\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} A function $f(x,y)$ is called {\it
homogeneous of degree} $k$ if the equation $(*)$ given by $f(tx,ty) =
t^k f(x,y)$ is true for all $x$, $y$, and $t$.

\medskip

\noindent a) Suppose $f(x,y) = x^3y - 5x^2y^2$.  Show that $f(x,y)$,
$f_x(x,y)$ and $f_y(x,y)$ are each homogeneous. What are the degrees
of homogeneity?  Also verify that $xf_x(x,y) + yf_y(x,y) = 4f(x,y)$.

\medskip

\noindent b) Suppose $f(x,y)$ is any function that is homogeneous of
degree $k$. Show that $f_x(x,y)$ is homogeneous of degree $k-1$.

\smallskip

\noindent {\bf Hint} Apply $\partial/\partial x$ to each side of $(*)$
and use the Chain Rule.

\medskip

\noindent c) Suppose $f(x,y)$ is any function that is homogeneous of
degree $k$. Show that
$ x f_x(x,y) + y f_y(x,y) = k f(x,y)$.

\smallskip

\noindent {\bf Hint} Apply $\partial/\partial t$ to each side of $(*)$
using the Chain Rule. Then set $t=1$.







\vfil\eject\end