Real Analysis/Arc Length
|←Total Variation||Real Analysis
Suppose we have a parametric curve in three dimensions, . 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, , and call this partition P. Take the sum of the distances , to get , and call this sum L(P). Now, take the supremum of the lengths, 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).
A continuous curve in three dimensions is rectifiable if and only if all of its component functions are functions of bounded variation.
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 .