# 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.