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

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\def\ds{\displaystyle}
\def\limi{\displaystyle{\lim_{n\to \infty}}}
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\noindent
{\scriptsize Calculus II, Spring 2024 
(http://www.math.nthu.edu.tw/\~\,wangwc/)} \hfill 

\begin{center}
\large
Homework 14   
\normalsize
\end{center}

\vspace{.3in}

%           The questions!

\begin{enumerate}

% 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.

% 15.8: Substitutions in Multiple Integrals.
\item
Section 15.8: Problems 5, 7, 9, 15, 19.

% 15.8: Substitutions in Multiple Integrals.
\item
Section 15.8: Compute the Jacobians for the substitutions
$x=r\cos\theta$, $y=r\sin\theta$ in double integral,
$x=r\cos\theta$, $y=r\sin\theta$, $z=z$ and
$x=\rho\sin\phi\cos\theta$, $y=\rho\sin\phi\sin\theta$
$z=\rho\cos\phi$
in triple integral, respectively.

% 15.8: Addtional and advanced exercises
\item
Chap 15, Additional and Advanced Exercises:
Problems 11, 12.

% 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 $P_1$?

% 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.

% Section 14.10: Partial Derivative with Constrained Variables
\item (Optional)

Section 14.10: 
Follow up on problem 3 of homework 12:

Evaluate the constrained second derivatives of $w$ at the points $(x_0, y_0, z_0)$ obtained in homework 12, problem 3.
That is, evaluate
$
\left( \ds{\partial \over \partial x } \left(\ds{\partial w \over \partial x }\right)_y \right)_y,
$
$
\left( \ds{\partial \over \partial y } \left(\ds{\partial w \over \partial y }\right)_x \right)_x
$
and
$
   \left( \ds{\partial \over \partial y } \left(\ds{\partial w \over \partial x }\right)_y \right)_x
 = \left( \ds{\partial \over \partial x } \left(\ds{\partial w \over \partial y }\right)_x \right)_y
$
at $(x_0, y_0, z_0)$
and use them to determine whether $(x_0,y_0,z_0)$ is a local min, local max, or saddle point of $w$ on the
constraint set $x^2+y^2+z^2 = 30$.

Hint:
First evaluate
$
\left( \ds{\partial \over \partial x } \left(\ds{\partial z \over \partial x }\right)_y \right)_y,
$
$
\left( \ds{\partial \over \partial y } \left(\ds{\partial z \over \partial y }\right)_x \right)_x
$
and
$
   \left( \ds{\partial \over \partial y } \left(\ds{\partial z \over \partial x }\right)_y \right)_x
 = \left( \ds{\partial \over \partial x } \left(\ds{\partial z \over \partial y }\right)_x \right)_y
$
at $(x_0, y_0, z_0)$ from the constraint $x^2+y^2+z^2 = 30$.  

Remark: Both methods (Lagrange Multiplier and Partial Derivative with Constrained Variables)
can be applied to find easily the critical points of $w$ on the constraint set. 
However, the second derivative test for Lagrange Multiplier is much more complicated (try 
google it and you will find it difficult even to understand the statement).
In contrast, the second derivative test for 
Partial Derivative with Constrained Variables is straight forward as outline above.

\end{enumerate}
 
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