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%==================== END PREAMBLE ==============
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\vspace*{-50pt}
{\bf  Math 361, Fall 2015\hfill Jerry L. Kazdan}
\medskip

\begin{center}
{\large\bf Problem Set 6}
\smallskip

\end{center}

%\comments{ 
{\sc Due:} Thurs. Oct.~22 in class. [Late papers will be
accepted (without penalty) until 1:00 PM Friday.] %} \bigskip
\bigskip

\centerline{\fbox{
{\bf Note:} The date of Exam 2 has been changed to Tuesday, Nov. 10.}}
\bigskip

 Please carefully read Chapter 7.1-7.2, 7.5-7.7 in the Marsden-Hoffman text. 
\medskip

The following short answer problems from Marsden-Hoffman are not
assigned, but you should know how to do them.  Some short answer
problems will be on our exams.

\medskip
p.~396,\#1, 2, 3, 5,\qquad p.~400 \#1, 2, 5,\qquad p.~411 \#1,\qquad p.~413 \#1, 2, 3,\qquad p. 444 \#35 

\bigskip

\begin{problems}

\item
\begin{parts}
\item Find a function $u(y,s)$ for $(y,s)\in \R^2$ such that $u_s=0$
and $u(y,0)=\sin 2y$.
\item Say we want to solve the partial differential equation
$u_x(x,t) - 2u_t(x,t)=0$.  One approach is to seek some change of
variable $y=ax+bt$, $s=cx+dt$ (where $a$, $b$, $c$, and $d$ are constants)
so that in the $y$, $s$ variables this equation has the simpler form
$u_s=0$ and then use the previous part.  Use this approach to solve
$u_x - 2u_t=0$ with $u(x,0)= x^2$.
\end{parts}


\item Let $w(x)$ and $u(x,y)$ be given smooth functions of the real
variables $x$ and $y$.
\begin{parts}
\item If $w(x)$ satisfies $w'' -c(x)w = 0$, where $c(x) >0$ is a given
function, show that $w$ cannot have a local positive maximum (that is, a local maximum where $w>0$).  Also show that $w$ cannot have a local negative minimum.
\item If $u(x,y)$ satisfies $ 4u_{xx}  + 3u_{yy} - 5u = 0 $,
show that it cannot have a local positive maximum.  Also show that $u$
cannot have a local  negative minimum.
\item If a function $u(x,y)$ satisfies $ 4u_{xx}  + 3u_{yy} - 5u = 0 $ in a bounded region $\Cal{D} \in \R^2$ and is zero on the boundary of the region,
show that $u(x,y)$ is zero throughout the region.
\end{parts}

\item The following equations define a map $F:(x,y,z)\mapsto (u,v,w)$:
\begin{align*}
 u(x,y,z) &= x + xyz^2 \\
 v(x,y,z) &= y + xy \\
 w(x,y,z) &= z + cx + 3z^2
\end{align*}
Clearly $F:(1,1,0)\mapsto (1,2,c)$.  Write $p=(1,1,0)$ and
$q=(1,2,c)$.
\begin{parts}
\item Compute the derivative $F'(p)$.
\item For which value(s) of the constant $c$ can the system of
equations: can be solved for $x$, $y$, $z$ as smooth functions of
$u$, $v$, $w$ near $(1,1,0)$?  Justify your assertion(s).
\item If $c$ is one of these ``good'' values, let $G:(u,v,w)\mapsto
(x,y,z)$ be the map inverse to $F$.  Compute the derivative $G'(q)$
and use it to compute $\partial y(u,v,w)/\partial v$ at $q$.
\end{parts}

\item{}  Show that the system of equations
\begin{align*}
3x+y-z+u^2=&0\\
x-y+2z+u=&0\\
2x+2y-3z+2u=&0
\end{align*}
can be solved for $x$, $y$, $u$ in terms of $z$,  for $x$, $z$, $u$ in
terms of $y$ for $y$, $z$, $u$ in terms of $x$, but {\it not} for
$x$, $y$, $z$ in terms of $u$.

\comments{
\item{}  Let $f(x,y_1,y_2)=x^2y_1 + e^x +y_2$ and
note that $f(0,1,-1)=0$.  Show that near $y_1=1$, $y_2=-1$ there is a
smooth function $x=g(y_1,y_2)$ with $g(1,-1)=0$ so that
$f(g(y_1,y_2),\,y_1,\,y_2)=0$.  Also, compute the gradient of $g$ at
$(1,-1)$.


\item Let $y = f(x,u)$ and $z = g(x,u,v)$ be smooth functions with, say,
$f(x_0,u_0)=y_0$ and $g(x_0,u_0,v_0)=z_0$.
\begin{parts}
\item Under what condition(s) can one eliminate $x$ from these
equations to express $z$ as $z = F(y,u,v)$ as a smooth function of $y$,
$u$, and $v$ near $x=u_0$, $y=y_0$, $v=v_0$ ?
\item Assuming this, then compute $\partial z/\partial u$ and
$\partial z/\partial y$ in terms of the derivatives of $f$ and $g$.
To make this computation more specific, assume that
$
f_x(x_0, u_0) = 1, \quad f_u(x_0, u_0) = -2,
\quad g_x(x_0, u_0, v_0) = -3, \quad g_u(x_0, u_0, v_0) = 4, \text{ and } 
\quad g_v(x_0, u_0,v_0) = -2.
$
\end{parts}
}

\item Let $p(x) := (x-1)(x-2)\cdots(x-6) = x^6 - 21x^5 + \cdots$ and
let $p(x,t)$ be the perturbed polynomial obtained by replacing $-21x^5$
by $-(21 +t)x^5$ (think of $t$ as being small).  Let $x(t)$ denote the perturbed
value of the root $x=4$, so $x(0)=4$.
\begin{parts}
\item Show that $x(t)$ is a smooth function of $t$ for all $\abs{t}$
sufficiently small.
\item  Compute the sensitivity of this root as one changes $t$,
that is, compute  $dx(t)/dt)\big|_{t=0}$.
\end{parts}

\item {} [Marsden-Hoffman p.~413 \#4] Let $f(x,y)= x^2 + y^2 + 3y^3 +8x^4 + x^2e^x\sin x +6$.  Show there are new coordinated $u$,$v$, where $u=u(x,y)$, $v=v(x,y)$ for which
$$ 
f(x,y)= u^2 + v^2+6.
$$
in a neighborhood of the origin.

\bigskip
\hspace*{-15pt}\vbox{{\bf Problems 7, 8, and 9 below are deferred until
  Homework Set 7}}

\item{} [Marsden-Hoffman p.~420 \#4] Find the extrema of $f(x,y,z)=x+y+z$ subject to the constraints: $x^2+y^2=1$, $2x+z=1$.

\item ]Marsden-Hoffman p.~444 \#35] Find the relative extrema of $f(x,y)=x^2+y^2$ subject to the constraint $x^2-y^2=1$.

\item Let $A$ be a real square symmetric matrix and let $v\in\R^n$
be a point on the unit sphere, $\norm{x}=1$, where $f(x)=\ip{x}{Ax}$ has its maximum.  Show that $v$ is an eigenvector of $A$.  What is the corresponding eigenvalue?

\item Let $A(t) = \bigl[a_{ij}(t)\bigr]$ be a square
matrix whose coefficients are real and depend smoothly on a real
parameter $t$.
\begin{parts} 
\item If $\lambda(0)$ is a simple eigenvalue (that is, its
algebraic multiplicity is one, so $\lambda(0)$ is a ``simple'' root of
the characteristic polynomial), show that $\lambda(t)$ 
%and the corresponding eigenvector $v(t)$ are 
is a smooth functions of $t$ for $t$ sufficiently small. 
\item If the above matrix $A(t)$ is self-adjoint (symmetric) 
with $A(0)v=\lambda(0)v$, derive the formula 
$$
\lambda'(0) = \frac{\ip{v}{A'(0)v}}{\norm{v}^2}
\qquad\left(\text{here}\quad {}'=\frac{d}{dt}\right). 
$$ 
\end{parts}

\end{problems}

%\comments{
\vskip 20pt

\begin{center}{\large \bf Bonus Problem}
            
[Please give these directly to Professor Kazdan]
\end{center}

\begin{problems}

\item [B-1] If $A$ is a square matrix that is sufficiently close
to the identity matrix, show that it has a square root, that is,  there is a
matrix $B$ with $B^2=A$.  Moreover this matrix $B$ is unique if it
is required to be near the identity matrix.

\item [B-2] If $h(x,y)= x^2 -2xy +5y^2$, since then
$h(x,y)= (x-y)^2 + 4y^2$, it is clear that under the change of
coordinates $u=x-y$, $v=2y$ we can write $h=u^2+v^2$ as a sum of
squares.  Prove that one can do this near the origin for any smooth
function $f(x,y):\R^2\to \R$ with the properties that
$$
f(0,0)=0, \qquad  f'(0,0)=0, \qquad  f''(0,0) \text{ is positive
  definite}.
$$
[Here $f'$ is the gradient and $f''$ the second derivative matrix.]

\item [B-3] [Marsden-Hoffman, p.~413 \#4]
\begin{parts}
\item If the smooth function $f$ has a non-degenerate critical point at $x_0\in\R^n$, show there is a neighborhood of $x_0$ containing no other critical points.
\item Find the critical points of $f(x,y)=x^2y^2$.
\end{parts}


 \end{problems}
%}




\medskip
[Last revised: \today]
\end{document}
