Definition:Smooth Path/Closed/Real Cartesian Space

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Let $\R^n$ be a real cartesian space of $n$ dimensions.

Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval.

Let $\rho: \left[{a \,.\,.\, b}\right] \to \R^n$ be a smooth path in $\R^n$.

$\rho$ is a closed smooth path if and only if $\rho$ is a closed path.

That is, if and only if $\rho \left({a}\right) = \rho \left({b}\right)$.