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%  Spring semester, 2023, Wei-Cheng Wang.


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\begin{document}


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\noindent
{\scriptsize Calculus II, Spring 2023 
(http://www.math.nthu.edu.tw/\~\,wangwc/)} \hfill 

\noindent
{\scriptsize Thomas' Calculus Early Transcendentals 13ed} \hfill 

\begin{center}
\large
Study guide for quiz 10
\normalsize
\end{center}

\noindent

\vspace{-.3cm}
Quiz problems include both the lecture contents and homework problems.

\begin{enumerate}

% 15.8: Substitutions in Multiple Integrals.
\item
Section 15.8: 

Study the meaning of the Jacobian and
memorize the formula
both in double and triple integration.

% 15.8: Substitutions in Multiple Integrals.
\item
Section 15.8: 

Assume that a change of variables between $(x,y,z)$
and $(u,v,w)$ is given, study how to change the
lower and upper limits of integration for $(x,y,z)$
into
lower and upper limits of integration for $(u,v,w)$.
See the examples in section 15.8.

% 16.1: Line Integrals.
% 16.2: Vector Fields and Line Integrals, Work, Circulation and Flux.
\item 
Section 16.1, 16.2: 
All exam problems will be expressed in explicit mathematical symbols.
So you do not need to memorize the definitions of 
first moments, center of mass and moments of inertia, etc. in Section 16.1
and work, circulation, flow and flux in Section 16.2
in preparing the exams.

% 16.1: Line Integrals.
% 16.2: Vector Fields and Line Integrals, Work, Circulation and Flux.
\item 
Section 16.1, 16.2:

Study the meanings of 
$$\displaystyle{ \int_C f(x,y,z)\, ds }, $$

$$\displaystyle{  \int_C \F(x,y,z) \cdot \T ds 
               = \int_C \F (x,y,z)\cdot d\r 
               = \int_C M(x,y,z)\,dx + N(x,y,z)\,dy + P(x,y,z)\, dz },$$
$$\displaystyle{  \int_C \F(x,y) \cdot \n \,ds 
               = \varointctrclockwise_C M(x,y) \,dy -N(x,y) \,dx }$$
and how to calculate them
using a properly chosen parametrization of 
$C$: $\r(t)$, $t_0 \leq t \leq t_1$.

A few points to pay attention:

Which of them is (are) independent of the orientation
of $C$? 
Which of them depend(s) on the orientation
of $C$?

How do you choose the parametrization $\r(t)$ so that  
the direction of $\T$ comply with the orientation of $C$?

How is the outward normal $\n$ related to
$\T$ if the parametrization of $C$ is
increasing in the counter-clockwise direction?

\end{enumerate}

\end{document}
