Math 6420 • PDE

\begin{enumerate}
\item[(a)] Consider a 2-dimensional map \((x,y)\to(u,v)\).

Explain: If the map is given by an analytic function [\(u+iv=f(x+iy)\)] and at some
point \(z_0=x_0+iy_0\) the derivative \(f'(z_0)\ne0\), then this map preserves small
shapes in the vicinity of \(z_0\).

\item[(b)] Does there exist a conformal transformation \(u+iv=f(x+iy)\) that maps triangle
\[
  \Delta=\{(x,y):0<x<1,\ 0<y<1,\ x+y<1\}
\]
to circle
\[
  D=\{(u,v):u^2+v^2<1\}\,?
\]
\end{enumerate}