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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 

\begin{center}
\large
Homework 12   
\normalsize
\end{center}

\vspace{.3in}

%           The questions!

\begin{enumerate}

% Section 14.10: Partial Derivative with Constrained Variables
\item
Section 14.10: 
% Problems 3, 7, 9, 12.
% 9: dx/dy*dy/dz*dz/dx = -1, appeared before. unify next time.
Problems 3, 7, 9, 12.

Hint for Problem 12: Lecture 23, page 4. 

% Section 14.10: Partial Derivative with Constrained Variables
\item
Section 14.10: 
Follow up on problem 12:

Derive a formula for
$\left( \partial u \over \partial x \right)_y$
if $u = U(x,y,z,w)$,
$f(x,y,z,w) = 0$ and $g(x,y,z,w) = 0$ 
% where $u = U(x,y,z,w)$ with the constraints
% $f(x,y,z,w) = 0$ and $g(x,y,z,w) = 0$
provided $f_z g_w - f_w g_z \neq 0$.

Hint: count the numbers of dependent and independent variables first.
Then use the result in Problem 12 to find 
$\left({\partial w \over \partial x }\right)_y$
and
$\left({\partial z \over \partial x }\right)_y$
first,
and then
$\left({\partial u \over \partial x }\right)_y$.

You can first take a specific example such as
$U(x,y,z,w) = x^2 + y^2 + z^2 + w^2$,
$f(x,y,z,w) = x +  y + z^2 + w^2$,
$g(x,y,z,w) = x^2 + y^2 + z + w$.
Then generalize the result to general $U$, $f$ and $g$.

% 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 $P_1$?

% Section 14.10: Partial Derivative with Constrained Variables
% \item
% Section 14.10: 
% Follow up on Section 14.8, problem 23:
% 
% It is known that the minimum and maximum for this problem occur at 
% $P_1=(1,-2,5)$ and $P_2=(-1,2,-5)$, respectively.
% \begin{enumerate}
% \item
% Compute 
% the partial derivatives with constrained variables
% $\left({\partial w \over \partial x }\right)_y$
% and
% $\left({\partial w \over \partial y }\right)_x$
% at $P_1$ and $P_2$, respectively
% if $w=x-2y+5z$ and
% $x^2+y^2+z^2=30$.
% \item
% Do the same for 
% $\left({\partial w \over \partial y }\right)_z$
% and
% $\left({\partial w \over \partial z }\right)_y$.
% \item
% Do the same for 
% $\left({\partial w \over \partial z }\right)_x$
% and
% $\left({\partial w \over \partial x }\right)_z$.
% \end{enumerate}

% Section 14.10: Partial Derivative with Constrained Variables
\item
Section 14.10: 
Compute the partial derivatives with constrained variables
$\left({\partial w \over \partial x }\right)_y$
and
$\left({\partial w \over \partial y }\right)_x$
if $w=x-2y+5z$ and
$x^2+y^2+z^2=30$.
Then find $(x_0, y_0, z_0)$ on 
$x^2+y^2+z^2=30$ such that
\begin{equation}\label{xy}
\left({\partial w \over \partial x }\right)_y(x_0, y_0, z_0) = 0, \quad
\left({\partial w \over \partial y }\right)_x(x_0, y_0, z_0) = 0.
\end{equation}

Remark: This is an alternative method for Section 14.8, problem 23. 
You can also change \eqref{xy} to
\begin{equation}\label{yz}
\left({\partial w \over \partial y }\right)_z(x_0, y_0, z_0) = 0, \quad
\left({\partial w \over \partial z }\right)_y(x_0, y_0, z_0) = 0,
\end{equation}
or
\begin{equation}\label{zx}
\left({\partial w \over \partial z }\right)_x(x_0, y_0, z_0) = 0, \quad
\left({\partial w \over \partial x }\right)_z(x_0, y_0, z_0) = 0
\end{equation}
and still get the same solution $(x_0, y_0, z_0)$ (why?).

% 15.1: Double and Iterated Integrals over Rectangles.
\item
Section 15.1: Problems 21, 33, 36.

% 15.2: Double Integrals over General Regions.
\item
Section 15.2: Problems 11, 19, 29, 35, 43, 47, 69.

Remark on problem 47:
A special trick will be  explained in Lecture 25, 240523.

% 15.3: Area by Double Integration.
% \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.

\end{enumerate}
 
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