Polya vector field

Given a complex function $f:\mathbb{C}\to \mathbb{C}$, the associated Polya vector field is

given in local coordinates by

Among other things it is interesting to interpret complex integration

That is, the integral of a complex function $f$ along a path $\gamma$ is a complex number such that:

• The real part $W_{\gamma }[\overline{f}]$ is the total work done by the Polya vector field along the curve.
• The imaginary part $F_{\gamma }[\overline{f}]$ is the total flux of the Polya vector field along $\gamma$.

The Polya vector field has null divergence an null rotational when the function $f$ is holomorphic, and this let us prove Cauchy's theorem.

Source: this video.