The arclength parameter s
Math 131 Multivariate Calculus
D Joyce, Spring 2014
Remark on notation. Throughout this discus-
sion we’ll be considering a moving point, that is,
a path x: [a, b]→Rn. We’ll take tto be the in-
dependent variable, which we’ll call time, and we’ll
use the prime notation to always mean the deriva-
tive with respect to t, so, for instance, x0=dx
dt .
Whenever we need the derivative with respect to
another variable, such as s, we’ll stick to Leibniz’
notation d
ds.
Arclength. Consider a path x: [a, b]→Rn. We
want to define the length Lof this path, also called
its arclength. It will be defined as the integral of
its speed. In order for the speed kx0(t)kto have an
integral, we’ll assume that xis C1(which means it
is differentiable and its derivative is continuous).
An argument by analogy for this definition is as
follows. In the one-dimensional case, the derivative
of position x(t) is velocity v(t) = x0(t), and, by
the fundamental theorem of calculus, the integral
Zb
a
v(t)dt of velocity is the difference in position,
Zb
a
v(t)dt =x(b)−x(a).
When the velocity is positive, the integral is the
distance travelled over the time interval [a, b], that
is, the length of the path along the x-axis. But
when the velocity is negative, the integral gives the
negation of the distance travelled. By replacing ve-
locity by its absolute value, that is by speed |x0(t)|,
we get the total distance travelled. Furthermore,
this integral of speed,
Zb
a|x0(t)|dt
always works to give the total distance Ltravelled
along the x-axis even when the object alternates
the direction it moves along the x-axis.
By analogy, you would guess that even when the
direction of travel is not restricted to the x-axis,
the distance Ltravelled along the curve should still
be the integral of the speed, and that works out to
be right.
A better argument than by analogy uses infinites-
imals. Imagine the time interval [a, b] to be divided
into infinitesimally short intervals [t, t +dt]. The dt
is called the differential of time t. During this in-
finitesimally short interval, the object moves from
position x(t) to x(t+dt). Let’s assume we’re in
dimension 2, so that x(t)=(x(t), y(t)). Then the
x-coodinate changes from x(t) to x(t+dt), a differ-
ence we can denote dx =x(t+dt)−x(t), called the
differential of x. Likewise, in the y-coordinate, we
get dy =y(t+dt)−y(t). Those are two sides of an
infinitely small right triangle with legs dx and dy,
called Leibniz’ differential triangle.
dx
ds dy
(x(t),y(t))
(x(t+dt),y(t+dt))
Let ds denote the hypotenuse of this infinitesimal
triangle, so that
ds2=dx2+dy2
and
ds =pdx2+dy2
Now, during the interval [t, t +dt] the object
moves a distance ds. If we sum all these infinitesi-
mal distances, we should get the total distance trav-
elled, that is, the length of the path is the integral
L=Zb
a
ds =Zb
a
ds
dt dt
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