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\begin{document}

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\noindent
\parbox{2in}{\bf Math 2130}
\hfill {\Large \bf Final Exam} \hfill
\parbox{2in}{\bf \hfill June $28^\mathrm{th}$\!,\  2019}

\vspace{0.2in}

\noindent {\large\bf Name:} \underline{\hskip 3.0 truein}
\hfill {\bf Be sure to show your work!}

\vspace{0.1in}

\noindent
{\bf\large 1. (7 points)} Let $\B F(x,y,z) = \la x^2+yz, y+e^z, \sin(xy^2) \ra$. Compute $\nabla \times \B F$ and $\nabla \bdot \B F$.\\ \force \hfill If $\B F$ conservative? {\Large \quad Yes \quad / \quad No}

\vfill

\noindent
{\bf\large 2. (11 points)} Let $\B F(x,y,z) = \left\la 2xyz+1, x^2z+z, x^2y+y+2z \right\ra$.  This is a conservative vector field (I've checked for you).

\begin{enumerate}[(a)]
\item Use the fundamental theorem of line integrals to compute $\int_C \B F \bdot d\B r$ where $C$ is the line segment from $(1,-1,0)$ to $(1,2,2)$.

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\item Recompute $\int_C \B F \bdot d\B r$ directly (i.e. parameterize $C$ etc.).

\vfill

\end{enumerate}

\newpage


\noindent
{\bf\large 3. (10 points)} Let $C$ be the boundary of a triangle with vertices $(0,-1)$, $(2,-1)$, and $(2,3)$ oriented counter-clockwise. 
Compute $\displaystyle \int_C \left(\cos(x^6+e^{3x})+y\right)\,dx+\left(x^2+\ln(y^8+3)\right)\,dy$.

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\node at (axis cs: 1.7,2.85) {$C_1$};
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\vspace{-1.7in}

\noindent
{\bf\large 4. (9 points)} $C_1$ is an upper-half of a circle of radius 3 (oriented clockwise), $C_2$ is a circle of radius 1 (oriented counter clockwise), and $C_3$ is a line segment closing off the semi-circle (oriented left to right). Let $\B F(x,y) = \langle M(x,y), N(x,y) \rangle$ be a vector field such that $M$ and $N$ have continuous first partials and in addition, $N_x-M_y = 4$ for all points in\\ region $R$. We also know $\displaystyle \int_{C_2} \B F \bdot d\B r = 2\pi$ and $\displaystyle \int_{C_3} \B F \bdot d\B r = \pi$. {\it Note:} The area of $R$ is $7\pi/2$.

\vspace{0.25in}

\noindent
Compute $\displaystyle \int_{C_1} M(x,y)\,dx + N(x,y)\,dy \quad = \quad \underline{\hspace{1.5in}}$.


\vspace{0.75in}

\noindent
{\bf\large 5. (12 points)} Let $S_1$ be the lower hemisphere: $x^2+y^2+z^2=1$, $z \leq 0$ and $S_2$ be the disk $x^2+y^2 \leq 1$ in the $xy$-plane. Orient both $S_1$ and $S_2$ upward. Let $\B F$ be a smooth vector field such that $\iint_{S_1} \B F \bdot \B n\,d\sigma= 10\pi$ and $\nabla \bdot \B F = 5x^2+5y^2+5z^2$.\\ \force \hfill Find $\iint_{S_2} \B F \bdot \B n\,d\sigma$.

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\newpage

\noindent
{\bf\large 6. (12 points)} Find the centroid of the part of the cone $z=3\sqrt{x^2+y^2}$ where $z \leq 6$.\\ \force \hfill Note: This is a {\bf surface}. You should be computing {\bf surface integrals}.

\[ m = \iint_{S_1} 1\,d\sigma \qquad M_{yz} = \iint_{S_1} x\,d\sigma \qquad M_{xz} = \iint_{S_1} y\,d\sigma \qquad M_{xy} = \iint_{S_1} z\,d\sigma \]


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\noindent
{\bf\large 7. (13 points)} Let $S_1$ be the surface parameterized by $\B r(u,v) = \langle u\cos(v),u\sin(v),9-u^2 \rangle$\\ \force \hfill where $1 \leq u \leq 2$ and $\pi \leq v \leq 2\pi$.

\begin{enumerate}[(a)]
\item Find both orientations for $S_1$.
   
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\item Set up but {\bf do not evaluate} the surface integral $\displaystyle \iint_{S_1} z\ln(x^2+y^2+1)\,d\sigma$. \quad [Don't worry about simplifying.] 

\vspace{1in}

\item Set up but {\bf do not evaluate} the flux integral $\displaystyle \iint_{S_1} \B F \bdot \B n\,d\sigma$ where $S_1$ is \underline{oriented downward} and $\B F(x,y,z) = \la xz,y,x+y \ra$. \quad [Don't worry about computing the dot product or any significant simplification.] 

\vspace{0.5in}

\end{enumerate}

\newpage

\noindent
{\bf\large 8. (11 points)} Let $S_1$ be the surface of the solid cylinder defined by $x^2+y^2 \leq 4$ and $-2 \leq z \leq 3$. Orient $S_1$ outward and let $\B F(x,y,z) = \left\la 4xy^2 +\cos(y^{15}+z), \ln(x^2+z^4+1)+2y, 4x^2z+e^{x^3y} \right\ra$.  Compute the flux integral $\iint_{S_1} \B F \bdot \B n\,d\sigma$.

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\noindent
{\bf\large 9. (15 points)} Let $S_1$ be the upper hemisphere: $x^2+y^2+z^2=4$ and $z \geq 0$. Orient $S_1$ downward.\\ \force \hfill  Verify Stokes' Theorem for the surface $S_1$, its boundary, and the vector field $\B F(x,y,z) = \la z,x,x \ra$. 

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