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Prove that if $u$ and $v$ are harmonic conjugate then $\nabla u\bot \nabla v$.

I really don't know if I am correct or not because it seems too trivial too me so I would appreciate an evaluation or correction.

I know that if $u$ and $v$ are conjugate harmonic then $u+iv$ is holomorphic (in my course it is often said to be *Analytic) in a certain domain (no domain was mentioned in the question.). Therefore Cauchy-Riemann equations hold satisfying $u_x=v_y,u_y=-v_x$ for all $z=(x,y)$ in the domain. That means $\nabla u=({\partial u\over \partial x},{\partial u\over \partial y})=(u_x,u_y),\nabla v=(v_x,v_y)$ and therefore $$\langle \nabla u, \nabla v \rangle=u_x\overline{v_x}+u_y\overline{v_y}=v_x{v_x}-v_y{v_x}=0.$$

Is there something wrong here or should it be that easy?

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    $\begingroup$ I think CR equations is indeed what you need. $\endgroup$ – Ilya Nov 11 '15 at 16:38
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    $\begingroup$ There is nothing wrong. Your approach is correct, and yes, it is just that easy. $\endgroup$ – Mark Viola Nov 11 '15 at 16:47
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To keep the question from being unanswered. Yes, you are correct.

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