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\begin{document}
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\vspace*{-.7in}

\noindent
\textbf{Math 5616H} \hfill \textbf{Homework 7}

\noindent
\textbf{Posted: 3/23; Updated: 3/24; Due: Friday, 3/27/2015}\\
The problem set is due at the beginning of the class on
Friday. \smallskip

\noindent
\textbf{Reading}: Chapter 9: pp. 221-228. \smallskip
\
\begin{prob}
  Let $U \subset \nr^n$ be an open set and $f : U \to \nr$. The
  function $f$ is said to be in $C^k(U)$, if $f$ has all continuous
  partial derivatives up to and including order $k$ on $U$. Let $l \le
  k$ and $i_1, i_2, \dots, i_l$ be a collection of integers between
  $1$ and $l$. Under these conditions, show that for any permutation
  $\sigma \in S_l$, we have
\[
D_{i_1, \dots, i_l} f = D_{\sigma (i_1), \dots, \sigma (i_l)} f.
\]
  \end{prob}

\begin{prob}
Let 
\[
f(x,y) = \begin{cases} 0 & \text{when $(x,y) = (0,0)$},\\
(x^2 + y^2) \sin \left( \frac{1}{\sqrt{x^2+y^2}} \right)& \text{otherwise}.
\end{cases}
\]
Show that $D_1 f$ and $D_2 f$ exist everywhere but are not continuous
at the origin.
\end{prob}

\begin{prob}
  Let $f: \nr^3 \setminus \{(0,0,0)\} \to \nr^3 \setminus \{(0,0,0)\}$
  be given by
\[
f(x,y,z) = \left( \frac{x}{x^2+y^2+z^2}, \frac{y}{x^2+y^2+z^2},
  \frac{z}{x^2+y^2+z^2} \right).
\]
Show that $f$ is locally invertible at every point in $\nr^3 \setminus
\{(0,0,0)\}$. Find an explicit formula for $f^{-1}$.
\end{prob}

\begin{prob}
Consider the equations
\[
ab^2 + cde + a^2 d = 3 \qquad \text{and} \qquad ace^3 + 2bd - b^2 e^2 = 2.
\]
Determine for which pairs of variables the Implict Function Theorem
implies that the two equations can be uniquely solved for in terms of
the other three near the point $(a,b,c,d,e) = (1,1,1,1,1)$.
\end{prob}

\begin{prob}
  Define a map from $\nr^2$ to itself by setting
\[
F(x,y) = (\sin x \cos y + \cos x \sin y, \cos x \cos y - \sin x \sin y).
\] 
Does there exist a point $(x_0, y_0)$ such that $F$ is locally
invertible in a neighborhood of $F(x_0, y_0)$?
\end{prob}

\end{document}