TITLE: Physical or geometric meaning of complex derivative
QUESTION [8 upvotes]: The derivative of a real-valued function at a point is the slope of the function at that point.
Similarly, what is the physical or geometric meaning of the derivative of a complex-valued function at a point?

REPLY [3 votes]: One clear physical meaning is given by harmonic function theory. In particular, if $f=u+iv$ is complex-differentiable on an open set then we have that both the component functions satisfy Laplace's Equation. In other words, they can be thought of as possible potential energy functions for a force field which is conservative on the domain, or, as voltage functions for an electrostatic field on the domain. However, there is more, the relation between $u$ and $v$ is that they define level-curves which intersect at a right angle. That is, $u$ and $v$ are orthogonal functions. Hence, if we think of one as the potential energy then the other will play the role of the force. The tangent lines to the force field are perpendicular to the equipotential lines of the potential. All of this is wrapped up in the simple assumption that $f'(z)$ exist and is nonzero.