Mathematics of Physics and Engineering

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

1854. 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