\magnification=\magstep1
\baselineskip=12pt
\parskip=6pt
\input amstex
\loadbold
\loadmsbm
\UseAMSsymbols
\def\bR{\bold R}
\def\bi{\bold i}
\def\bj{\bold j}
\def\bk{\bold k}
\def\bn{\bold n}
\def\br{\bold r}
\def\bE{\bold E}
\def\bF{\bold F}
\def\bx{\bold x}
\def\ba{\bold a}
\def\bb{\bold b}
\def\bT{\bold T}
\def\bt{\bold t}
\def\ds{\displaystyle}
\line{MA 174\hfil FINAL EXAM\hfil 5/7/99}
\medskip
\line{D.~Gottlieb\hfil Name\ {\hbox to 2.5truein{\hrulefill}}}
\bigskip\noindent
We will use the following three vector fields in this exam.
The position vector field:
$$
\br=x\bi+y\bj+z\bk.
$$
The Coulomb vector field:
$$
\bE={\br\over \|\br\|^3}={1\over (\sqrt{x^2+y^2+z^2})^3}\ (x\bi+y\bj+z\bk).
$$
The rotation vector field:
$$
\bR=-y\bi+x\bj+0\bk.
$$
 
\vfill\eject
\item{1.}
\itemitem{(5 pts)\quad a)}For suitable $\bF$ and region $D$, state the divergence theorem.

\itemitem{(5 pts)\quad b)}For suitable $\bF$ and surface $S$, state Stokes' theorem.
\vfill\eject
ERRATA


\item{\quad 2.}Recall $\br$ is the position vector field.
%\itemitem{(5 pts)\quad a)}Show $\boldsymbol\nabla\times \br=\bold 0$.

%\itemitem{(5 pts)\quad b)}Show $\boldsymbol\nabla \cdot \br=3$.

%\itemitem{(5 pts)\quad c)}Find a function $f(x,y,z)$ so that $\boldsymbol\nabla f=\br$.

%\itemitem{(5 pts)\quad d)}Find $\ds\int\limits_C \br\cdot d\br=\int\limits_c \br\cdot \bT ds$ where $C$ is the line running from the origin to the\newline
%point $(1,2,3)$ in three space.

\itemitem{(5 pts)\quad e)}If the volume enclosed by a closed surface $S$ equals $3\pi$, what is the flux $\ds\iint_S\br \cdot \bn d\sigma$ equal to?

\itemitem{(5 pts)\quad f)}Show there is no closed surface $S$ whose normal vector field $\bn$ is always perpendicular to $\br$.

An alternative problem for 5c), worth five points:

Find three real numbers whose sum is $9$ and the sum of whose squares is
as small as possible.
\vfill\eject

\item{\quad 3)}Recall $\bE\colon = \ds{\br\over \|\br\|^3}$ is the coulomb vector field.

\itemitem{(10 pts)\quad a)}Show $\boldsymbol\nabla \cdot \bE=0$.

\itemitem{(10 pts)\quad b)}Show $\ds\boldsymbol\nabla \bigg({-1\over r}\bigg)={\br\over r^3}=\bE$ where $r=\sqrt{x^2 +y^2 +z^2}=\|\br\|$.

\itemitem{(5 pts)\quad c)}Find the flux $\ds\iint_S \bE\cdot \bn d\sigma$ for a closed surface $S$ not containing $\bold 0$.

\itemitem{(5 pts)\quad d)}Find the flux for $S$ containing $\bold 0$.

\itemitem{(5 pts)\quad e)}Find $\ds\int\limits_C \bE\cdot \bn ds$ where $C$ is a circle in the $x$--$y$ plane about the origin with radius 1.

\vfill\eject
\item{4.}Suppose $\bE_1=\ds{(\br-\ba)\over \|\br-\ba\|^3}$ where $\ba =(1,0,0)$ and let $\bE_2=\ds{\br-\bb\over \|\br-\bb\|^3}$ where $\bb=(0,2,0)$.

\itemitem{(0 pts)\quad a)}True or False:\ The fact that the divergence of $\bE$ is zero leads you to believe the same for $\bE_1$ and the fact that the flux of $\bE$ through a surface $S$ depends only on whether $\bold 0$ is enclosed in $S$ or not enclosed in $S$ leads you to believe the flux of $\bE_1$ through $S$ only depends on whether $\ba$ is enclosed by $S$ or not.

\itemitem{(0 pts)\quad b)}True or False:\ Your calculus skills are such that you could verify the mathematical statements in part a).

\itemitem{(5 pts)\quad c)}Evaluate $\boldsymbol\nabla \cdot (\bE + \bE_1+\bE_2)$.

\itemitem{(5 pts)\quad d)}What is the flux of $\bE+\bE_1+\bE_2$ through a sphere $S$ of radius $1/2$ centered at $(0,0,1)$?

\vfill\eject
\item{5.}
\itemitem{(5 pts)\quad a)}Show $\boldsymbol\nabla \cdot (a \bF_1+ b \bF_2)=a\boldsymbol\nabla \cdot \bF_1+b\boldsymbol\nabla\cdot \bF_2$ where $\bF_1$ and $\bF_2$ are plane vector fields and $a$ and $b$ are constants.

\itemitem{(5 pts)\quad b)}Show $\boldsymbol\nabla (f\cdot g)=g(\boldsymbol\nabla f)+f(\boldsymbol\nabla g)$ where $f$ and $g$ are differentiable functions of two variables $x$ and $y$.

\itemitem{(5 pts)\quad c)}Suppose $F(x,y)=f(x,y) g(x,y)$ and suppose that $F(x,y)$ has a maximum at $(1,2)$.
If $g(x,y)$ has a minimum at $(1,2)$ and if $f$ has no critical point at $(1,2)$, what must $g(1,2)$ equal?

\itemitem{(5 pts)\quad d)}Evaluate $\boldsymbol\nabla \cdot (\br+3\bE)$.

\vfill\eject
\item{(10 pts)\quad 6.}Suppose the surface $S$ is bounded by the curves $C_1,C_2$ and $C_3$.
Suppose $\boldsymbol\nabla \times \bF=\bold 0$\newline
\null\hskip2truein on $S$ and suppose the circulations $\ds\int\limits_{C_1}\bF\cdot d\br=1$\newline
\null\hskip2truein and $\ds\int\limits_{C_2} \bF\cdot d\br=-2$ where the integrals are taken in the\newline
\null\hskip2truein direction of the arrows in the diagram.\newline
What is $\ds\int\limits_{C_3} \bF\cdot d\br$?

\vfill\eject
\item{7.}
\itemitem{(10 pts)\quad a)}Find the work done by moving a particle of mass 1 around a circle of radius 1 around the origin of the $x$--$y$ plane opposite to the direction of the force field $\bR=-y\bi+x\bj$.

\itemitem{(5 pts)\quad b)}Calculate the curl, $\boldsymbol\nabla\times \bR$, of $\bR$ and then use Stokes' theorem to get a circulation integral around the boundary of a bounded region $A$ in the $x$--$y$ plane which gives the area of $A$.

\vfill\eject
\item{8.}$\ds\int\limits_1^4 \int\limits_0^{\sqrt{x}} {3\over 2}\ e^{y/\sqrt{x}} dy dx$.

\itemitem{(10 pts)\quad a)}Sketch the region of integration.

\itemitem{(10 pts)\quad b)}Reverse the order of integration.

\vfill\eject
\item{9.}
\itemitem{(10 pts)\quad a)}What is the unit outward pointing normal vector at the $(1,1,1)$ point on the ellipsoid $S$ given by $x^2+2y^2+3z^2=6$?

\itemitem{(10 pts)\quad b)}What is the equation of the tangent plane to $S$ at $(1,1,1)$?

\itemitem{(10 pts)\quad c)}$S$ intersects the sphere $(x-1)^2+y^2+z^2=2$ (of radius $\sqrt{2}$ and center at $(1,0,0)$), in a curve $C$.
Show $(1,1,1)$ is on this curve $C$.

\itemitem{(10 pts)\quad d)}What is the unit tangent $\bT$ to $C$ at $(1,1,1)$?

\vfill\eject
\item{10.}
\itemitem{(10 pts)\quad a)}Consider the set of all vectors $\bx$ in space so that $\|\bx-\ba\|=b$ where $\ba$ is a constant vector and $b$ is a positive number.
What quadric surface does this set represent.
Can you describe the roles of $\ba$ and $\bb$?

\item{}OR

\itemitem{(alt.~10 pts)\quad b)}Where does the line $t(1,-1,1)+(1,1,1)$ intersect the ellipsoid $3x^2+4y^2+2z^2=10$?

\vfill\eject



\vfill\eject
\vfil\eject
\bye





























































\vfill
\item{(10)\quad 3)}Find the distance from $(2,-3,4)$ to the plane $x+2y+2z=13$.

\vfill
\item{(10)\quad 4)}Find the angle between the planes $x+y=1,\ 2x+y-2z=2$.

\vfill
\item{(10)\quad 5)}Find the arc length of the section of the curve
$\bold r(t)=(\ln (t^2+1))\bold i+(t-2\tan^{-1} t)\bold j+\bold k,\ 1\leq t\leq 5$.

\vfill
\item{(10)\quad 6)}Find the curvature of $\bold r(t)=(2+t)\bold i+(t+2 t^2)\bold j+(1+t^2)\bold k$ at $t=0$.

\vfill
\item{(10)\quad 7)}If $w=x^2+y^2+z^2$ and $z^3-xy+yz+y^3=1$, find $\ds\bigg({\partial w\over \partial y}\bigg)_x$ at $(x,y,z)=(2,-1,1)$.

\vfill\eject
\item{(10)\quad 8)}Let $f(x,y,z)=x^2+2xy-y^2-z^2$.
Find the points on the surface $f(x,y,z)=10$ where the tangent plane is perpendicular to $2\bold i+\bold j+\bold k$.

\vfill
\item{(15)\quad 9)}Let $f(x,y)=x^3+3xy+y^3$.
Find and classify all critical points of $f$.

\vfill
\item{(10)\quad 10)}Find the maximum value of $f(x,y,z)=x^3 y^2 z$ on the plane $3x+2y+z=12$.

\vfill
\item{(10)\quad 11)}How accurately may $V=\ds{x^3\over y^2}$ be calculated from measurements of $x$ and $y$ in error at most 1\%.

\vfill
\item{(10)\quad 12)}Let $D$ be the region in the first octant cut from the cylinder $x^2+4y^2\leq 4$ by the $xy$--plane and the plane $x+2z=2$.
Express the volume $V$ of $D$ with the indicated order.
$$
V=\int_{_{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}^{^{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}} 
\int_{_{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}^{^{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}
\int_{_{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}^{^{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}\ dz dx dy
$$

\vfill\eject
\item{(15)\quad 13)}Let $D$ be the region in the half space $z\geq 0$ bounded by $x^2+y^2+z^2=4$ and the cone $z^2=3(x^2+y^2)$.
Assume that the density $\delta(x,y,z)=\ds\sqrt{x^2+y^2+z^2}$.
Express the mass $M$ in the indicated coordinates.
\medskip
\itemitem{(a)}$\ds{M=\int\limits_{_{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}^{^{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}
\int\limits_{_{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}^{^{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}
\int\limits_{_{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}^{^{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}
\ \boxed{\vbox{\vskip.25truein}\hbox{\hskip1.5truein}}}\
d\rho d\phi d\theta $
\vfill
\itemitem{(b)}$\ds{M=\int\limits_{_{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}^{^{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}
\int\limits_{_{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}^{^{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}
\int\limits_{_{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}^{^{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}
\ \boxed{\vbox{\vskip.25truein}\hbox{\hskip1.5truein}}}\
dz dr d\theta $

\vfill
\item{(10)\quad 14)}Express the circulation of the vector field $\bold F=6z\bold i+y^2\bold j+12x\bold k$ along the curve $\bold r(t)=(\sin t)\bold i+(\cos t)\bold j+{t\over 6}\bold k,\ 0\leq t\leq 2\pi$.
$$
\text{Circulation }=\int_{_{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}^{^{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}\ \boxed{\vbox{\vskip.25truein}\hbox{\hskip1.5truein}}\ dt
$$

\vfill
\item{(10)\quad 15)}Express the flux of the vector field $\bold F=z^2\bold i+x\bold j-3z\bold k$ through the surface\newline
 $\bold r(u,v)=u\bold i+v\bold j+(4-v^2)\bold k,\ 0\leq u\leq 1,\ -2\leq v\leq 2$.
$$
\text{Flux }=
\int\limits_{_{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}^{^{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}
\int\limits_{_{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}^{^{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}
\ \boxed{\vbox{\vskip.25truein}\hbox{\hskip1.5truein}}\ du dv
$$

\vfill\eject
\item{(10)\quad 16)}Let $C$ be the entire boundary of the semidisk $x^2+y^2\leq 1$ and $y\geq 0$.
Express the line integral as a double integral.
$$
\oint_C y^2 dx+x^2 dy=
\int_{_{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}^{^{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}
\int_{_{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}^{^{\boxed{\vbox{\vskip.125truein}\hbox{\hskip.125truein}}}}
\ \boxed{\vbox{\vskip.25truein}\hbox{\hskip1.5truein}}\ dy dx
$$

\vfill
\item{(10)\quad 17)}Use the divergence theorem to compute the outward flux of the vector field
$$
\bold F=(2x+yz)\bold i+(3y+zx)\bold j+(4z+xy)\bold k
$$
across $x^2+y^2+z^2=4$.

\vfill
\item{(10)\quad 18)}Compute
$$
\int_{(1,1,1)}^{(0,0,1)} (1+z)dx+zdy+(x+y+2z)dz.
$$

\vfill
\item{(10)\quad 19)}Let $S$ be the semisphere $x^2+y^2+z^2=4,\ y\geq 0$ and $\bold n$ its upward unit normal.
Let
$$
\bold F=(z^2-y)\bold i+(x+z^3)\bold j+2\bold k.
$$
Use Stoke's theorem to compute $\ds\iint\limits_S (\nabla\times\bold F)\cdot \bold n \ d \sigma$.

\vfill\eject
\bye
