For example,$ \begin{align} \vec{f} &= \begin{bmatrix} u(x,y) \\ v(x,y) \\ \end{bmatrix} \end{align}$ could be recongnized as $f(z)=u(x,y)+iv(x,y),z \in \mathbb{C}$,In vector calculus:its derivative is$ \begin{align} D\vec{f} &= \begin{bmatrix} u_{x}& u_{y} \\ v_{x}& v_{y} \\ \end{bmatrix} \end{align}$,but in complex analysis，its derivative is:$f^{'}(z)=u_{x}+iv_{x}$.Both are derived from $ \lim_{z \to z_{0}} \frac{f(z)-f(z_{0})}{z-z_{0}}$ .why does it become so different?Do they have some relations?
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Cauchy-Riemann... – Charles Hudgins Aug 03 '23 at 15:03
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Scalar field derivative. Real vs Complex might also be helpful. – Mark S. Aug 03 '23 at 20:23
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@CharlesHudgins I'd prefer to say that Cauchy-Riemann is more like a consequence of definitions of the complex derivative, not the reason they become different. – Mark S. Aug 03 '23 at 20:25