Polya Vector Field -

The of (f) is defined as the vector field in the plane given by

Let [ f(z) = u(x,y) + i,v(x,y) ] be an analytic function on a domain (D \subset \mathbbC). polya vector field

[ \nabla u = (u_x, u_y) = (v_y, -v_x). ] The of (f) is defined as the vector

[ \mathbfV_f = (u,, -v). ]

The Pólya field (\mathbfV_f) is exactly (w) — so it is a (gradient of a harmonic function, also curl-free and divergence-free locally). y) + i

[ \mathbfV_f(x,y) = \big( u(x,y),, -v(x,y) \big). ]