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 {\bf Name~}\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\hfill
 %{Class Time~}\underline{~~~~~~~~}
 \newline


\begin{center}
\bf Math 3150 Problems \\
{\small\bf Chapter 1} 
\end{center}

{\small\sf  {\bf Due date}: See the internet due date. Problems are
collected once a week. Records are locked when the stack is
returned. Records are only corrected, never appended.

{\bf Submitted work}. Please submit one package per problem
set. Label each problem with its corresponding problem number, e.g.,
\fbox{Prob3.1-4} or \fbox{Xc1.2-4}. Kindly label extra credit
problems with label \fbox{\bf Extra Credit}. You may attach this
printed sheet to simplify your work.

{\bf Labeling}. The label \fbox{Probx.y-z} means the problem is for
chapter \fbox{x}, section \fbox{y}, problem \fbox{z}. When $y=0$,
then the problem does not have a textbook analog, it is a
\BF{background problem}. Otherwise, the problem number should match
a corresponding problem in the textbook. The same labeling applies
to extra credit problems, e.g., \fbox{Xc1.0-4}, \fbox{Xc1.1-2}.}

\section*{Chapter 1: 1.1-1.2 -- Introduction}

\Prob{Prob1.0-1}{(Partial Derivatives)} Compute the partial
derivatives.
$$
\begin{array}{|l|l|l|}
\hline
 \partial_x (1+t+x^2) & \partial_x (1+xt+(tx)^2) & \partial_x (\sin xt+t^2+tx^2)
 \\ \hline
 \partial_t (\sinh t \sin 2x) & \partial_t (e^{2t+x}\sin(t+x)) & \partial_t (e^{3t+2x}(e^t\sin x+e^x\cos t))
 \\ \hline
 \end{array}
$$

\Prob{Prob1.0-2}{(Jacobian)} Find the Jacobian matrix $J$ and the
Jacobian
determinant $|J|$ for the transformation $\.F(x,y)=\colvector{f_1\\
f_2}$, where $f_1(x,y)=\cos (2y) \sin (3x)$,
$f_2(x,y)=e^y\sin(y+x)$.

{\small\sf \BF{Background}: $J$ is the augmented $2\times 2$ matrix
of column vectors $\partial_x \.F(x,y$ and $\partial_y \.F(x,y)$.
For a $2\times 2$ matrix $A=\twobytwo{a&b}{c&d}$, the determinant is
defined by $|A|=ad-bc$.}

\Prob{Prob1.0-3}{(Directional Derivative)} Compute the gradient
vector $\grad(f)$ and the directional derivative of $f$ in the
direction of $\.v=2\vec{\imath}-\vec{\jmath}$ at $x=y=1$.

{\small\sf \BF{Background}: $\grad(f)=\colvector{\partial_x f\\
\partial_y f}$, $DD(f)=\grad(f)\cdot \.v/|\.v|$.}

$$
\begin{array}{|l|l|l|}
\hline
 f=x+y+1& f=x^2+y^2& f=ye^{x-1}
 \\ \hline
 \end{array}
$$

\Prob{Prob1.0-4}{(First Order ODE)} Solve the following first order
ODE problems. Reference: Appendix A of Asmar's text. Edwards-Penney
DE+LA Chapter 1. The first four are linear, the remaining separable.

$$
\begin{array}{|l|l|l|}
\hline
 y'-y=0 & y'+2y=0 & y'+2y=4
 \\ \hline
 xy'+y=x+1 & y'=y^2 & y'=xy
 \\ \hline
 \end{array}
$$


\Prob{Prob1.0-5}{(Wronskian)} Compute the Wronskian of the given
functions at $x=0$.

{\small\sf \BF{Reference}: Appendix A of Asmar's text, {\em PDE and
BVP}. Edwards-Penney, {\em DE and LA}, Chapter 5.}

$$
\begin{array}{|l|l|l|}
\hline
 1,x & 1,e^x & \cos x, \sin x\\ \hline
 e^x,e^{2x},e^{3x}& 1,\cosh x,\sinh x&1,\ln|1+x|,x\ln|1+x|
 \\ \hline
 \end{array}
$$

\Prob{Prob1.0-6}{(Newton's Laws)} ~\newline (a) State Newton's
second law $F=ma$ in calculus terms for a motion with position
$x(t)$.

(b) Use the statement in (a) to derive the spring-mass ODE
$mx''(t)+kx(t)=0$. Assume no external forces and a Hooke's restoring
force on the spring of magnitude $kx(t)$.

{\small\sf \BF{References}: Asmar section 1.2 and Edwards-Penney
DE+LA Chapter 5. The ODE in (b) is called the {\bf harmonic
oscillator}.}

\Prob{Prob1.1-3}{(PDE Solution)} Verify that $u(x,t)=e^{-(x-t)^2}$
is a solution of the initial value problem $\dd\frac{\partial
u}{\partial t}+\frac{\partial u}{\partial x} = 0$,
$u(0,t)=e^{-t^2}$. Draw four (4) waveform snapshots of the solution
$u$, for $x=0,1,2,3$. Each snapshot is a 2-dimensional plot in the
$tu$-plane with $t$-range $-2$ to $5$.

\Prob{Prob1.1-6}{(Change of Variables)} Show the details in solving
the PDE $\dd 2\frac{\partial u}{\partial t}+3\frac{\partial
u}{\partial x} = 0$ by using a change of variables. Check your
answer.

{\small\sf \BF{Problem notes}: Let $r=at+bx$, $s=ct+dx$ be the
change of variables $(t,x)\to (r,s)$ and let $w(r,s)=u(t,x)$. The
problem is to determine $a,b,c,d$ such that the PDE is changed into
the simpler PDE $\dd\frac{\partial w}{\partial r}=0$.

\BF{Reference}: Example 3, section 1.1 of Asmar's text.

\BF{Answer}: $u(x,t)=f(2x-3t)$. }

\Prob{Prob1.1-12}{(Method of Characteristics)} Show the details in
solving $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial
y} = 0$ by the method of characteristics. Check your answer.

{\small\sf \BF{Problem notes}: The ODE for characteristics is the
linear ODE $\dd\frac{dy}{dx}=\frac{y}{x}$. The answer is
$u=f(y/x)$.}


\Prob{Prob1.2-4a}{(d'Alembert's Solution)} Let $w(r,s)=u(x,t)$,
$r=x+ct$, $s=x-ct$, where $u(x,t)$ is a solution to the wave
equation $\dd\frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2
u}{\partial x^2}$. Show the details in the derivation of the
equation $\dd\frac{\partial^2 w}{\partial r \partial s}=0$.

\Prob{Xc1.2-4b}{(d'Alembert's Solution)} Let $w(r,s)=u(x,t)$,
$r=x+ct$, $s=x-ct$, where $u(x,t)$ is a solution to the wave
equation $\dd\frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2
u}{\partial x^2}$. Assume  $\dd\frac{\partial^2 w}{\partial r
\partial s}=0$. Solve this equation to verify
$u(x,t)=F(x+ct)+G(x-ct)$ where $F$ and $G$ are sufficiently smooth
functions.

\Prob{Prob1.2-5}{(Vibrating Infinite String)} Solve the wave
equation $\dd\frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2
u}{\partial x^2}$ with initial conditions $\dd
u(x,0)=\frac{1}{1+x^2}$, $u_t(x,0)=0$, $-\infty < x < \infty$, $t
\ge 0$. Use d'Alembert's solution.

\Prob{Prob1.2-17a}{(Vibrating Finite String)}
  Solve the wave
equation $\dd\frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2
u}{\partial x^2}$ with boundary and initial conditions
$u(0,t)=u(L,t)=0$, $u(x,0)=\frac12\sin\frac{\pi
x}{L}+\frac14\sin\frac{3\pi x}{L} + \frac25\sin\frac{7\pi x}{L}$,
$u_t(x,0)=0$, $0 < x < L$, $t \ge 0$. Use the series formula
$u(x,t)=\sum_{n=1}^\infty b_n\sin\frac{n\pi x}{L}\cos\frac{n\pi
ct}{L}$.

{\small\sf {\bf References}: Asmar's text, {\em PDE and BVP},
section 1.2 and Edwards-Penney, {\em DE and BVP}, section 9.6 (2280
textbook).}

%\Prob{Xc1.2-17b}{(Total Energy)} Compute the total energy
%$E=\frac{\pi^2T}{4L}\sum_{n=1}^\infty (nb_n)^2$, where $T$ is the
%tension and $L$ is the length of the string (the answer contains
%symbols $L$ and $T$).
%
%{\bf References}: Prob1.2-17a.



 \Prob{Xc1.2-17b}{(Loudness)} The fraction of the loudness
associated with the fundamental tone ($b_1$-term in the series) is
the quotient
$$F_1=\frac{\left.\left(n^2b_n^2\right)\right|_{n=1}}{\sum_{k=1}^\infty k^2b_k^2}$$
Find an approximation to the percentage $100F_1$.

{\small\sf {\bf References}: Prob1.2-17a. The discussion of music in
E\&P includes a derivation of the formula for the percentage
loudness $100F_n$.}
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