2D + Vector Field}

$\left( \dfrac{1}{\sqrt{\left( 1+\left| y^{2}-t^{2}\right| ^{2}\right) }},%
\dfrac{y^{2}-t^{2}}{\sqrt{\left( 1+\left| y^{2}-t^{2}\right| ^{2}\right) }}%
\right) $

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\end{example}

\begin{quote}
$\blacktriangleright $ To plot a three-dimensional vector field\label%
{Plot 3D, vector field}
\end{quote}

\begin{enumerate}
\item[1.]  Type three three-variable expressions, representing the $x$-, $y$%
-, and $z$-components of a vector field, into a vector.

\item[2.]  Leave the insertion point in the vector.

\item[3.]  From the \textsf{Plot 3D} submenu, choose \textsf{Vector Field}%
.\medskip 
\end{enumerate}

\begin{quote}
$\blacktriangleright $ \textsf{Plot 3D + Vector Field}
\end{quote}

\begin{quotation}
$[yz,xz,xy]$
\end{quotation}

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The three-dimensional version is often a challenge to visualize. Multiple
views can be helpful. Three or four views of the vector field $%
f(x,y,z)=\left( yz,xz,xy\right) $ can provide a reasonable graphical
representation. Boxed axes can also help the visualization.

\begin{quote}
$\blacktriangleright $ To change the view
\end{quote}

\begin{enumerate}
\item[1.]  Click the frame until a small box appears in the upper-right
corner of the frame.

\item[2.]  With the left mouse button held down, rotate the plot.
\end{enumerate}

\begin{center}
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\end{center}

\subsection{\ Plots of Gradient Fields\label{Plot 2D, gradient}}

Scalar-valued functions of two variables can be visualized in several ways.
Given the function $f(x,y)=xy\sin xy$, choosing \textsf{Rectangular} from
the \textsf{Plot 3D} submenu produces a surface represented by the function
values. Another way to visualize such a function is to choose \textsf{%
Gradient }from 
\index{Plot 2D@Plot 2D!gradient@gradient} the\textsf{\ Plot 2D }submenu.
This choice produces a plot of the vector field that is the gradient of this
expression, plotting vectors at grid points whose magnitude and direction
indicate the steepness of the surface and the direction of steepest ascent. 
\index{Gradient@Gradient}\label{Gradient field}

The vector field that assigns to each point $\left( x,y\right) $ the
gradient of $f$ at $\left( x,y\right) $ is called the \textsl{gradient field}
associated with the function $f$.\medskip

\begin{quote}
$\blacktriangleright $ To plot a gradient field
\end{quote}

\begin{enumerate}
\item[1.]  Type an expression $f(x,y)$.

\item[2.]  Leave the insertion point in the expression, and from the \textsf{%
Plot 2D} submenu, choose \textsf{Gradient}.\medskip 
\end{enumerate}

For example, type the expression $x^{2}+2y^{2}$, and choose \textsf{Gradient}
from the \textsf{Plot 2D} submenu. This procedure produces a plot of the
vector field that is the gradient of this expression. The following shows
the relative steepness on the left, the surface on the right.\medskip

\begin{quote}
$\blacktriangleright $ \textsf{Plot 2D + Gradient, Plot 3D + Rectangular}
\end{quote}

\begin{quotation}
$x^{2}+2y^{2}$
\end{quotation}

\begin{center}
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\end{center}

The gradient field for a scalar-valued function $f(x,y,z)$ of three
variables is a three-dimensional vector field where each vector represents
the direction of maximal increase. The surface represented by the function
values is embedded in four-dimensional space, so you must use indirect
methods such as plotting the gradient field to help you visualize this
surface.\medskip

\begin{quote}
$\blacktriangleright $ \textsf{Plot 3D + Gradient\label{Plot 3D, gradient}}
\end{quote}

\begin{quotation}
$xz+xy+yz$
\end{quotation}

\begin{center}
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As with other three-dimensional vector fields, multiple views convey
graphical information more effectively than does a single view.

\subsection{\ Related topics}

\begin{itemize}
\item  \hyperref{Plotting Curves and Surfaces}{}{}{DM6-1.tex#Plotting}

\item  \hyperref{2D Plots of Functions and Expressions}{}{}{DM6-4.tex#Plot
expression}

\item  \hyperref{3D Plots of Functions and Expressions}{}{}{DM6-6.tex#3D
Plots}

\item  \hyperref{Vectors}{}{}{DM9-1.tex#Vectors}

\item  \hyperref{Gradient, Divergence, and Curl}{}{}{DM9-2.tex#Gradient}

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

\item  \hyperref{Matrix-Valued Operators}{}{}{DM9-5.tex#Hessian}

\item  \hyperref{Plots of Complex Functions}{}{}{DM9-6.tex#Conformal plot}

\item  \hyperref{Exercises and Solutions}{}{}{DM9-7.tex#Exercises}
\end{itemize}

\rule{0.67in}{0.01in}

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"1";cropbottom "0";filename 'index.wmf';file-properties "XNPEU";}}}{}{}{%
DMIndex.tex#Math Index}\quad {\small Index entries: \hyperref{Plotting 2D}{}{%
}{DMIndex.tex#Plotting 2D}, \hyperref{Plotting 3D}{}{}{DMIndex.tex#Plotting
3D}, \hyperref{Vector calculus}{}{}{DMIndex.tex#Vector calculus}}

\end{document}
