Definition:Absolute Value of Complex Differential

Let $I$ be a real interval.

Let $\gamma : I \to \C$ be a smooth path in $\C$.

Let $f : \C \to \C$ be a complex function.

Let the contour integral of $f$ along $\gamma$ exist.

In other words:

$\ds \int_\gamma \map f z \rd z$

or:

$\ds \int_I \map f {\map \gamma t} \map {\gamma'} t \rd t$

exists.

Then:

$\cmod {\rd z} = \cmod {\map {\gamma'} t} \rd t$