Notes
Arc Length Integrals and
the line element ds
(and an introduction to the vector
differential d~s )
sec 13.3
19 February 2014
Goals.
Notes
We will:
I
Define the line element and the vector differential d~s .
I
Use the line element to define an integral that measures
the length of a curve.
I
Define and compute the arc length function.
Review: ~r (t), ~r 0(t), ~r 00(t)
Find ~r 0 (t) and ~r 00 (t) on a parabola y = x 2 parameterized by:
~r (t) = tı̂ + t 2 ̂
At what time(s) t is ~r 0 (t) orthogonal to ~r (t)?
Notes
Warm-up: Acceleration on the Parabola
Notes
For how many values of t will acceleration be orthogonal to
velocity on the parabola parametrized by ~r (t) = tı̂ + t 2 ̂ ?
1. Acceleration and velocity will be orthogonal at exactly
one time t.
2. Acceleration and velocity will be orthogonal for more than
one, but finitely many, times t.
3. Acceleration and velocity will always be orthogonal (all
times t).
4. Acceleration and velocity will never be orthogonal (no
times t).
5. I don’t understand the question.
Exercise: Try this at home.
Notes
Here is an example of a parameterization for a circle for which
~r (t) and ~r 0 (t) are orthogonal (always true on a circle!), but
~r 0 (t) and ~r 00 (t) are not.
~r (t) = cos(t 2 )ı̂ + sin(t 2 )̂ .
I
Show that ~r (t) is constant (in fact, k~r (t)k = 1).
I
Show that velocity ~r 0 (t) is orthogonal to position ~r (t) for
all times t (hint: use the dot product).
I
Show that speed k~r 0 (t)k is not constant — that is, speed
depends on t.
I
Show that acceleration ~r 00 (t) is orthogonal to velocity
~r 0 (t) for only one value of t.
Warm-up: Integrals
Notes
What
Z b do you think of when you think of a definite integral
f (t) dt?
a
1. The (signed) area under a curve.
2. The limit of Riemann sums lim
n→∞
n
X
f (ti )∆ti
i=0
3. The limit of a sum of rectangles with increasingly smaller
bases.
4. A process of “chopping and adding”, where the pieces are
chopped progressively smaller and smaller.
Lengths of Curves
Notes
To compute the length of a curve, follow a process similar to
computing definite integrals via Riemann sums:
I
Chop the curve into small pieces.
I
Approximate each piece by the line segment joining the
endpoints.
I
Add up the lengths of the line segments to get an
approximation of the length of the curve.
I
Repeat this process using smaller and smaller pieces.
The length of the curve is the limit (if it exists) of these
approximations.
Lengths of Curves & the Line Element ds
Notes
The length of a curve from P to Q is denoted by the integral :
Z
Q
s=
ds
P
The differential ds is called the line element (or element of arc
length). It can be imagined to be an infinitesimally small distance
Z Q
along the curve. In this case, the integral
ds reads: “chop up
P
the curve between P and Q into very, very small parts, each of
length ds, then add up the lengths of each parts”.
Disclaimer: This is a very informal way of looking at integration.
It is not mathematically correct — the mathematical definition
involves limits of Riemann sums — but it gives an intuitively
accessible way of understanding the informational content of the
integral.
Computing ds & Lengths of Curves
Suppose a curve is parameterized by ~r (t), where ~r 0 (t) is
continuous and ~r 0 (t) 6= ~0 for all t. Then:
ds = k~r 0 (t)kdt
(You are responsible for knowing this equation, even though it
is not explicitly in the text.)
−→
−→
If ~r (a) = OP and ~r (b) = OQ, then:
Z
Q
s=
Z
ds =
P
a
b
k~r 0 (t)kdt
Notes
Example: Computing Length of a Curve
Notes
Find the length of the curve ~r (t) = (t 2 + 1) ı̂ + t 3 ̂ between
the points P = (1, 0) and Q = (5, 8).
Interpreting ds: Distance = Speed × Time
Notes
Suppose ~r (t) gives the position at time t of a particle
traveling along a curve. Then k~r 0 (t)k is the speed of the
particle along the curve.
How far does the particle travel along the curve during the
time increment dt?
ds = infinitesimal distance
= speed × infinitesimal time increment
= k~r 0 (t)kdt
Interpreting ds: Distance = Magnitude of
Displacement
A vector differential d~s can be imagined as an “infinitesimal
displacement vector”.
If ~r = x ı̂ + y ̂ + z k̂ , its vector differential is:
d~s = dxı̂ + dy ̂ + dz k̂
The line element ds is the magnitude of the vector differential
d~s :
p
ds = kd~s k = dx 2 + dy 2 + dz 2
This is an infinitesimal version of the Pythagorean theorem!
Notes
Computing d~s and ds Along Curves
Notes
If ~r (t) = x(t) ı̂ + y (t) ̂ + z(t) k̂ is differentiable, then:
d~s = x 0 (t)dt ı̂ + y 0 (t)dt ̂ + z 0 (t)dt k̂
h
i
= x 0 (t) ı̂ + y 0 (t) ̂ + z 0 (t) k̂ dt
= ~r 0 (t) dt
So (as before):
ds = kd~s k = k~r 0 (t)kdt
Notes on d~s and ds
I
We will be seeing more of d~s and ds in chapters 16 and
17.
I
We have worked with differentials before. Recall the
method of u-substitution in integration: if u = f (t), then
du = f 0 (t)dt = df
dt.
dt
I
An excellent introduction to d~s (also called d~r ) and ds
can be found on the Bridge Book wiki. (← this is a link.)
Notes
Notes