[ \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). polya vector field
[ \nabla u = (u_x, u_y) = (v_y, -v_x). ] [ \mathbfV_f = (u,, -v)
We want (\mathbfV_f = (u, -v) = (\partial \psi / \partial y,; -\partial \psi / \partial x)). From the first component: (\partial \psi / \partial y = u). From the second: (-\partial \psi / \partial x = -v \Rightarrow \partial \psi / \partial x = v). ] We want (\mathbfV_f = (u, -v) =
The of (f) is defined as the vector field in the plane given by
Thus the Pólya field rotates the usual representation of (f) by reflecting across the real axis. Write (f(z) = u + i v). Then:
Thus (\nabla \psi = (v, u)). Check integrability: (\partial_x (v) = v_x = u_y) and (\partial_y (u) = u_y) — they match. So (\psi) exists (since domain simply connected). So: