Gauss's Theorem 151
EXERCISE 3.2.2.B (a) Verify all equalities in (3.2.2). (b) Verify that, in
polar coordinates x = r cos 6, y = r sin 6, we have
m(G) = \i A
Jd
r^2 d9.
dG
Hint: verify that, in polar coordinates, xdy — ydx = r^2 dd.
FOR EXAMPLE, let us compute the area under one arc of the cycloid,
the trajectory of a point on the rim of a wheel rolling without slippage
along a straight line. If a is the radius of the wheel, and the wheel is rolling
to the right so that, at time t = 0, the point on the rim is at the origin,
then the vector parametric equation of the cycloid is
r(t) = a{t - sin t)i + a(l - cos t) j. (3.2.3)
One revolution of the wheel corresponds to t = 2n. The region G under
one arc is bounded by the x-axis on the bottom and the arc on the top. By
(3.2.2), keeping in mind the orientation of the boundary, the area of the
region is m(G) = - §dG ydx — - fQ* Odx - a^2 /27r(l - cost)^2 dt = Sna'
On the arc, x'(t) = a(l — cost), y(t) = a(l — cost), and, of the three
possible formulas for the area, the one involving only ydx results in the
easiest expression to integrate.
EXERCISE 3.2.3. B (a) Verify that (3.2.3) is indeed a vector parametric
equation of the trajectory of the point on the rim of a rolling wheel. Hint:
consider 0 < t < ir. Denote by A the point on the rim, A', the current point of
contact between the wheel and the ground, and C, the center of the wheel. Then
r(t) = OA'+A'C+CA, \OA'\ = at, and the angle between CA' and CA is t; draw
the picture, (b) Draw the picture of the cycloid and verify the integration.
There is a hardware device, called planimeter, which implements the
formula tn(G) = (1/2) §dG xdy — ydx and measures the area of a planar fig-
ure by traversing the perimeter. The Swiss mathematician JACOB AMSLER
(1823-1912) is credited with the invention of a mechanical planimeter in
- Today, both mechanical and electronic planimeters are in use.
3.2.2 The Divergence Theorem of Gauss
Depending on the source, the following result is known as Gauss's
theorem, the Gauss-Ostrogradsky theorem, or the divergence theorem.
This theorem can be motivated by the following physical argument. Let v