Polya Vector Field 🔥 Extended

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

[ \nabla u = (u_x, u_y) = (v_y, -v_x). ] polya vector field

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