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Orthogonality of level curves of harmonic conjugates - an example of 3rd degree

Orthogonality of level curves - harmonic functions

Take f(z)=z^3, where z is a complex variable. In Cartesian form we have f(z)=u(x,y)+iv(x,y) where u(x,y)=x^3-3xy^2 and v(x,y)=3x^2y-y^3. The level curves of u and v are orthogonal curves, i.e. at their points of intersection they have orthogonal tangents. Here the level curves have in general 3 points of intersection. We illustrate the orthogonality at one of the points of intersection. Try at the other ones.