TITLE: If f(x)dx is a rectangle with height f(x) and width dx, what is f(z)dz in complex analysis
QUESTION [6 upvotes]: I am trying to intuitively understand the multiplication $f(z)dz$ in complex analysis. For instance, $f(x)dx$, we are all aware, is a rectangle with height $f(x)$ and width $dx$ so its multiplication is an area of this rectangle. Is there a similar way to visualize $f(z)dz$ also?

REPLY [2 votes]: You can interpret the function $f$ as a "complex force field" defined in the region $\Omega$; the function value $f(z)$ is then the force vector attached to the point $z$. On the other hand $dz$ represents an infinitesimal movement from $z$ to $z+dz$. The product $f(z)\>dz$ is then the "complex work" done when a mass point moves under the influence of $f$ from $z$ to $z+dz$.
Note that the notion of line integral picks up this intuition:
$$\int_\gamma f(z)\>dz=\lim_\ldots \sum_{k=1}^N f(z_k)(z_k-z_{k-1})\ ,$$
where
$$\gamma:\quad t\mapsto z(t)\qquad(a\leq t\leq b)$$
is a curve in $\Omega$, and $z_k=z(t_k)$ for a partition $a=t_0<t_1<\ldots<t_N=b$.