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\begin{document}


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\noindent
{\scriptsize Thomas' Calculus Early Transcendentals 12ed} \hfill 

\begin{center}
\large
Study Guide for Chap 14
\normalsize
\end{center}

\noindent
% \noindent
\vspace{.5cm}

\begin{enumerate}

% Try to make up some problems for each of the following.
% If you can do that, that means you do understand it.
\vspace{.5cm}

\begin{enumerate}

\item
Review and memorize definitions of limit
and continuity (using $\varepsilon$ and $\delta$) for functions
of two or more variables. Review related examples in section 14.2
and understand the reasons for the examples where the limit
does not exist.
 
\item 
Study the definition of differentiability 
for functions of two or more variables in section 14.3.
Study the relation of differentiability with the tangent plan,
wih linear approximations,
and why exisitence of partial derivatives and directional 
derivatives at point does not imply differentiability at a point.

\item 
Study why differentiability at a point implies continuity at that point.

\item 
Study the relation between Chain rule and differentiability 
for functions of two or more variables in section 14.4.
Practice the chain rule, for example, between
Cartesian coordinate $(x,y)$ and polar coordinate $(r,\theta)$.

\item 
Review the definition of the directional derivatives.
Practice with some examples.

\item 
Review the definition and properties of the gradient vector
and its relation with tangent and normal line/plane.

\item 
Review all cases for the method of the Lagrangian multipliers.
For example, extrems values of
\begin{itemize}
\item
functions of 2 variables, with 1 constraint.
\item
functions of 3 variables, with 1 constraint.
\item
functions of 3 variables, with 2 constraints.
\end{itemize}

\item 
Review the derivation of Taylor's formula for functions of two
or more variables.

\item 
Review partial derivatives with constrained variables.
For example, partial derivatives of 
\begin{itemize}
\item
functions of 3 variables, with 1 constraint.
\item
functions of 4 variables, with 2 constraints.
\item
functions of 3 variables, with 2 constraints.
\end{itemize}

%% \item 
%% For multiple integrals, focus on the interchanging between
%% $\int\int dx dy $ and $\int \int dy dx$ when the domain
%% $R$ is not a rectangle. How do you do that?


\end{enumerate}
\end{document}