Vector Field — Polya

[ u_x = v_y, \quad u_y = -v_x. ]

Let (\phi = u) (potential). Then

The field ((v, u)) appears as the Pólya field of (-i f(z)). Connection to harmonic functions Since (f) is analytic, (u) and (v) are harmonic and satisfy the Cauchy–Riemann equations: polya vector field

[ \nabla u = (u_x, u_y) = (v_y, -v_x). ] [ u_x = v_y, \quad u_y = -v_x

The of (f) is defined as the vector field in the plane given by Connection to harmonic functions Since (f) is analytic,

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

Indeed, the stream function (\psi) such that (\mathbfV_f = ( \psi_y, -\psi_x )) can be taken as (\psi = -v). Check: [ \psi_y = -v_y = -(-u_x) = u_x? \text Wait carefully. ] Better: Let (\psi = -v). Then (\nabla^\perp \psi = (\psi_y, -\psi_x) = (-v_y, v_x)). But by Cauchy–Riemann, (v_x = u_y), (v_y = -u_x), so ((-v_y, v_x) = (u_x, u_y)) — that’s (\nabla u), not (\mathbfV_f). So that’s not correct. Let's derive cleanly: