
%  This is a LaTex file.

%  Homework for the course "Calculus II", 
%  Spring semester, 2024, Wei-Cheng Wang.

% \documentclass[12pt]{article}
% \usepackage{amsmath}
% \usepackage{amsfonts}
% \usepackage{amssymb}
\documentclass[12pt]{article}
\usepackage{amsmath,amsfonts,amssymb,amstext}
\DeclareSymbolFontAlphabet{\Bbb}{AMSb}


%          The page format, somewhat wider and taller page than in art12.sty.

\topmargin  0.1in \headsep 0in \textheight 8.9in \footskip 0.6in
\oddsidemargin 0in  \evensidemargin 0in  \textwidth 6.5in

\def\V{\mathcal V}
\def\L{\mathcal L}
\def\S{\mathcal S}
\def\T{\mathcal T}
\def\U{\mathcal U}
\def\A{\boldmath A}
\def\M{\boldmath M}
\def\l{\mbox{\boldmath $l$}}
\def\r{\mbox{\boldmath $r$}}
\def\v{\mbox{\boldmath $v$}}
\def\ds{\displaystyle}
\def\limi{\displaystyle{\lim_{n\to \infty}}}
\def\sumi{\displaystyle{\sum_{n=1}^\infty}}
\def\sumzi{\displaystyle{\sum_{n=0}^\infty}}
\def\sumk{\displaystyle{\sum_{n=1}^k}}
\def\sumkzi{\displaystyle{\sum_{k=0}^\infty}}
\def\limk{\displaystyle{\lim_{k\to \infty}}}
\def\vF{\text{\boldmath $F$}}
\def\vG{\text{\boldmath $G$}}
\def\vH{\text{\boldmath $H$}}
\def\vi{\text{\boldmath $i$}}
\def\vj{\text{\boldmath $j$}}
\def\vk{\text{\boldmath $k$}}



%          The page format, somewhat wider and taller page than in art12.sty.

\topmargin -0.1in \headsep 0in \textheight 8.9in \footskip 0.6in
\oddsidemargin 0in  \evensidemargin 0in  \textwidth 6.5in

\begin{document}

%          The title and header.

\noindent
{\scriptsize Calculus II, Spring 2024 
(http://www.math.nthu.edu.tw/\~\,wangwc/)} \hfill 

\begin{center}
\large
Homework 15   
\normalsize
\end{center}

\vspace{.3in}

%           The questions!

\begin{enumerate}

% Section 14.10: Partial Derivative with Constrained Variables
% \item
% Follow up on example 5 of section 14.8:
% Let $f(x,y,z)=x^2+y^2+z^2$
% and compute ${df\over dx}$ at
% $P_1 = ( {\sqrt{2} \over 2}, {\sqrt{2} \over 2} ,1-\sqrt{2} )$,
% subject to the two constraints $g_1(x,y,z) = x^2+y^2-1= 0$
% and $g_2(x,y,z) = x+y+z-1= 0$.
% That is, evaluate
% ${d\over dx}f(x,y(x),z(x))$ at $x={\sqrt{2}\over 2}$,
% where $y=y(x)$ and $z=z(x)$ are implicitly given by
% the constraints $g_1= 0$ and $g_2 = 0$.
% Can you explain why you get ${df\over dx} = 0$ at $P_1$?

% 16.1: Line Integrals.
\item
Section 16.1: Problems 15, 23, 25, 29.

% 16.2: Vector Fields and Line Integrals, Work, Circulation and Flux.
\item
Section 16.2: Problems 19, 25, 29(b), 35(a).

See equation (5) in p978 and (6) in p979 for definition of flow, circulation and flux.

% 16.3: Path Independence, Conservative Fields and Potential Functions.
\item
Section 16.3: Problems 1, 3, 5, 9, 21, 26, 29.

\item
Section 16.3:

Let 
$\vF = \displaystyle{ x \over \sqrt{x^2+y^2}} \,\vi + \displaystyle{ y \over \sqrt{ x^2+y^2}} \,\vj + 0 \,\vk$
and
$\vG = \displaystyle{-y \over x^2+y^2} \,\vi + \displaystyle{ x \over x^2+y^2} \,\vj + 0 \,\vk$.
\begin{enumerate}
\item
Show that both $\vF$ and $\vG$ satisfy the component test.
\item
The natural domain of both $\vF$ and
$\vG$ is $\{(x,y,z), \, x^2+y^2 \neq 0 \}$
(that is where $\vF$ and $\vG$ are defined).
Show that $\vF$ is conservative in this domain by finding
its potential function.
\item
Show that $\vG$ is NOT conservative in this domain 
(see Example 5 on p990).
\item
If given another $\vH$ satisfying the component test in this
domain, how do you determine whether $\vH$ is conservative?
\end{enumerate}

% 16.3: Path Independence, Conservative Fields and Potential Functions.
\item
Section 16.3: 

Let $\vF = \displaystyle{ x \over \sqrt{x^2+y^2+z^2}} \,\vi +
           \displaystyle{ y \over \sqrt{x^2+y^2+z^2}} \,\vj +
           \displaystyle{ z \over \sqrt{x^2+y^2+z^2}} \,\vk$.
What is the natural domain of $\vF$? 
Show that $\vF$ satisfies the component test in this domain.
Is this domain simply connected?
Is $\vF$ conservative in this domain? 

\end{enumerate}
 
\end{document}

