Arclength

When finding the length of a curve by integration, you are ultimately integrating an expression such as √(dx² + dy²). This particular expression applies in 2 dimensions; in words, it is the principal square root of the sum of the square of the differential of x and the square of the differential of y. An expression like this, containing differentials, is called a differential form; the textbook mentions differential forms briefly in Section 15.3, but they are really all over the place in this course, sometimes hidden just under the surface, sometimes out in the open without being acknowledged. I'll be pointing them out throughout the course.

This particular differential form is called the arclength element and is traditionally written ds (although that notation is misleading for reasons that I will return to later). A simpler way to think of ds, which works in any number of dimensions, is as |dP|, the magnitude of the differential of the position. Remember that dP is a vector when P is a point, so it has a magnitude; in fact, dP is the same as dr, so you can also think of ds as |dr|, the magnitude of the differential of r. In 2 dimensions, where P = (x, y) and r = ⟨x, y⟩, dr = dP = ⟨dx, dy⟩, whose magnitude is the arclength element that I've been talking about.

When working with a parametrized curve, every variable (x, y, and z if it exists, whether individually or combined into P or r) is given as a function of some parameter t. By differentiating these, their differentials come to be expressed using t and dt. The absolute value |dt| will naturally appear in the integrand; but if you set up the integral so that t is increasing, then dt is positive, so |dt| = dt. Then you can write |dP| as |v| dt, where v = dP/dt = dr/dt is the velocity, as given in the textbook. More explicitly, this is √((dx/dt)² + (dy/dt)²) dt (in 2 dimensions), which is also given in the textbook. But while you might integrate this in practice to perform a specific calculation, you are most fundamentally integrating a differential form in which t does not appear. This is why the result ultimately does not depend on how you parametrize the curve. (Later on, we'll discuss what it means, in general, to integrate a differential form along a curve, including why and to what extent this is independent of the parametrization.)

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This web page was written in 2016 by Toby Bartels, last edited on 2016 April 5. Toby reserves no legal rights to it.

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