Orthogonality of level curves of harmonic conjugates

Check orthogonality

Express a complex function f of one complex variable in Cartesian form: you obtain two real functions u and v of two real variables x,y. If f is analytic over a certain domain, then u and v are "harmonic conjugates". The level curves of u (curves whose equations are u(x,y)=C1) are orthogonal to the level curves of v (curves whose equations are v(x,y)=C2). This applet checks this property for f(z)=z^2.