Polya Vector Field -

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

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

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). The Pólya field (\mathbfV_f) is exactly (w) —

[ \nabla u = (u_x, u_y) = (v_y, -v_x). ] From the first component: (\partial \psi / \partial y = u)

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: