Theorem 11.4 (Orthogonal Families of Level Curves).  Let be harmonic in a domain D.  Let be the harmonic conjugate and let     be the complex potential.  Then the two families of level curves given in Equations  (10-16) and (10-17), respectively, are orthogonal in the sense that if is a point in common to the two curves    and  ,  and  if    ,  then these two curves intersect orthogonally.

Exploration for Theorem 11.4.

Use the Cauchy Riemann equations    and ,  which are implemented by using the Mathematica  substitutions:

The normal vectors to the curves    are

We will show that   is orthogonal to    by forming the dot product:

Using the Cauchy Riemann equations, this is simplified and we obtain:

Therefore   is orthogonal to   .

(c) 2006 John H. Mathews, Russell W. Howell