Is the following definite integral necessarily zero:
$$\oint_0^{s'} \oint_0^s \vec{ds} . \vec {ds'}$$
where $s$ and $s'$ are arc lengths of closed curves.
If yes, then for what reason?
2026-03-27 10:09:10.1774606150
Is the following definite integral necessarily zero?
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Possible interpretation of your question. If $\gamma$ is a closed curve with arclength parameter $s$,then the line integral of the scalar function $1$ $$\oint_\gamma\,ds$$ is the length of the curve. BTW, $\gamma$ being closed is irrelevant. Integrating this constant again along another curve...