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

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

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\large
Homework 13   
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\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 $P_1$?

\item
Section 15.3: Problems 13, 17. % and more on interchanging order

% 15.4: Double Integrals in Polar Form.
\item
Section 15.4: Problems 11, 21, 23, 26, 41, 42, 43.

% 15.5: Triple Integrals in Rectangular Coordinates.
\item
Section 15.5: Change the order of integration in 
Example 3 to $dxdydz$ 
% (the same example on
% page 9 of Lecture 22 and page 1-2 of Lecture 23).

% 15.5: Triple Integrals in Rectangular Coordinates.
\item
Section 15.5: Problems 21, 35, 41, 43, 47.

% % 15.7: Triple Integrals in Cylindrical and Spherical Coordinates.
% \item
% Section 15.7: Problems 9, 11, 13, 14, 15, 19, 21, 31, 39, 43, 45.
\end{enumerate}
 
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