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.


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