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Thread: Question about line integral

  1. #1
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    Question about line integral

    let $\displaystyle \kappa$ denote the curvature of a closed spacecurve $\displaystyle \alpha$. Find $\displaystyle \oint_{\alpha} \frac{1}{\kappa} \ d\kappa$

    Looking at this, I would imagine that this would either be $\displaystyle 0$ since $\displaystyle \alpha$ is closed or $\displaystyle T$, the tangent Vector, but I'm not to sure on how to achieve the desired result.

    My other idea would have been to write it down as follows:

    $\displaystyle \oint_{\alpha} \frac{\|\alpha'\|^3}{\|\alpha' \times \alpha''\|} \cdot \|\alpha'\| \ dt$

    But these attempts don't seem to lead to anything.
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  2. #2
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    $\displaystyle \oint_{\alpha} \frac{1}{\kappa} \ d\kappa = \oint_{\alpha} d \ln | \kappa | = 0$
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