Complex Derivative Field

There is a direct analogy relating the real and imaginary components of the complex derivative to divergence and curl (respectively) of a 2D vector field.

Let $f$ be a complex-differentiable complex function and $\mathbf{F}$ be the corresponding 2D vector field:

Complex Number Surface	Vector Field
$z=x+iy$	$\mathbf{z}=\begin{bmatrix} x & y \end{bmatrix}$
$f=u+iv$	$\mathbf{F}=\begin{bmatrix} u & v \end{bmatrix}$
$\Re\frac{df}{dz}=\frac{du}{dx}=\frac{dv}{dy}$	$\nabla\cdot\mathbf{F}=\frac{du}{dx}+\frac{dv}{dy}$
$\Im\frac{df}{dz}=\frac{dv}{dx}=-\frac{du}{dy}$	$\nabla\times\mathbf{F}=\frac{dv}{dx}-\frac{du}{dy}$ ^(note)

So

$\Re f^\prime(z)=\frac{1}{2}\nabla\cdot\mathbf{F}(\mathbf{z})$
$\Im f^\prime(z)=\frac{1}{2}\nabla\times\mathbf{F}(\mathbf{z})$

(note) Here curl is a scalar, referring to the component orthogonal to the x-y plane, which is the only non-zero component in the 2D case.

complex derivative, complex differentiation, curl, divergence, vector field, vector operator

Addition in Other Domains

Itō’s Insight

Leave a Reply Cancel reply

Cheap NFL Jerseys Cheap NFL Jerseys goedkope air max online Basket Air Jordan 11 Christian Louboutin Outlet Cheap Jerseys Wholesale Jerseys Michael Kors Outlet Online Ray Ban Sunglasses Jerseys Wholesale Coach Outlet Store michael jordan air jordan 11 Coach Outlet Cheap NFL Jerseys Kate Spade Outlet Sale air max 90 pas cher junior Wholesale Jerseys Jerseys Cheap nba jerseys vegas Jerseys Wholesale China