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 by the way of the Cauchy-Riemann differential equations. 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 $u_y = -v_x$, the Newton field components $N_x$ and $N_y$ are given by:

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

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, the curl of $N$ is determined to be expressed by:

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

The curl of $N$ vanishes at points where $F$ or $F'$ vanishes; that is, the curl of F vanishes where it is

• purely real on the real axis or
• purely imaginary on the imaginary axis