[ \mathbfV_f(x,y) = \big( u(x,y),, -v(x,y) \big). ]
Let [ f(z) = u(x,y) + i,v(x,y) ] be an analytic function on a domain (D \subset \mathbbC).
So (\mathbfV_f) is (solenoidal) — it has a stream function. 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).
[ u_x = v_y, \quad u_y = -v_x. ]
The Pólya field (\mathbfV_f) is exactly (w) — so it is a (gradient of a harmonic function, also curl-free and divergence-free locally).
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: [ \mathbfV_f(x,y) = \big( u(x,y),, -v(x,y) \big)
Let (\phi = u) (potential). Then
