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%\fancyhead[CE]{\fontsize{9}{11}\selectfont F. GOTTI AND M. GOTTI}
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%	\mbox{}
%	\title{Multivariable Calculus}
%	
%	\maketitle

\section*{Stokes' Theorem and Divergence Theorem}
\bigskip

\begin{problem} [Stewart, Example~16.8.1]
	Find the line integral of the vector field $F = \langle -y^2, x, z^2 \rangle$ over the curve $C$ of intersection of the plane $x + z = 2$ and the cylinder $x^2 + y^2 = 1$ knowing that $C$ is oriented counterclockwise when viewed from above. [Answer: $\pi$]
\end{problem}

\begin{problem} [Stewart, Example~16.8.1]
	Find the flux of the vector field $F = \langle xz, yz, xy \rangle$ across the part of the sphere $x^2 + y^2 + z^2 = 4$ that lies inside the cylinder $x^2 + y^2 = 1$ and above the $xy$-plane.
\end{problem}

\begin{problem} [Exercise~16.8.10]
	Evaluate $\int_C F \cdot dr$ below if $C$ is oriented counterclockwise as viewed from above.
	\begin{enumerate}
		\item $F(x,y,z) = \langle xy, yz, zx \rangle$ and $C$ is the boundary of the part of the paraboloid $z = 1 - x^2 - y^2$ in the first octant.
		\vspace{5pt}
		\item $F(x,y,z) = \langle 2y, xz, x+y \rangle$ and $C$ is the intersection of the plane $z = y+2$ and the cylinder $x^2 + y^2 = 1$.
	\end{enumerate}
\end{problem}

\begin{problem} [Stewart, Exercise~16.8.16]
	Let $C$ be a simple closed smooth curve that lies in the plane $x + y + z = 1$. Show that the line integral
	\[
		\int_C z \, dx - 2x \, dy + 3y \, dz
	\]
	depends only on the area of the region enclosed by $C$ and not on the shape of $C$ or its location in the plane.
\end{problem}

\begin{problem} [Stewart, Exercise~16.8.17]
	A particle moves along line segments from the origin to the points $(1,0,0), (1,2,1), (0,2,1)$, and back to the origin under the influence of the force field
	\[
		F(x,y,z) = \langle z^2, 2xy, 4y^2 \rangle.
	\]
	Find the work done.
\end{problem}

\begin{problem} [Stewart, Exercise~16.8.18]
	Evaluate
	\[
		\int_C (y + \sin x)dx + (z^2 + \cos y) dy + x^3 dz,
	\]
	where $C$ is the curve with parametrization $r(t) = (\sin t, \cos t, \sin 2t)$ for $0 \le t \le 2\pi$.
\end{problem}

%\section{The Divergence Theorem}

\begin{problem} [Stewart, Example~16.9.1]
	Fin the flux of the vector field $F(x,y,z) = \langle z,y,x \rangle$ over the unit sphere $x^2 + y^2 + z^2 = 1$. [Answer: $4\pi/3$]
\end{problem}

\begin{problem} [Stewart, Example~16.9.2]
	Find the flux of the vector field
	\[
		F(x,y,z) = \langle xy, (y^2 + e^{xz^2}, \sin(xy) \rangle
	\]
	over the surface of the region bounded by $z = 1 - x^2$ and the planes $z = 0$, $y = 0$, and $y+z = 2$. [Answer: $184/35$]
\end{problem}

\begin{problem} [Stewart, Exercise~16.9.24]
	Use the Divergence Theorem to evaluate
	\[
		\iint_S (2x + 2y + z^2) \, dS,
	\]
	where $S$ is the sphere $x^2 + y^2 + z^2 = 1$.
\end{problem}

\begin{problem} [Stewart, Exercise~16.9.18]
	Find the flux of the vector field
	\[
		F(x,y,z) = \langle z \tan^{-1}(y^2), z^3 \ln(x^2 + 1), z \rangle
	\]
	across the part of the paraboloid $z^2 + y^2 + z = 2$ that lies above the plane $z=1$ and is oriented upward.
\end{problem}

\begin{problem} [Stewart, Exercises~16.9.(26,29)]
	Argue the following identities assuming that $S$ and $E$ satisfy the conditions of the Divergence Theorem and the scalar functions and the vector fields have continuous second-order partial derivatives.
	\begin{enumerate}
		\item $V(E) = \frac 13 \iint_S F \cdot dS$, where $F(x,y,z) = \langle x,y,z \rangle$.
		\vspace{5pt}
		\item $\iint_S (f \nabla g) \cdot n \, dS = \iiint_E (f \nabla^2 g + \nabla f \cdot \nabla g) \, dV$.
	\end{enumerate}
\end{problem}

\begin{problem} [UC Berkeley Final]
	Evaluate the flux
	\[
		\int \! \! \int_S F \cdot dS,
	\]
	where
	\[
		F(x,y,z) = \langle z^2x + e^{z^2 - y^2}, y^3/3 + x^2y + \sin(z + x^2), x^2 \rangle
	\]
	and $S$ is the top half of the sphere $x^2 + y^2 + z^2 = 1$ oriented upward. [Answer: $13/20$]
\end{problem}

\begin{problem} [UC Berkeley Final]
	A fisherman's net has a rim, which is a circle of radius $5$. He fixes it in the sea in such a way that the rim is in the $xz$-plane with center at the origin. The velocity of water is given by the vector field
	\[
		F(x,y,z) = \langle x^4 + 2y^2, 3 - y^2, 2yz - 4 x^3 z \rangle.
	\]
	Find the flux of the water across the net. [Answer: $75\pi$]
\end{problem}

\begin{problem} [UC Berkeley Final]
	Let $S_r$ denote the sphere of radius $r$ with the center at the origin and outward orientation. Suppose that $E$ is a vector field well-defined on all of $\rr^3$ and such that
	\[
		\int \! \! \int_{S_r} E \, dS = ar + b,
	\]
	for some fixed constant $a$ and $b$.
	\begin{enumerate}
		\item Compute in terms of $a$ and $b$ the following integral
		\[
			\int \! \! \int \! \! \int_D \emph{div} \, E \, dV,
		\]
		where $D = \{(x,y,z) \in \rr^3 \mid 25 \le x^2 + y^2 + z^2 \le 49\}$. [Answer: $2a$]
		\item Suppose that in the above situation $E = \emph{curl} \, F$ for some vector field $F$. What conditions, if any, does this place on $a$ and $b$?
	\end{enumerate} [Answer: $a = b = 0$]
\end{problem}

\begin{problem} [Stewart, Exercise~16.9.31]
	Suppose that $S$ and $E$ satisfy the conditions of the Divergence Theorem and $f$ is a scalar function having second-partial derivatives. Prove that
	\[
		\iint_S fn \, dS = \iiint_E \nabla f\, dV.
	\]
	These surface and triple integrals of vector functions are vectors defined by integrating each component function. [Hint: Start by applying the Divergence Theorem to $F = f c$, where $c$ is an arbitrary constant vector.]
\end{problem}

\begin{problem} [UC Berkeley Final]
	Let $C$ be the curve of intersection of the cylinder $x^2 + y^2 = 1$ and the plane $z = 2x + 3y$, oriented counterclockwise when viewed from above. Let
	\[
		F = \langle x^{2018} + y, y^{2018} + z, z^{2018} + x \rangle.
	\]
	Calculate $\int_C F \cdot dr$.
\end{problem}

\begin{problem} [UC Berkeley Final]
	Calculate the flux $\iint_S F \cdot dS$, where $S$ is the hemisphere $x^2 + y^2 + z^2 = 1$ with $z \ge 0$ oriented upwards, and $F = \langle x + \sin y, y + \cos z, z+1 \rangle$.
\end{problem}
	
\vspace{20pt}

\begin{thebibliography}{20}
	
	\bibitem{jS15} J.~Stewart: \emph{Calculus} 8th Edition, Cengage Learning, Boston 2016.
	
\end{thebibliography}
	
\end{document}