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So I've had this assignment in which I had to proof that two level curves of a function and one of its harmonic conjugates intersect each other orthogonally.

The proof itself wasn't that difficult, but I wondered: why does this happen?

What is the underlying cause for this phenomenon?

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This is a consequence of the Cauchy-Riemann equations: $$ u_x=v_y,\\u_y=-v_x, $$ which imply that $$ \nabla u \,\perp\, \nabla v. $$

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