# Curves in complex plane

• Nov 4th 2012, 01:03 PM
flametag3
Curves in complex plane
If a function $f(z) = u(x,y) + iv(x,y)$ is analytic in some domain in the complex plane.

How can we show that families of the level curves $u(x,y) = c_1$ and $v(x,y) = c_2$ are orthogonal.

Also, point z_0 of intersection of two curves $u(x,y) = c_1$ and $v(x,y) = c_2$ that the tangent lines are perpendicular if f'(z) is not 0.

For this do I just find u' and v' and then apply Cauchy - Riemann eqns??
• Nov 4th 2012, 06:07 PM
chiro
Re: Curves in complex plane
Hey flametag3.

You can indeed do that and the formality is associated with what is known as the inverse function theorem which says that if the derivative is zero at a point when it has no inverse in the neighbourhood at that point.

Since it is zero everywhere for du/dv and for dv/du then there is no dependency at all between the two functions and they are as such, orthogonal and independent.

This idea is at the heart of geometry and tensors are able to encapsulate this not just for curved space-times, but isomorphically for functions depending on how you look at it.