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%  Spring semester, 2024, Wei-Cheng Wang.


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\begin{document}


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%          The title and header.

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

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

\begin{center}
\large
Study guide for Final Exam
\normalsize
\end{center}

\noindent

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

\begin{enumerate}

% \item
% Review study guide for midterm 01 and midterm 02.
% Review and correct your mistakes in midterm 01 and midterm 02.

\item
Review study guide for quiz 09-12. % (Section 14.10-15.8).

% 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 
$$\displaystyle{ \int\limits_C f(x,y,z)\, ds }, $$

$$\displaystyle{  \int\limits_C \F(x,y,z) \cdot \T ds 
               = \int\limits_C \F (x,y,z)\cdot d\r 
               = \int\limits_C M(x,y,z)\,dx + N(x,y,z)\,dy + P(x,y,z)\, dz },$$
$$\displaystyle{  \int\limits_C \F(x,y) \cdot \n \,ds 
               = \oint\limits_C \F(x,y) \cdot \n \,ds 
               = \varointctrclockwise\limits_C M(x,y) \,dy -N(x,y) \,dx }$$
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?

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

Read 
"Note on Conservative Fields and Simply Connected Domains
(Check\_conservativeness\_v01.pdf)"
on the course homepage.

Study and memorize the definitions of 
'path independent', 'conservative' and 'potential function' (p984).

Study the statement and proof of Theorem 1:'Fundamental Theorem of Line Integrals' (p985).

Study the definition and examples (Figure 16.22) 
of 'connected domain' and 'simply connected domain' (p985).

Study the relation between 
conservative fields and
\begin{enumerate}
\item
'Gradient Fields' (Theorem 2, p986).
\item
'Loop Property' (Theorem 3, p987) 
\item
'Component test' (p988), 
\end{enumerate}
Which of them are equivalent to 'Conservative Fields'?

Which of them are equivalent to 'Conservative Fields' only on simply connected domains?
% What happens homework 13, probelm 2 for the case when the domain is not simply connected.
% If the domain is not simply connected,

Which of the implications 
'$\Longleftarrow$', 
'$\Longrightarrow$' remains valid
even if the domain is not simply connected?
Review homework 15, problem 4,5 for relevant examples.

% which of them are eqrequires the domain to be simply connected?
% 
% Regarding the two theorems on equivalent characterization of 
% 
% Study the proof of Theorem 2: 'Conservative Fields are Gradient Fields' (p986).
% 
% Regarding the two theorems on equivalent characterization of 
% conservative fields,
% Theorem 3: 'Loop Property' (p987) and 'Component test' (p988), 
% which of them requires the domain to be simply connected?
% If the domain is not simply connected,
% which of the implications 
% '$\Longleftarrow$', 
% '$\Longrightarrow$' remains valid? which one may probably be false?
% Is Example 5 a counter example of which case?

For given functions $M(x,y,z), N(x,y,z), P(x,y,z)$
satisfying the component test, how does one find the 
potential function (if it exists) by way of direct
integration? See Example 3, page 98 for details.

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

Skip "Exact Differential Forms" on page 991-992.

% 16.4: Green's Theorem in the Plane.
% \item 
% Section 16.4:

% Study and memorize definitions of divergence and curl.
% It is easier to start with  
% $${\rm div} \F(x,y,z) = \nabla \cdot \F(x,y,z),
% \quad \text{and} \quad
% {\rm curl} \F(x,y,z) = \nabla \times \F(x,y,z),$$
% then reduce them to
% $${\rm div} \F(x,y) \quad 
% \text{and} \quad
% {\rm curl} \F(x,y) \cdot \k.$$

\end{enumerate}

\end{document}
