Let $f, g: \mathbb{R}^2 \to \mathbb{R}$ be differentiable functions such that $g_x = f_y$ and $g_y = -f_x$. The flux lines of $f$ are defined to be the curves that are orthogonal to the level curves of $f$. Why is it that the level curves of $g$ are the flux lines of $f$?
I know it has everything to do with the fact that the gradient is orthogonal to level curves, but I haven't quite been able to formulate an airtight proof yet. I'd appreciate some help.
Note that the gradient vectors $$< f_x, f_y>$$ and $$< g_x, g_y>$$ are perpendicular to each other.
That implies that the level curves meet at a $ 90$ degrees angle.
Thus the level curves of one is the flux curves of the other.