Real Analysis/Arc Length

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Real Analysis Arc Length

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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: