Definition of smooth and piecewise smooth curves

[gotjrgkr]

Hi!
I want to know the precise definition of smooth curve and piecewise smooth curve if the curve indicates a continuous function from an interval [a,b] into a set of real numbers.

 

[Klockan3]

A smooth curve is a curve for which all of its derivatives exists, which gives that all derivatives must be continuous and smooth. A piecewise smooth curve is a curve for which there exists a set of intervals where the curve is smooth on each of those intervals. As an example, a curve might not be smooth on [a,b] but it is still piecewise smooth if it is smooth on both [a,c] and (c,b].

 

[HallsofIvy]

It would be better to say "tangent vector" than "all of its derivatives". Also, we require that the tangent vector never be 0.

 

[gotjrgkr]

A smooth curve is a curve for which all of its derivatives exists, which gives that all derivatives must be continuous and smooth. A piecewise smooth curve is a curve for which there exists a set of intervals where the curve is smooth on each of those intervals. As an example, a curve might not be smooth on [a,b] but it is still piecewise smooth if it is smooth on both [a,c] and (c,b].

In the statement 'the curve is smooth on each of those intervals', can i interpret it as 'each restriction of the curve to a subinterval of the interval is smooth'??