1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 391

problems in mathematical physics that depend on the solution of equations

of the same type.

Other authors have considered the extension of analytic function theory

to two-dimensional surfaces or to spaces of dimension higher than two.

In what follows we describe briefly some of the theories that have been

proposed, referring the reader to the corresponding papers or books for

further details.

(a) In 1868, E. Beltrami [7] extended the Cauchy-Riemann equations to

any surface S defined in parametric form by three equations

x = fi(p, q), y = h(p, q), z = fa(p, q)

where the fk ( k = 1, 2, 3) are twice-differentiable functions of their

parameters. In vector notation we may write

r = xi+yj+zk

Then we have

ds^2 = E dp^2 + 2F dp dq + G dq^2

where

E = xP^2 + Yp^2 + zP^2 = rp • rp

F = XpXq + YpYq + ZpZq = rp · rq

G = xq^2 + Yq^2 + zq^2 = rq · rq

If u and v are functions of p and q such that

du^2 + dv^2 = >.(Edp^2 + 2Fdpdq + Gdq^2 )

where >. is independent of the differentials, Beltrami obtains, letting H =
(EG - F2)1/2,

du + idv = μ ( ../Edp + F ~H dq)

with μji = >., and it follows that

Up + ivp = μ../E

. F+iH

Uq + ZVq = μ VE

By eliminating μ, one obtains

E(uq + ivq) = (F + iH)(up + ivp)

which can be written in the form

1

Vp = H (Fup - Euq),

(6.24-1)

(6.24-2)