- #1

lugita15

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If you ask a child to measure the length of a curve, a circle for instance, most likely the child will take a piece of string, lay it out so that it exactly corresponds to the curve, and mark the endpoints of the curve on the string. Then they will straighten the string out, and measure the length between the two marked points with a ruler.

How would we express this concept mathematically? Well, we can say that the length of a curve C is equal to the length of the line segment D which C can be continuously deformed into. But we must restrict the class of allowable deformations, so as to exclude deformations which stretch or shrink the curve. In other words, we can only allow "length-preserving deformations." I'm not sure what the technical name for these are, but since a transformation which leaves the metric of a space undisturbed is called an isometry, maybe they are called isometric deformations. The (admittedly nonrigorous) way I am envisioning it is that we split the curve into infinitesimal pieces. Then we rotate the infinitesimal pieces until they are all facing the same direction, and then line them up end to end, so that they form a line segment. I believe that the definition of a continuous length-preserving definition of a curve can be made precise using topology, but I don't know for sure because my knowledge of topology is limited to a course I took in Real Analysis. Regardless, let us call it the "topological definition" of arc length.

My question is, can it be proven that the usual calculus definition of arc length is equivalent to my topological definition for all (rectifiable, smooth, and well-behaved) curves? If so, how can it be proven?

Any help would be greatly appreciated.

Thank you in Advance.

P.S. I suppose that a similar topological definition can be applied to the area, surface area, volume, n-dimensional hypervolume, etc. In general take a "curved" region, apply a topological deformation which preserves the property you want to calculate, so that the problem reduces to calculating the property for an equivalent "flat" region.