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%	\mbox{}
%	\title{Multivariable Calculus}
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%	\maketitle
	
\section*{Line Integrals and Green's Theorem}
\bigskip

\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} [UC Berkeley Final]
	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} [UC Berkeley Final]
	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}

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