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%TCIDATA{<META NAME="Title" CONTENT="Exercises">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 17:13:32 1997}
%TCIDATA{Language=American English}
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\begin{document}


\section{Exercises\label{Exercises}}

\begin{enumerate}
\item[1.]  \label{q1}\label{Directional derivative}Evaluate the directional
derivative of $f\left( x,y,z\right) =3x-5y+2z$ at $\left( 2,2,1\right) $ in
the direction of the outward normal to the sphere $x^{2}+y^{2}+z^{2}=9$%
.\hfill \hyperref{\TCIIcon{BITMAPSETAnswer}{0.2603in}{0.2421in}{0in}}{}{}{a1}

\item[2.]  \label{q2}Find a vector $v$ normal to the surface $z=\sqrt{%
x^{2}+y^{2}}+\left( x^{2}+y^{2}\right) ^{3/2}$ at the point $\left(
x,y,z\right) \neq \left( 0,0,0\right) $ on the surface.\hfill \hyperref{%
\TCIIcon{BITMAPSETAnswer}{0.2603in}{0.2421in}{0in}}{}{}{a2}

\item[3.]  \label{q3}Let $f\left( x,y,z\right) =\dfrac{mM}{\sqrt{%
x^{2}+y^{2}+z^{2}}}$ denote 
\index{Newton's gravitational potential@Newton's gravitational potential}%
Newton's gravitational potential. Show that the gradient is given by 
\[
\nabla f\left( x,y,z\right) =-%
\dfrac{mM}{\left( x^{2}+y^{2}+z^{2}\right) ^{3/2}}\left[ 
\begin{array}{c}
x \\ 
y \\ 
z
\end{array}
\right] 
\]

\ \ \medskip \hfill \hyperref{\TCIIcon{BITMAPSETAnswer}{0.2603in}{0.2421in}{%
0in}}{}{}{a3}
\end{enumerate}

\section{Solutions\label{Solutions}}

\begin{enumerate}
\item[1.]  \label{a1}The directional derivative is given by $D_{u}f\left(
x,y,z)\right) =\nabla f\left( x,y,z\right) \cdot u$, where $u$ is a unit
vector in the direction of the outward normal to the sphere $%
x^{2}+y^{2}+z^{2}=9$. The vector 
\[
\nabla \left( x^{2}+y^{2}+z^{2}\right) =\,\left[ 
\begin{array}{c}
2x \\ 
2y \\ 
2z
\end{array}
\right] 
\]
is normal to the sphere $x^{2}+y^{2}+z^{2}=9$, and at $\left( 2,2,1\right) $
this normal is $\left( 4,4,2\right) $. A unit vector in the same direction
is given by 
\[
u=\left[ 
\begin{array}{c}
4 \\ 
4 \\ 
2
\end{array}
\right] \div \left\| \left[ 
\begin{array}{c}
4 \\ 
4 \\ 
2
\end{array}
\right] \right\| =\,\frac{1}{6}\left[ 
\begin{array}{c}
4 \\ 
4 \\ 
2
\end{array}
\right] =\,\left[ 
\begin{array}{c}
2/3 \\ 
2/3 \\ 
1/3
\end{array}
\right] 
\]
Hence, $\nabla f\left( x,y,z\right) \cdot u$ evaluated at $(2,2,1)$ is$\,-%
\frac{2}{3}$.\hfill \hyperref{\TCIIcon{BITMAPSETProbSolvHint}{0.2006in}{%
0.243in}{0in}}{}{}{q1}

\item[2.]  \label{a2}A normal vector is given by 
\[
\begin{array}{l}
\nabla \left( \sqrt{x^{2}+y^{2}}+\left( x^{2}+y^{2}\right) ^{3/2}-z\right)
\\ 
\qquad \qquad =\left( \dfrac{1}{\sqrt{\left( x^{2}+y^{2}\right) }}x+3\sqrt{%
\left( x^{2}+y^{2}\right) }x,\dfrac{1}{\sqrt{\left( x^{2}+y^{2}\right) }}y+3%
\sqrt{\left( x^{2}+y^{2}\right) }y,-1\right) \\ 
\qquad \qquad =\left( x\dfrac{1+3x^{2}+3y^{2}}{\sqrt{\left(
x^{2}+y^{2}\right) }},y\dfrac{1+3x^{2}+3y^{2}}{\sqrt{\left(
x^{2}+y^{2}\right) }},-1\right) \\ 
\qquad \qquad =\dfrac{1}{\sqrt{\left( x^{2}+y^{2}\right) }}\left( x\left(
1+3x^{2}+3y^{2}\right) ,y\left( 1+3x^{2}+3y^{2}\right) ,-\sqrt{\left(
x^{2}+y^{2}\right) }\right)
\end{array}
\]

\begin{itemize}
\item  Hence, any scalar multiple of \hfill \hyperref{\TCIIcon{%
BITMAPSETProbSolvHint}{0.2006in}{0.243in}{0in}}{}{}{q2} 
\[
\left[ x\left( 1+3x^{2}+3y^{2}\right) ,y\left( 1+3x^{2}+3y^{2}\right) ,-%
\sqrt{\left( x^{2}+y^{2}\right) }\right] 
\]
is also normal to the given surface.
\end{itemize}

\item  \label{a3}Evaluate the expression $\nabla \left( \dfrac{mM}{\sqrt{%
x^{2}+y^{2}+z^{2}}}\right) $. Then delete the first two rows of the vector,
because $m$ and $M$ are constant parameters. \hfill \hyperref{\TCIIcon{%
BITMAPSETProbSolvHint}{0.2006in}{0.243in}{0in}}{}{}{q3} 
\[
\begin{array}{l}
\nabla \left( \dfrac{mM}{\sqrt{x^{2}+y^{2}+z^{2}}}\right) \vspace{8pt} \\ 
\qquad =\left( -m\dfrac{M}{\left( \sqrt{\left( x^{2}+y^{2}+z^{2}\right) }%
\right) ^{3}}x,-m\dfrac{M}{\left( \sqrt{\left( x^{2}+y^{2}+z^{2}\right) }%
\right) ^{3}}y,-m\dfrac{M}{\left( \sqrt{\left( x^{2}+y^{2}+z^{2}\right) }%
\right) ^{3}}z\right) \vspace{8pt} \\ 
\qquad =-\;\dfrac{mM}{\left( x^{2}+y^{2}+z^{2}\right) ^{3/2}}\left(
x,y,z\right)
\end{array}
\]
This gives the Newtonian gravitational force between two objects of masses $%
m $ and $M$, with one object at the origin and the other at the point $%
\left( x,y,z\right) $.\medskip
\end{enumerate}

\subsection{\ Related topics}

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

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

\item  \hyperref{Plots of Vector Fields and Gradients}{}{}{DM9-3.tex#Vector
field}

\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}

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DMIndex.tex#Math Index}\quad {\small Index entries: \hyperref{Vector calculus%
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\end{document}
