= Orthogonality of regular real and imaginary level curves
{title2=$\nabla u\cdot\nabla v=0,\quad |\nabla u|=|\nabla v|=|f'|$}
For a <holomorphic function> $f=u+iv$, the <Cauchy-Riemann equations> give the displayed <gradient> identities. Where $f'\ne0$, both <level sets> are regular curves; their nonzero <normal vectors> are perpendicular, so their tangents are perpendicular as well. The qualification is necessary. At a <critical point> both <gradients> vanish and the zero <level sets> may have several tangent branches. For $f(z)=z^2$ at zero, $u=0$ consists of two diagonals while $v=0$ consists of the two coordinate axes; there is no general orthogonality assertion at that point.
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