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%\fancyhead[CE]{\fontsize{9}{11}\selectfont F. GOTTI AND M. GOTTI}
%\fancyhead[CO]{\fontsize{9}{11}\selectfont ATOMICITY AND BOUNDEDNESS OF MONOTONE PUISEUX MONOIDS}
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\begin{document}
	
	\mbox{}
	\title{Multivariable Calculus}
	
	\maketitle
	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%\section{Introduction} \label{sec:intro}
	%\noindent {\footnotesize Note: Most of the problems were taken from the textbook} \cite{jS15}.
	
\section{Line Integrals and Green's Theorem}

\begin{problem} [Stewart, Exercise~16.1.(25,26)]
	Find and sketch the gradient vector field of the following functions:
	\[
		{\it (1)} \ f(x,y) = \frac 12(x-y)^2 \quad \quad \quad \quad {\it (2)} \ f(x,y) = \frac 12(x^2 - y^2).
	\]
\end{problem}

\begin{problem} [Stewart, Exercise~16.2.(5,11,14)]
	Evaluate the following line integrals:
	\begin{enumerate}
		\item $\int_C (x^2y + \sin x) \, dy$, where $C$ is the arc of the parabola $y = x^2$ from $(0,0)$ to $(\pi, \pi^2)$;
		\vspace{5pt}
		\item $\int_C  xe^{yz} \, ds$, where $C$ is the line segment from $(0,0,0)$ to $(1,2,3)$;
		\vspace{5pt}
		\item $\int_C  y \, dx + z \, dy x \, dz$, where $C = (\sqrt{t}, t, t^2)$ for $1 \le t \le 4$.
	\end{enumerate}
\end{problem}

\begin{problem} [Stewart, Exercise~16.2.41]
	Find the work done by the force field
	\[
		F(x,y,z) = \langle x-y^2, y-z^2, z-x^2 \rangle
	\]
	on a particle that moves along the line segment from $(0,0,1)$ to $(2,1,0)$.
\end{problem}

\begin{problem} [Stewart, Exercise~16.2.(5,11,14)]
	Let $C$ be a smooth curve given by a vector function $r(t)$ for $a \le t \le b$, and let $v$ be a constant vector
	\begin{enumerate}
		\item Show that $\int_C v \cdot dr = v \cdot (r(b) - r(a))$.
		\vspace{5pt}
		\item Show that $\int_C r \cdot dr = \frac 12 \big(|r(b)|^2 - |r(a)|^2 \big)$.
	\end{enumerate}
\end{problem}

\begin{problem} [Stewart, Exercise~16.3.(14,18)]
	Find a function $f$ such that $F = \nabla f$ and use it to evaluate $\int_C F \cdot dr$ along the given curve.
	\begin{enumerate}
		\item $F(x,y) = \langle (1 + xy)e^{xy}, x^2 e^{xy} \rangle$, where $C : r(t) = \langle \cos t, 2 \sin t \rangle $ for $0 \le t \le \pi/2$.
		\vspace{5pt}
		\item $F(x,y,z) = \langle \sin y, x \cos y + \cos z, -y \sin z \rangle$, where $C : r(t) = \langle \sin t, t, 2t \rangle$ for $0 \le t \le \pi/2$.
	\end{enumerate}
\end{problem}

\begin{problem} [Stewart, Exercise~16.3.(19,20)]
	Show that the following line integrals are independent of paths evaluate them.
	\begin{enumerate}
		\item $\int_C 2xe^{-y}\,dx + (2y - x^2e^{-y})\,dy$, where $C$ is any path from $(1,0)$ to $(2,1)$.
		\vspace{5pt}
		\item $\int_C \sin y \,dx + (x \cos y - \sin y) \,dy$, where $C$ is any path from $(2,0)$ to $(1,\pi)$.
	\end{enumerate}
\end{problem}

\begin{problem} [Stewart, Exercise~16.3.(19,20)]
	Determine whether or not the given subsets of $\rr^2$ are open, connected, and/or simply connected:
	\begin{enumerate}
		\item $\{(x,y) \mid 0 < y < 3\}$;
		\vspace{5pt}
		\item $\{(x,y) \mid 1 < |x| < y\}$;
		\vspace{5pt}
		\item $\{(x,y) \mid 1 \le x^2 + y^2 \le 4, y \ge 0\}$;
		\vspace{5pt}
		\item $\{(x,y) \mid (x,y) \neq (2,3)\}$.
	\end{enumerate}
\end{problem}

\begin{problem} [Cal Final, Summer 2018W]
	Suppose a parametrized curve
	\[
		r(t) = \langle x(t), y(t), z(t) \rangle \quad \text{for} \quad 0 \le t \le 1
	\]
	satisfies the equation $x x'(t) + y y'(t) + z z'(t) = 0$ for all $t$. If $x(0) = y(0) = z(0) = 3$ and $x(1) = y(1) = 2$, find $|z(1)|$.
\end{problem}

\begin{problem} [Stewart, Exercises~16.4.(1,7)]
	Evaluate the following integral via Green's Theorem:
	\begin{enumerate}
		\item $\oint_C y^2 \, dx + x^2y \, dy$, where $C$ is the rectangle with vertices $(0,0)$, $(5,0)$, $(5,4)$, and $(0,4)$;
		\vspace{5pt}
		\item $\int_C (y + e^{\sqrt{x}}) \, dx + (2x + \cos y^2) \, dy$, where $C$ is the boundary of the region enclosed by the parabolas $y = x^2$ and $x = y^2$.
	\end{enumerate}
\end{problem}

\begin{problem} [Stewart, Exercises~16.4.18]
	A particle starts at the origin, move along the $x$-axis to $(5,0)$, then along the quarter-circle $x^2 + y^2 = 25$ where $x \ge 0$ and $y \ge 0$ to the point $(0,5)$, and then down to the $y$-axis back to the origin. Find the work done on this particle by the force field $F = \langle \sin x, \sin y + xy^2 + \frac 13 x^3 \rangle$.
\end{problem}

\begin{problem} [Stewart, Exercises~16.4.19]
	Let $D$ be the region bounded by a positively-oriented, piecewise-smooth, simple closed curve $C$.
	\begin{enumerate}
		\item Argue that $\emph{Area}(D) = \oint_C x \, dy = - \oint_C y \, dx = 1/2 \oint_C x \,dy - y \, dx$.
		\vspace{5pt}
		\item Use the previous part to find the area under one arch of the cycloid $(t - \sin t, 1 - \cos t)$.
	\end{enumerate}
\end{problem}

If the {\bf density} of a solid occupying the region $E$ is given by $\rho(x,y,z)$, then its mass can be computed by
\[
	m = \iiint_E \rho(x,y,z) \, dV
\]
and its {\bf center of mass} is $(\bar{x}, \bar{y}, \bar{z})$, where
\[
	\bar{x} = \iiint_E x \rho(x,y,z) \, dV; \quad \bar{y} = \iiint_E y \rho(x,y,z) \, dV; \quad \bar{z} = \iiint_E z \rho(x,y,z) \, dV.
\]
If the density is constant, the center of mass is also called {\bf centroid}.

\begin{problem} [Stewart, Exercises~16.4.22]
	Let $D$ be a region bounded by a simple closed path $C$ in the $xy$-plane. Argue that the coordinate of the centroid $(\bar{x}, \bar{y})$ of $D$ are
	\[
		\bar{x} = \frac{1}{2A} \oint_C x^2 \, dy \quad \text{ and } \quad \bar{y} = \frac{1}{2A} \oint_C y^2 \, dx,
	\]
	where $A$ is the area of $D$. [Hint: Use Green's Theorem and $\rho = m/A$.]
\end{problem}

\begin{problem} [Cal Final, Summer 2018W]
	Let $f$ be a differentiable function on $\rr^2$ such that
	\[
		\frac{\partial f}{\partial x}(x,y) = \frac{\partial f}{\partial y}(x,y)
	\]
	for all $x,y$. Suppose also that $f(2,3) = 6$. Compute $f(4,1)$.
\end{problem}

	
\section{Surface Integrals and Flux}

\begin{problem} [Stewart, Exercises~16.5.(13,15,18)]
	Determine whether or not the following vector fields are conservative. For each conservative, find a function $f$ such that $F = \nabla f$.
	\begin{enumerate}
		\item $F(x,y,z) = \langle y^2z^3, 2xyz^3, 3xy^2z^2 \rangle$.
		\vspace{5pt}
		\item $F(x,y,z) = \langle z \cos y, 2xyz^3, xz \sin y, x \cos y \rangle$.
		\vspace{5pt}
		\item $F(x,y,z) = \langle e^x \sin yz, z e^x \cos yz, ye^x \cos yz \rangle$.
	\end{enumerate}
\end{problem}

\begin{problem}[Stewart, Exercises~16.5.(19,20)]
	Is there a vector field $G$ on $\rr^3$ satisfying the following condition.
	\begin{enumerate}
		\item $\emph{curl}(G) = \langle x \sin y, \cos y, z - xy \rangle$. Explain why?
		\item $\emph{curl}(G) = \langle x,y,z \rangle$. Explain why?
	\end{enumerate}
\end{problem}

The {\bf Laplace operator} $\nabla^2$ is defined by $\nabla^2 f = f_{xx} + f_{yy} + f_{zz}$, where $f$ is a function having continuous second partial derivatives. The Laplace operator can be applied also to a vector field $F = \langle P, Q, R \rangle$ as follows: $\nabla^2 F =  \langle \nabla^2 P, \nabla^2 Q, \nabla^2 R \rangle$.

\begin{problem} [Stewart, Exercises~16.5.(27,28,29)]
	For vector fields $F$ and $G$ on $\rr^3$, argue the following identities:
	\begin{enumerate}
		\item $\emph{div} (F \times G) = G \cdot \emph{curl} \, F - F \cdot \emph{curl} \, G$,
		\vspace{5pt}
		\item $\emph{div} (\nabla F \times \nabla G) = 0$,
		\vspace{5pt}
		\item $\emph{curl}(\emph{curl} \, F) = \emph{grad}(\emph{div} \, F) - \nabla^2 F$.
	\end{enumerate}
\end{problem}

\begin{problem} [Stewart, Exercises~16.5.(30,31)]
	For the vector field $r = \langle x,y,z \rangle$, argue the following identities:
	\begin{enumerate}
		\item $\nabla \cdot (|r| r) = 4|r|$,
		\vspace{5pt}
		\item $\nabla^2 (|r|^3) = 12 r$,
		\vspace{5pt}
		\item $\nabla(1/|r|) = - r/|r|^3$,
		\vspace{5pt}
		\item $\nabla(\ln |r|) = r/|r^2|$.
	\end{enumerate}
\end{problem}

\begin{problem} [Stewart, Exercises~16.5.(33,34)]
	Let $C$ be a positively-oriented, piecewise-smooth, simple closed curve in $\rr^2$ given by $r(t) = \langle x(t), y(t) \rangle$ with $a \le t \le b$, and let $n(t) = 1/|r'(t)| \langle y'(t), - x'(t) \rangle$.
	\begin{enumerate}
		\item Use Green's Theorem to argue that
		\[
			\oint_C F \cdot n \, ds = \iint_{\emph{int}(C)} \emph{div} \, F(x,y) \, dA,
		\]
		for any smooth vector field $F$ on $\rr^2$.
		\vspace{5pt}
		\item Use the previous part to argue Green's First Identity:
		\[
			\iint_{\emph{int}(C)} f \nabla^2g \, dA = \oint_C f(\nabla g) \cdot n \, ds - \iint_{\emph{int}(C)} \nabla f  \cdot \nabla g \, dA,
		\]
		for any functions $f$ and $g$ whose appropriate partial derivatives exist and are continuous.
		\vspace{5pt}
		\item Use the previous part to argue Green's Second Identity:
		\[
			\iint_{\emph{int}(C)} (f \nabla^2g - g \nabla^2f) \, dA = \oint_C (f \nabla g - g \nabla f) \cdot n \, ds,
		\]
		for any functions $f$ and $g$ whose appropriate partial derivatives exist and are continuous.
	\end{enumerate}
\end{problem}

\begin{problem} [Stewart, Exercise~16.6.34]
	Find the equation of the tangent plane to the surface parameterized by $r(u,v) = (u^2 + 1, v^3 + 1, u + v)$ at $(5,2,3)$.
\end{problem}

\begin{problem} [Cal Final, Summer 2018W]
	Find the tangent plane to the parametrized surface $r(u,v) = \langle u^2 -1, uv, v^3 \rangle$ at the point $(3,4,8)$.
\end{problem}

\begin{problem} [Stewart, Exercise~16.6.61]
	Find the area of the part of the sphere $x^2 + y^2 + z^2 = 4z$ that lies inside the paraboloid $z = x^2 + y^2$.
\end{problem}

\begin{problem}
	Deduce the formula for the flux of a vector field $F = \langle P,Q,R \rangle$ across a surface $S$ that is the graph of a smooth function $g \colon U \subset \rr^2 \to \rr$.
\end{problem}

\begin{problem} [Stewart, Exercises~16.7.(15,26)]
	Evaluate the following surface integrals.
	\begin{enumerate}
		\item $\iint_S y \, dS$, where $S$ is the surface $y = x^2 + 4z$ for $(x,z) \in [0,1] \times [0,1]$.
		\vspace{5pt}
		\item $\iint_S F \cdot dS$, where $F(x,y,z) = \langle y, -x, 2z \rangle$ and $S$ is the hemisphere $x^2 + y^2 + z^2 = 4$ and $z \ge 0$ oriented downward.
	\end{enumerate}
\end{problem}

\begin{problem} [Stewart, Exercise~16.7.43]
	A fluid has density $870~kg/m^3$ and flows with velocity $v(x,y,z) = \langle z, y^2, x^2 \rangle$, where $x,y$, and $z$ are measured in meters and the components of $v$ in meters per second. Find the rate of flow outward through the cylinder $x^2 + y^2 = 4$ for $0 \le z \le 1$. [Hint: the rate of flow outward is the flux $\iint_S (\rho v) \cdot n \, dS$.]
\end{problem}

\begin{problem} [Stewart, Exercise~16.7.49]
	Let $F$ be an inverse square field, that is, $F(r) = cr/|r|^3$ for some constant $c$, where $r = \langle x,y,z \rangle$. Show that the flux of $F$ across a sphere $S$ with center the origin is independent of the radius of $S$.
\end{problem}

\begin{problem} [Cal Final, Fall 09]
	Evaluate the surface integral
	\[
		\int \! \! \int_S z \, dS,
	\]
	where $S$ is 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. [Answer: $2\pi$]
\end{problem}

\begin{problem} [Cal Final, Fall 09]
	Evaluate the flux
	\[
		\int \! \! \int_T F \cdot dS
	\]
	of the vector field $F(x,y,z) = \langle -x, xy, zx \rangle$ across the triangle $T$ with vertices $(1,0,0)$, $(0,2,0)$, and $(0,0,2)$ with downward orientation. [Answer: $1/3$]
\end{problem}

\begin{problem} [Cal Final, Summer 2018W]
	Let $S$ be a surface which is contained in the plane $z = x+y$, oriented upward. Suppose that $S$ has area $2018$. Consider the constant vector field $F = \langle 3,4,5 \rangle$. Calculate the flux of $F$ across the surface $S$. [Answer: $-4036/\sqrt{3}$]
\end{problem}

\section{Stokes' Theorem and Divergence Theorem}

\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} [Cal Final, Fall 09]
	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} [Cal Final, Fall 09]
	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} [Cal Final, Fall 09]
	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} [Cal Final, Summer 2018W]
	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} [Cal Final, Summer 2018W]
	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}