Smooth Curve

Topics: Calculus - Curve - Vector-Valued Function

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(definition)

We say that the curve traced out by the vector-valued function $r(t)=⟨r_1(t),r_2(t),\dots ,r_n(t)⟩$ on an interval $l$ is smooth if $r^{\prime }(t)$ (its derivative) is continuous on $l$ and $r^{\prime }(t)\ne 0$, except possibly at any endpoints of $l$.

Not All Zeroes

Notice that this says that the curve is smooth provided $r_1^{\prime }(t),r_2^{\prime }(t),\dots ,r_n^{\prime }(t)$ are all continuous on $l$, but they are not all zero at the same point in $l$.