\documentclass[12pt]{article}
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\usepackage{enumerate}
\usepackage{ifthen}
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\begin{document}

\begin{center}
{\Large \sc Supplemental Vector Calculus Problems} 
\end{center}

\vspace{0.1in}

\noindent {\Large\bf Green's with Holes \& Path Dependance:}

%\vspace{0.1in}

\hspace*{-0.3in}
\begin{minipage}[b]{3.5in}
Let ${\bf F}(x,y) = \la M(x,y), N(x,y) \ra$ be a vector field where
$M(x,y)$ and $N(x,y)$ have continuous first partials such that
$M_y=N_x$ except at the points $P_1$, $P_2$, and $P_3$.

\vspace{0.15in}

Suppose that\dots 

\vspace{0.15in}

$\displaystyle{\int_{C_1} {\bf F} \,{\bf \cdot}\, d{\bf r} = 1}$, \qquad $\displaystyle{\int_{C_2} {\bf F} \,{\bf \cdot}\, d{\bf r} = 2}$, 

$\displaystyle{\int_{C_3} {\bf F} \,{\bf \cdot}\, d{\bf r} = 3}$, \quad and \quad
$\displaystyle{\int_{C_4} {\bf F} \,{\bf \cdot}\, d{\bf r} = 4}$

\vspace{0.2in}

where $C_1$, $C_2$, \dots $C_8$ are pictured to the right.

\vspace{0.3in}

Compute:
\begin{enumerate}
\setcounter{enumi}{\value{mycounter}}
\item $\displaystyle{\int_{C_5} {\bf F} \,{\bf \cdot}\, d{\bf r}}$ [Answer: 6]
\item $\displaystyle{\int_{C_6} {\bf F} \,{\bf \cdot}\, d{\bf r}}$ [Answer: 3]
\item $\displaystyle{\int_{C_7} {\bf F} \,{\bf \cdot}\, d{\bf r}}$ [Answer: 0]
\item $\displaystyle{\int_{C_8} {\bf F} \,{\bf \cdot}\, d{\bf r}}$ [Answer: 0]
\setcounter{mycounter}{\value{enumi}}
\end{enumerate}

\vspace{1in}

\end{minipage} \hfill \hfill
\begin{minipage}[b]{3in}
\begin{tikzpicture}[
declare function={goofyCircleX(\a,\as,\b,\bs,\sc,\sft,\x)=(((1+cos(\a*\x)*\as+sin(\b*\x)*\bs)*cos(\x))*\sc+\sft);},
declare function={goofyCircleY(\a,\as,\b,\bs,\sc,\sft,\x)=(((1+cos(\a*\x)*\as+sin(\b*\x)*\bs)*sin(\x))*\sc+\sft);},
declare function={goofyLineX(\xs,\xe,\x)=((\xe-\xs)*\x+\xs);},
declare function={goofyLineY(\ys,\ye,\x)=((\ye-\ys)*\x+1.3*sin(\x*360)+\ys);},
declare function={rX(\t,\x,\y)=(\x*cos(\t)-\y*sin(\t));},
declare function={rY(\t,\x,\y)=(\x*sin(\t)+\y*cos(\t));},
]

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]


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% Pin label to circle node
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\node[circle,fill=red,scale=0.5,label=30:{$B$}] at (axis cs:4,3) {};

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\addplot[black, thick, domain=0:0.3, smooth] ({x*rX(110,1,0)+goofyLineX(-3,4,0.5)}, {x*rY(110,1,0)+goofyLineY(-1.5,3,0.5)});
\addplot[black, thick, domain=0:0.275, smooth] ({x*rX(200,1,0)+goofyLineX(-3,4,0.5)}, {x*rY(200,1,0)+goofyLineY(-1.5,3,0.5)});

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\node[label=0:{$C_3$}] at (axis cs:1.55,-1.25) {};
\node[label=0:{$C_4$}] at (axis cs:0.9,-0.05) {};

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\begin{tikzpicture}[
declare function={goofyLineX(\xs,\xe,\x)=((\xe-\xs)*\x-sin(\x*180)+\xs);},
declare function={goofyLineY(\ys,\ye,\x)=((\ye-\ys)*\x+2.75*sin(\x*180)+2*(\x-\x^2)+\ys);},
declare function={rX(\t,\x,\y)=(\x*cos(\t)-\y*sin(\t));},
declare function={rY(\t,\x,\y)=(\x*sin(\t)+\y*cos(\t));},
]

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\addplot[black, thick, domain=0:0.25, smooth] ({x*rX(25,1,0)+1.25}, {x*rY(25,1,0)+3.175});
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\begin{tikzpicture}[
declare function={rX(\t,\x,\y)=(\x*cos(\t)-\y*sin(\t));},
declare function={rY(\t,\x,\y)=(\x*sin(\t)+\y*cos(\t));},
]

\begin{axis}[
  axis x line=none, axis y line=none,  % No axes as opposed to alternate style "middle"
  width=9cm, height=7cm,     % size of the image = rigged to keep fixed scaling
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]

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\addplot[black, thick, domain=0:0.25, smooth] ({x*rX(180,1,0)+1.25}, {x*rY(180,1,0)-0.85});
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\vspace{-0.75in}

\noindent
{\bf \Large Green's Theorem:}
\begin{enumerate}
\setcounter{enumi}{\value{mycounter}}
\item Let $C$ be the square with vertices $(0,0)$, $(1,0)$, $(1,1)$, and $(0,1)$ (oriented counter-clockwise). Compute
          the line integral: $\displaystyle{\int_C y^2\,dx+x^2\,dy}$ two ways. First, compute the integral directly by parameterizing
          each side of the square. Then, compute the answer again using Green's Theorem.
\item Compute $\displaystyle{\int_C {\bf F} \,{\bf \cdot}\, d{\bf r}}$ where ${\bf F}(x,y)=\la y-\ln(x^2+y^2), 2\,\mbox{arctan}(y/x) \ra$
          and $C$ is the circle $(x-2)^2+(y-3)^2=1$ oriented counter-clockwise. [Answer: $-\pi$]
\item Let $C$ be the boundary of the part of the disk $x^2+y^2 \leq 16$ which lies in the first quadrant (oriented counter-clockwise). Compute
          $\displaystyle{\int_C x^2y\,dx-y^2x\,dy}$. [Answer: $-32\pi$]
\item Let $C$ be the line segments from $(1,0)$ to $(1,1)$, $(1,1)$ to $(0,1)$, and $(0,1)$ to $(0,0)$. Compute
          $\displaystyle{\int_C \sqrt{1+y^3}\,dx + (x^2+e^{-y^2})\,dy}$.\hfill
          {\small {\it Hint:} Use Green's to swap $C$ with the line from $(1,0)$ to $(0,0)$.}
\setcounter{mycounter}{\value{enumi}}
\end{enumerate}

\vspace{0.1in}

\newpage

\noindent
{\bf \Large Surface \& Flux Integrals:}
\begin{enumerate}
\setcounter{enumi}{\value{mycounter}}
\item Find the centroid of the upper-hemisphere of $x^2+y^2+z^2=9$. [Answer: $(0,0,3/2)$]
\item Find the centroid of $z=x^2+y^2$ where $z \leq 1$.
\item Let ${\bf r}(u,v) = \la u\cos(v),u\sin(v),v \ra$ where $0 \leq u \leq 1$ and $0 \leq v \leq 2\pi$. Let $S_1$ be the surface parameterized by ${\bf r}$. Graph $S_1$ in Maple. Then evaluate 
$\displaystyle{\iint_{S_1} \la x,y,z-2y \ra \,{\bf \cdot}\, {\bf n}\,d\sigma}$ if $S_1$ is given the orientation $\displaystyle{{\bf n} = \frac{{\bf r}_u \times {\bf r}_v}{|{\bf r}_u \times {\bf r}_v|}}$.  [Answer: $\pi^2$] % Colley page 435 example 4.
\item Let $S_1$ be the part of the paraboloid $z=1-x^2-y^2$ which lies above the $xy$-plane ($z \geq 0$) and orient $S_1$ upward.
Evaluate $\displaystyle{\iint_{S_1} \la x,y,z \ra\,{\bf \cdot}\, {\bf n}\,d\sigma}$. [Answer: $3\pi/2$] % Anton page 963.
\item Let $S_1$ be the upper-hemisphere of $x^2+y^2+z^2=9$ oriented upward. Evalute $\displaystyle{\iint_{S_1} \la -y,x,-1 \ra \,{\bf \cdot}\, {\bf n}\,d\sigma}$. [Answer: $-9\pi$] % Colley 7.2 #19.
\item Let $S_1$ be the surface of the cylinder bounded by $x^2+y^2=4$, $z=0$, and $z=5$ oriented outward (include the top and the bottom of the cylinder). Evaluate $\displaystyle{\iint_{S_1} \la x^3,y^3,0 \ra \,{\bf \cdot}\, {\bf n}\,d\sigma}$. If you know the Divergence Theorem, use it to recompute your answer. [Answer: $120\pi$] % Colley page 445.
\item Let $S_1$ be the part of the plane $x+y+z=1$ which lies in the first octant ($x\geq 0$, $y \geq 0$, $z \geq 0$) and oriented {\bf downward}.
Evaluate $\displaystyle{\iint_{S_1} (xze^y{\bf i}-xze^y{\bf j}+z{\bf k}) \,{\bf \cdot}\, {\bf n}\,d\sigma}$. [Answer: $-1/6$] % Stewart 16.7 #21
\item Let $S_1$ be the cube with vertices $(\pm 1, \pm 1, \pm 1)$ oriented outward. Evaluate $\displaystyle{\iint_{S_1} \la x,2y,3z \ra \,{\bf \cdot}\, {\bf n}\,d\sigma}$.
If you know the divergence theorem, recalculate this integral using the theorem. [Answer: 48] % Stewart 16.7 #27
\setcounter{mycounter}{\value{enumi}}
\end{enumerate}

\vspace{0.1in}

\noindent
{\bf \Large Stokes' Theorem:}
\begin{enumerate}
\setcounter{enumi}{\value{mycounter}}
\item Let $S$ be the lower half of the sphere $x^2+y^2+z^2=4$ oriented downward with boundary $C$. 
          Also, let ${\bf F}(x,y,z)=\la 2y-z,x+y^2-z,4y-3x \ra$.
          Verify Stokes' Theorem by computing both sides of $\displaystyle{\int_C {\bf F}\,{\bf \cdot}\,d{\bf r} = \iint_S (\nabla \times {\bf F}) \,{\bf \cdot}\, {\bf n}\,d\sigma}$.
\item Verify Stoke's theorem for ${\bf F}(x,y,z) = \la y^2,x,z^2 \ra$ and $S_1$ the part of the circular paraboloid $z=x^2+y^2$
which lies below $z=1$. Give $S_1$ the upward orientation. % Stewart 16.8 #13.
\item Let $C$ be the circle parameterized by ${\bf r}(t)=\la \cos(t),\sin(t),0 \ra$ where $0 \leq t \leq 2\pi$. 
Evaluate $\displaystyle{\int_C \la z^2,2x,-y^3 \ra \,{\bf \cdot}\, d{\bf r}}$. [Answer: $2\pi$] % Anton 18.8 #1.
\item Let $C$ be the rectangular boundary of the part of the plane $z=y$ which lies above $0 \leq x \leq 1$ and $0 \leq y \leq 3$.
Give $C$ the counter-clockwise orientation when viewed from above. Evaluate $\displaystyle{\int_C \la x^2,4xy^3,y^2x \ra \,{\bf \cdot}\, d{\bf r}}$. 
[Answer: $90$] % Anton page 978.
\item Let $S_1$ be the part of the graph of $z=e^{-(x^2+y^2)}$ which lies above $x^2+y^2 \leq 1$. Orient $S_1$ upward. Let ${\bf F}(x,y,z) = (e^{y+z}-2y){\bf i} + (xe^{y+z}+y){\bf j} + e^{x+y}{\bf k}$. Evaluate $\displaystyle{\iint_{S_1} (\nabla \times {\bf F}) \,{\bf \cdot}\, {\bf n}\,d\sigma}$.  [Answer: $2\pi$]\\ {\small {\it Hint:} Use Stokes' theorem to replace $S_1$ with another surface which shares the same boundary.} % Colley example 2 page 452.
\item $S_1$ be the four sides and the top of the cube with vertices $(\pm 1, \pm 1, \pm 1)$ oriented outward.
Let ${\bf F}(x,y,z) = xyz{\bf i}+xy{\bf j}+x^2yz{\bf k}$. Evaluate $\displaystyle{\iint_{S_1} (\nabla \times {\bf F}) \,{\bf \cdot}\, {\bf n}\,d\sigma}$
by using Stokes' theorem to exchange $S_1$ with an ``easier'' surface whose boundary is the same as $S_1$.
[Answer: 0] % Stewart 16.8 #5.
\item Compute the work done by the force field ${\bf F}(x,y,z)=\la x^x+z^2,y^y+x^2,z^z+y^2 \ra$ on a particle moving
around the edge of the part of the sphere $x^2+y^2+z^2=4$ which lies in the first octant and is oriented
in a counter-clockwise direction when viewed from above. [Answer: 16] % Stewart 16.8 #17.
\item Let $C$ be the curve of intersection of $x+y+z=1$ and $x^2+y^2=9$. Orient $C$ counter-clockwise when viewed from above.
Evaluate $\displaystyle{\int_C \la x^2z,xy^2,z^2 \ra\,{\bf \cdot}\, d{\bf r}}$.
[Answer: $81\pi/2$] % Stewart 16.8 #11.
\setcounter{mycounter}{\value{enumi}}
\end{enumerate}

\vspace{0.1in}

\noindent
{\bf \Large The Divergence Theorem:}
\begin{enumerate}
\setcounter{enumi}{\value{mycounter}}
\item Let $E$ be the solid bounded above by $z=9-x^2-y^2$ and below by $z=0$, let $S$ be the boundary of $E$, and let ${\bf F}(x,y,z) = \la x,y,z \ra$. Verify the Divergence Theorem by computing both sides of $\displaystyle{\iint_S {\bf F} \,{\bf \cdot}\, {\bf n}\,d\sigma = \iiint_E (\nabla \,{\bf \cdot}\,{\bf F}) \,dV}$.
\item Let $S_1$ be the unit sphere $x^2+y^2+z^2=1$ oriented outward.\\ {$\mbox{\;}$} \hfill Evaluate $\displaystyle{\iint_{S_1} \la z,y,x \ra\,{\bf \cdot}\, {\bf n}\,d\sigma}$.
[Answer: $4\pi/3$] % Stewarts example 1 page 1129.
\item Verify the divergence theorem when ${\bf F}(x,y,z)=\la xy,yz,zx \ra$ and $E$ is the cylindrical region bounded by $x^2+y^2=1$, $z=0$, and $z=1$.
% Stewarts 16.9 #5
\item Let $S_1$ be the boundary of the upper-half of $x^2+y^2+z^2 \leq 1$ (the upper-half of the unit ball). Orient $S$ outward.
Evaluate $\displaystyle{\iint_{S_1} \la x^3,y^3,z^3 \ra \,{\bf \cdot}\, {\bf n}\,d\sigma}$. [Answer: $6\pi/5$] % Anton page 971.
\item Let $S_1$ be the surface of the circular paraboloid bounded by $z=x^2+y^2$ and $z=1$. Orient $S_1$ outward. 
Evaluate $\displaystyle{\iint_{S_1} \la x^2,y^2,z^2 \ra \,{\bf \cdot}\, {\bf n}\,d\sigma}$. [Answer: $2\pi/3$] % Anton 18.7 #11.
\item Let $S_1$ be the surface of the cylinder bounded by $y^2+z^2=1$, $x=-1$, and $x=2$ (including the front and back) oriented outward.
Evaluate $\displaystyle{\iint_{S_1} \la 3xy^2,xe^z,z^3 \ra \,{\bf \cdot}\,{\bf n}\,d\sigma}$. [Answer: $9\pi/2$] % Stewart 16.9 #9
\item Let $S_1$ be the upper-hemisphere of $x^2+y^2+z^2=1$ oriented upwards (this is just the upper shell do not include the bottom).
Also, let ${\bf F}(x,y,z)=\la z^2x,y^3/3+\tan(z),x^2z+y^2 \ra$. Evaluate $\displaystyle{\iint_{S_1} {\bf F}\,{\bf \cdot}\,{\bf n}\,d\sigma}$. 
Hint: use the theorem to exchange $S_1$ with an ``easier'' surface. [Answer: $13\pi/20$] % Stewarts 16.9 #19.
\item Let $S_1$ be the part of the graph of $z=(1-x^2-y^2)e^{1-x^2-3y^2}$ which lies above the $xy$-plane oriented upward.
Also, let ${\bf F}(x,y,z) = \left\la e^y\cos(z),\sqrt{x^3+1}\sin(z),x^2+y^2+3 \right\ra$. Evaluate $\displaystyle{\iint_{S_1} {\bf F} \,{\bf \cdot}\,{\bf n}\,d\sigma}$ by using
the theorem to exchange $S_1$ with an ``easier'' surface. [Answer: $7\pi/2$] % Colley example 4 page 455.
\item Prove Green's Formulas: Let $f$ and $g$ be smooth functions and let $S$ be the surface (oriented outward) of a simple solid region $E$.
\[ \iiint_E \nabla f \,{\bf \cdot}\, \nabla g + f \, \nabla \,{\bf \cdot}\, (\nabla g) \,dV = \iint_S f\,\nabla g \,{\bf \cdot}\, {\bf n}\,d\sigma \]
\[ \iiint_E \left(f \, \nabla \,{\bf \cdot}\, (\nabla g) - g\, \nabla \,{\bf \cdot}\, (\nabla f)\right)\,dV = \iint_S (f\,\nabla g - g\,\nabla f) \,{\bf \cdot}\, {\bf n}\,d\sigma \]
Hint: First, prove the following ``product'' rule: $\nabla \,{\bf \cdot}\, (f \, {\bf F}) = \nabla f \,{\bf \cdot}\, {\bf F} + f \,\nabla \,{\bf \cdot}\, {\bf F}$.
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