NewtonFlowCurl

The curl of the Newton flow $N$ of the function $F(z) = F(x + iy) = u(x, y) + iv(x, y)$ can be determined via the Cauchy-Riemann differential equations. We compare the curl of $N$ to electrostatics and magnetostatics. Let $F'(z) = u_x + iv_x$, where the subscript $x$ denotes the partial derivative with respect to $x$. Using Cauchy-Riemann equations $u_x = v_y$ and $u_y = -v_x$, we find the Newton field components $N_x$ and $N_y$:

$$N_x = -\frac{F_x}{2D}, \quad N_y = -\frac{F_y}{2D}$$

where $D = u_x^2 + v_x^2$. The curl of $N$ is given by:

$$\text{curl } N = N_{y,x} - N_{x,y}$$.

Using the Cauchy-Riemann and Laplace equations, we arrive at the following expression for the curl of $N$:

$$ \text{curl } N = \frac{2}{D^2} \left[ (2u u_x v_x + v v_x^2 - u_x^3) u_{xx} - (2v u_x v_x + u v_x^2 - u_x^2 v_x) v_{xx} \right] $$

The curl of $N$ vanishes at points where $F$ or $F'$ vanishes. If $F$ is real on the real axis or purely imaginary on the imaginary axis, the curl vanishes there.