Complex Trkalian fields and solutions to Euler's equations for the ideal fluid.

We consider solutions to the complex Trkalian equation, ~ ∇×~c = ~c, where ~c is a 3 component vector function with each component in the complex field, and may be expressed in the form ~c = e ~ ∇F, with g real and F complex. We find, there are precisely two classes of solutions; one where g is a Cartesian variable and one where g is the spherical radial coordinate. We consider these flows to be the simplest of all exact 3-d solutions to the Euler’s equation for the ideal incompressible fluid. Pictures are presented. The novel approach we use in solving for these classes of solutions to these 3-dimensional vector pdes involves differential geometric techniques: one may employ the method to generate solutions to other classes of vector pdes.