This particular differential form
is called the **arclength element**
and is traditionally written d*s*
(although that notation is misleading for reasons that I will return to later).
A simpler way to think of d*s*,
which works in *any* number of dimensions,
is as |d*P*|, the magnitude of the differential of the position.
Remember that d*P* is a vector when *P* is a point,
so it has a magnitude;
in fact, d*P* is the same as d**r**,
so you can also think of d*s* as
|d**r**|, the magnitude of the differential of **r**.
In 2 dimensions, where *P* = (*x*, *y*)
and **r** = ⟨*x*, *y*⟩,
d**r** = d*P* = ⟨d*x*, d*y*⟩,
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 d*t*.
The absolute value |d*t*| will naturally appear in the integrand;
but if you set up the integral so that *t* is increasing,
then d*t* is positive, so |d*t*| = d*t*.
Then you can write |d*P*| as |**v**| d*t*,
where **v** = d*P*/d*t* = d**r**/d*t*
is the velocity,
as given in the textbook.
More explicitly,
this is
√((d*x*/d*t*)^{2} +
(d*y*/d*t*)^{2}) d*t*
(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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