Real Analysis/Arc Length

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Real Analysis
Arc Length

Suppose we have a parametric curve in three dimensions, f(t)=(x_1(t),x_2(t),x_3(t)). Of course, it would be required that all three functions be continuous. This essentially defines a curve, since it is a continuous image of the real numbers onto the real 3-space.

Now, we can define the arc length of this curve over an interval. Say the interval is [a,b]. Now divide [a,b] into partitions, a=a_0<a_1<a_2<a_3<...<a_n=b, and call this partition P. Take the sum of the distances |f(a_n)-f(a_{n-1})|, to get \sum_{i=1}^n \sqrt{\sum_{j=1}^3 (x_j(a_i)-x_j(a_{i-1}))^2}, and call this sum L(P). Now, take the supremum of the lengths, \sup\{L(P)\in R|P is a partition\}. If this number is finite, we call it a rectifiable curve.

Now we establish a sufficient and necessary condition for a curve in 3-space to be rectifiable (note: this can easily be extended to an n-space through an analogous argument).

Theorem:
A continuous curve in three dimensions is rectifiable if and only if all of its component functions are functions of bounded variation.
Proof:

Theorem:
If a curve f(x) in 3-space is continuously differentiable in all 3 components, then it is rectifiable and the length from f(a) to f(b) is \int_a^b |f'(x)| dx.
Proof: