MHB What Does the Integral Expression $\int_{\gamma} \rho(z) |dz|$ Represent?

[poissonspot]

I don't know if I've ever encountered a differential term with a modulus around it (or if I have, ignored it). Here's an example: $ \int\limits_{\gamma}{\rho(z)}{|{dz}|} $

If it was simply $ \int\limits_{\gamma}{|{dz}|} $ I imagine this is the length of the curve $\gamma $, but what might the above denote, if for instance $ \gamma $ is a rectifiable curve, $ \rho $ a measurable metric?

(for context look here: http://www.math.niu.edu/~fletcher/Fthesis.pdf pg 17)

Thanks

Edit: Also, does the idea $ \int\limits_{\gamma}{|{dz}|} $ being a line integral in the complex plane yielding the length of curve $ \gamma $ sound reasonable?

 

[Euge]

I don't know if I've ever encountered a differential term with a modulus around it (or if I have, ignored it). Here's an example: $ \int\limits_{\gamma}{\rho(z)}{|{dz}|} $

If it was simply $ \int\limits_{\gamma}{|{dz}|} $ I imagine this is the length of the curve $\gamma $, but what might the above denote, if for instance $ \gamma $ is a rectifiable curve, $ \rho $ a measurable metric?

(for context look here: http://www.math.niu.edu/~fletcher/Fthesis.pdf pg 17)

Thanks

Edit: Also, does the idea $ \int\limits_{\gamma}{|{dz}|} $ being a line integral in the complex plane yielding the length of curve $ \gamma $ sound reasonable?

A line integral involving $|dz|$ indicates integration with respect to arclength. Suppose $C$ is a differentiable curve parameterized by $\gamma : [0,1] \to \Bbb C$. Let $f$ be an integrable complex function. Then

$\displaystyle \int_\gamma f(z)\, dz := \int_0^1 f(\gamma(t)) \gamma'(t)\, dt$

and

$\displaystyle \int_C f(z) \, |dz| := \int_0^1 f(\gamma(t)) |\gamma'(t)| \, dt$.

In particular, if $C$ is a rectifiable curve in $\Bbb C$, then $\int_C |dz|$ is the arclength of $C$, as you expected.