I stumbled with this problem in an old book that has been bothering me last days. Could you help me with this?
Be $f(x + iy) = u(x,y) + iv(x,y)$ an holomorphic function. Prove that level curves $u(x,y) = c, v(x,y)=k$ are two families of orthogonal trajectories.
Assume $\frac{\partial u}{\partial y}\neq0$ on some interval. You have $$ \frac{\partial u}{\partial x}\times 1+\frac{\partial u}{\partial y}\times y'(x)=0, $$ so (writing $y_1$ to represent the first curve) $$ y_1'(x)=-\frac{\frac{\partial u}{\partial x}}{\frac{\partial u}{\partial y}}. $$ Similarly, and using Cauchy-Riemann, $$ y_2'(x)=-\frac{\frac{\partial v}{\partial x}}{\frac{\partial v}{\partial y}}=-\frac{-\frac{\partial u}{\partial y}}{\frac{\partial u}{\partial x}}=-\frac1{y_1'(x)}. $$