# Showing two families of curves are orthogonal.

by EricVT
Tags: curves, families, orthogonal, showing
 P: 163 Let the function f(z) = u(x,y) + iv(x,y) be analytic in D, and consider the families of level curves u(x.y)=c1 and v(x,y)=c2 where c1 and c2 are arbitrary constants. Prove that these families are orthogonal. More precisely, show that if zo=(xo,yo) (o is a subscript) is a point in D which is common to two particular curves u(x,y)=c1 and v(x,y)=c2 and if f '(zo) is not equal to zero, then the lines tangent to those curves at (xo,yo) are perpendicular. I really have absolutely no idea how to show this. It gives the suggestion that $$\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y}\frac{dy}{dx} = 0$$ and $$\frac{\partial v}{\partial x} + \frac{\partial v}{\partial y}\frac{dy}{dx} = 0$$ So the total derivatives with respect to x of u and v are both zero. Should I equate these and look for some relationship between the partials? Since the function is analytic we know $$u_x = v_y$$ $$u_y = -v_x$$ So this can be rewritten in several different ways, but I really just don't know what I am looking for. Can anyone please offer some advice?