142 Functions of Several Variables

where the correspondence between the X and Q-coordinates is not invert-
ible: the polar angle 6 is not defined when x = y — 0. We call such points
special and allow their existence as long as there are not too many of them;
at this point, we rely on intuition to decide how many is "many" and will
not go into the details.

EXERCISE 3.1.25.B Let 91 = r, 92 = 6, 93 — if be the usual spherical
coordinates so that Xi = rcosO siny, x-i = rsin# simp, X3 = rcosip.
Express qi,q2,q3 as functions of Xk, k = 1,2,3. Find all special points.

A coordinate curve in system Q is obtained by fixing two of the Q
coordinates, and using the remaining one as a parameter. For every non-
special point P with Q coordinates (cj, 02,03), there are three coordinate
curves passing through P. For example, we can fix 92,93 and vary qi.
The corresponding coordinate curve C\ is given in cartesian coordinates
by xk = x(<7i>c2,C3), k = 1,2,3. The coordinate curves Ci and C3 are
obtained in a similar way. The coordinate system Q is called orthogonal
if the coordinate curves are smooth and are mutually orthogonal at every
non-special point. The unit tangent vectors qk, k = 1,2,3 to these curves
at P define the local orthonormal basis at P; we choose the orientation and
the ordering of the curves so that the basis is right-handed: q-^ x q 2 — q 3.
Figure 3.1.1 shows the coordinate curves and unit tangent vectors for the
polar coordinates x = r cos 0,y = r sin 6.

0 = §2 y r = q j

2.

o

l

Fig. 3.1.1 Polar Coordinates

In the Q coordinates, the coordinate curves are also represented as in-

tersections of pairs of surfaces: {(91,92,93) : 9n = cn,qm = cm}, where

m, n = 1,2,3, m < n, and cm, cn are real numbers. If we assume that, at

time t = 0, each curve passes through the point P, then, in the Q coordi-

nates,

• The 91 coordinate curve with 92 = 02, 93 = 03, is the collection of points
  (ci +£,c 2 ,c 3 ), teR;


• The 92 coordinate curve with qi= c\, 93 = 03, is the collection of points