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.