Differential Geometry/Arc Length

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The length of a vector function on an interval is defined as

If this number is finite, then this function is rectifiable.

For continuously differentiable vector functions, the arc length of that vector function on the interval would be equal to .

Proof

Consider a partition , and call it . Let be the partition with an additional point, and let , and let be the arc length of the segments by joining the of the vector function. By the mean value theorem, there exists in the nth partition a number such that

Hence,

which is equal to

The amount

shall be denoted . Because of the triangle inequality,

Each component is at least once continuously differentiable. There exists thus for any , there is a such that

when .

Therefore, if then , so that

which approaches 0 when n approaches infinity.

Thus, the amount

approaches the integral since the right term approaches 0.

If there is another parametric representation from , and one obtains another arc length, then

indicating that it is the same for any parametric representation.

The function where is a constant is called the arc length parameter of the curve. Its derivative turns out to be .