My differential geometry text defined a curve as a function from an open interval of $\mathbb R$ onto $\mathbb R^3$, but it defines arc length as the integral in terms of the velocity along a closed interval. Why "closed"? Would the velocity not be well-defined if we used an open interval?
2026-04-08 16:14:23.1775664863
Why is arc length defined on a closed interval?
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