354 Chapter6andv2 x +v2 y-/3 7 ab 1
= -U-x Vx+....;_u_y_V_y = u~ + u~ = r 2 J 2 = iJT
( d) Also, prove that
1 u2 + u2 + v2 + v2
k+-= x y x y
k IJI
k = max IJMz)I
min IJ~(z}Itan2i9 = _l:_p_
7 - Oi( l~I)
112
:::; lf~(z)I:::; (klJl)^112- (a) Let f = u +iv E '.D(A) and let 7: z = z(t), Oi :5 t :5 f3 be a regular
arc with 7 C A. In addition, suppose that u and v have continuous
partial derivatives up to the second order at a point z E 7, and
that z^11 (t) exists for that value oft which maps into z. Show that
fe'(z) = [fe(z)]~ = fzz + 2fzze_^2 ;^6 + fzze-^4 i^9 - 2iI<fze-aiB
where I< denotes the curvature of"/,* at z. Deduce that if 7* is a
straight line, then
fe'(z) = fzz + 2fzze-^2 i^6 + fzze-^4 iB = Uz + fze-^2 i^9 )<2>
'I'his is the so-called second rectilinear directional derivative.
(b) Under the same assumption on fas in part (a), show that
fe~(z) = [fe(z)]~ = fzz + fzz(e-^2 ;^6 + e-^2 iw) + fzze-^2 i(o+w)
both directional derivatives being taken along rectilinear paths.
(E. Kasner [67])- (a) Let O" = fzz + 2fzze-^2 i^6 + fzze-^4 iB and assume z fixed. Show that
as () varies from 0 to 27r, the point O" describes twice in the clockwise
direction a lima<;on with base circle 0"1 = fzz + fzze-^4 i^9 •
(b) If lfzzl = lfzzl, show that the lima<;on specializes into a cardioid.
( c) If f zz = O, the lima<;on specializes into a circle described four times
in the clockwise direction.- Let w = f(z) and z = g((), where f is differentiable at z, g differen-
tiable at (, and the range of g is contained in the domain of f. Prove
the chain rule for directional derivatives, namely,