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\noindent {\bf Problem statement} Solutions of this equation

$${\rm (HE)\ \ \ \ \ }{{\partial^2 H} \over {\partial x^2}} +
{{\partial^2 H} \over {\partial y^2}} = 0 \ \ \ \ $$


\noindent describe {\it steady state heat f{l}ow} in a thin plate. The
partial differential equation is called the {\it Heat Equation}.

\medskip

\noindent a) Verify that if $H(x,y)= 3x^2 - 3y^2 +5xy + 4x - 2y + 7$ ,
then this $H$ is a solution of the equation $\rm (HE)$. What are all
the critical points of this $H$?  What type of critical point (max,
min, saddle) is each of them?

\medskip

\noindent b) Verify that if $H(x,y)= \cos x \sinh y${\parindent=0in \footnote*{The
function ``sinh $y$'' (sinh is pronounced like ``cinch'') is the
hyperbolic sine of y, and is defined by $\sinh y={{e^y-e^{-y}}\over
2}$.}}, then this $H$ is a solution of the equation $\rm (HE)$. What
are all the critical points of this $H$?  What type of critical point
(max, min, saddle) is each of them?

\medskip

\noindent c) Make a guess about the kind of critical point of
solutions of the equation $\rm (HE)$ can have. Prove that {\bf if} the
second derivative test can be applied, then any critical point must be
one of the type you asserted. Your assertion corresponds to a
physically ``reasonable'' property of heat flow: it doesn't focus or
lump up in the absence of heat sources.










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