Exercise $2.$ Show that if f is smooth in $\backslash$Omega $,$ then there is a C so that

max $|$fT$($x$,$y$)$f$($x$,$y$)|<=$Ch$2.($x$,$y$)$ in $\backslash$Omega

Suppose T is a triangular element with vertices $($a$,$b$,$c$).$ The standard tri$-$

angular element, called T$0,$ is the unit right triangle with vertices $\backslash$alpha $=(0,0),$

$\backslash$beta $=(1,0),$ and $\backslash$gamma $=(0,1).$ The condition number of T is the condition number

ofthelinear$($affine$,$actually$)$mapwitha$->\backslash$alpha $,$b$->\backslash$beta $,$andc$->\backslash$gamma $.$ Themap x

is affine in the sense that it takes $($x$,$y$)$ to M y $+$r with r $=0$ in general.

The map is linear if r $=0.$ The condition number is the condition number of M: $\backslash$kappa $($T$)=\backslash$kappa $($M$).$ Note a little sloppiness here: The condition number $\backslash$kappa $($T$)$ defined here depends a little bit on which vertex of T goes to which vertex of T$0.$ You could fix that by making T$0$ a unit isosceles triangle $($all edges have length $1)$ or by noting that knowing the condition number to within a factor of $2$ is good enough for this exercise. Making T$0$ an isosceles triangle makes gradient estimation more complicated.