Polya vector field

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

$$ V: \mathbb{C} \to T \mathbb{C} $$

given in local coordinates by

$$ z\mapsto (z,\overline{f(z)}) $$

Among other things it is interesting to interpret complex integration

$$ \int_{\gamma} f(z) d z=W_{\gamma}[\bar{f}]+i F_{\gamma}[\bar{f}] $$

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

The real part $W_{\gamma}[\bar{f}]$ is the total work done by the Polya vector field along the curve.

The imaginary part $F_{\gamma}[\bar{f}]$ 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.

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Author of the notes: Antonio J. Pan-Collantes

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