1550251515-Classical_Complex_Analysis__Gonzalez_

·Differentiation 377

Fig. 6.19

As in case I, the maximum and minimum values of the ratio d,,P /dB are

attained when tan2B = 2F/(E - G) if E -:f. G, or when cos2B = 0 if

E = G, F -:f. 0. The resulting values of(} give the principal directions, i.e.,

those corresponding to the endpoints of the diameter of the Kasner circle

that contains the origin. The principal directions are indeterminate when

fz = 0. In this case E = G and F = 0, as well as in the case fz = 0,

which we have ruled out.

By a computation similar to that in the case J > 0 we find, assuming

that fz -:f. 0,

(

d,,P) -2lfzl

max dB = lfzl + lfzl

. ( d,,P) -2lfzl
mm d(} = lfzl -lfzl

If fz = 0, fz -:f. 0, we get d,,P/dB = -2 (a constant).

6.18 NONCONFORMAL MAPPINGS. TISSOT'S

THEOREM FOR THE PLANE

In 1878, Tissot [118] proved that at an ordinary point of a surface the prin-

cipal directions form the only pair of orthogonal directions that is mapped

into an orthogonal pair by a nonconformal (yet differentiable) transforma-

tion of the surface into another. More generally, if the Jacobian of the

transformation does not vanish at a point, then for every direction there

is an associated direction which is such that the corresponding oriented