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Let $\Gamma$ be a smooth curve in $\mathbb{C}^d$. Since $\mathbb{C}^d$ can be seen as $\mathbb{R}^{2d}$, one can define the line integral of functions $f:\Gamma\to \mathbb{C}$ using for instance arclength parametrization.

However, when $d=1$, one can use the usual complex line integral of complex analysis instead. Is there a natural extension of this definition when $d \geq 2$ ?

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  • $\begingroup$ The most natural extension would have f mapping to $C^d$ rather than $C$. $\endgroup$ – Michael Renardy Aug 27 at 12:38
  • $\begingroup$ Yes, sure. But i'm dealing with functions defined on curves and the arclength definition of integration does not seem adequate. $\endgroup$ – Chr Aug 27 at 13:25
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    $\begingroup$ The line integral of complex analysis is the integral of a 1-form, so we can immediately generalize to integration of 1-forms along curves, which is defined in any manifold. The line integral of complex analysis is not an arc length integral of a function. $\endgroup$ – Ben McKay Aug 27 at 15:06
  • $\begingroup$ Unfortunatlely I'm not familiar with 1-form. Is the function f, defined on $\Gamma$, a 1-form ? $\endgroup$ – Chr Aug 27 at 16:55
  • $\begingroup$ @Chr: no, a function on a curve is not a 1-form. You can read Spivak, Calculus on Manifolds, for a nice introduction to differential forms. But your question does not appear to be a question of research; perhaps this is the wrong website for your question. $\endgroup$ – Ben McKay Aug 27 at 20:58

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