Guys, I need your kind assistance. I am studying arcs length. Suppose a vectorial function with domain [a, b] (interval in R) and range in RxR. This range is a curve in the RxR plane.(adsbygoogle = window.adsbygoogle || []).push({});

Take a partition P of [a, b]: a= t0, t1, t2,...., tn = b.

We have a straight line which goes from F(t0) to F(t1), another straight line that goes from F(t1) to F(t2), etcetera. Thus we build a "polygonal". By definition, the supremum of set A = { lengths of the polygonals corresponding to any partition} is the arc length.

But all books (in their drawings) assume that, for example, F(t2) can not be located in the curve that was "cut off" by the straight line which goes from F(t0) to F(t1). In other words: they assume that F(t2) can not go backwards and settle between F(t0) and F(t1).

I simply do not understand where is their basis for such assumption.

In two books they say that both component functions of F (lets call them f and g) have continuous derivatives and that there is not "t" in [a,b] such that (f'(t), g'(t)) = (0,0). I know that this implies that, for example, if we take F'(t1), at least one of the original component functions (f or g) is monotone in a interval that contains t1, but I fail to see how this conect with my question of the previous paragraph.

Can you help me? Thanks.

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# Arc length or a smooth curve

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