574 Partial derivatives
continuous. When we proved Theorem 11.23, we knew about the mean-value
theorem and had enough hypotheses to enable us to apply it.
22 This problem provides preliminary information about a way in which
surfaces can be determined and studied. The circle in the xy plane having its
center at the origin and radius a has the simple rectangular equation x2 + y2 =
a2. We have seen, however, that it is often convenient to use the parametric
equations(1) x = a cos it, y=asinu
and to recognize that, when r is the sector running from the origin to P(x,y),
we have(2) r = (a cos u)1 + (a sin u)j
and the tip of r runs once in the positive direction around the circle as u increases
over the interval 0 < it 5 27r. This remark, in which u has been used where we
ordinarily use 0 or 0, is designed to lead us gently to the idea that if ff, f2, f3 are
suitable functions of two parameters u and v, then the vector r defined by
(3) r = f,(u,v)i +f2(u,v)j + f3(u,v)k
will be the vector running from the origin to the point P(x,y,z) on a surface S
for which x = fl(u,a), y = f2(u,a), z = f3(u,a). For example, when we use
spherical coordinates r, 0, 6 as in Section 10.1, the equation of the sphere S
having its center at the origin and radius a has the spherical equation r = a.
The formulas of Problem 3 of Section 10.1 then show that the point P on S
having spherical coordinates r, 0, 0 has rectangular coordinates x, y, z, where
(4) x = a cos 0 sin 0, y = a sin 4> sin 0, z = a cos B.
Except that the parameters are called 0 and 0 instead of u and v, we obtain a
special case of (3) by setting
(5) r = a(cos 0 sin 01 + sin 0 sin Oj + cos 8k),
and (5) is a two-parameter parametric equation of the sphere S a part of which
is shown in Figure 11.292. When 0 < Co < 7r
and 0 = 90, (5) is the parametric equation
of a circle Cp(O0), a geographic parallel, on
S which the tip of r traces as ¢ increases from
-7r to 7r. When -7r < ¢1o 7r and 0 = 4>0i
a=90y [a /i \ .-A (5) is the parametric equation of the semi-Figure 11.292circle CM(¢0), a geographic meridian, which
the tip of r traces as B increases from 0 to
7r. We make only a few calculations to illus-
trate the utility of these things. When
0 < 0 < 7r, the foward tangent tl to Cr(00)
at the point PO for which 0 = 4>o and 0 = Oo
is obtained by putting 0 = Bo in (5), differ-
entiating with respect to ¢, and putting 95 = 4)0 in the result. Thus,
(6) tj =a(- sin 4o sin 00i + cos 4)0 sin O j).
Similarly, when -7 < 4o < 7r, the forward tangent t2 to Cat(¢o)at the point