1550251515-Classical_Complex_Analysis__Gonzalez_

398 Chapter^6

and

Ux =A [Vy Wyl' Vx = A [ Wy Uy] , Wx =A [Uy Vy l

Vz Wz Wz Uz Uz Vz

Uy=,;\ [ Vz Wz], Vy = ,;\ [ Wz Uz] , Wy = ,\ [ Uz Vz l

Vx Wx Wx Ux Ux Vx

Uz =A [ Vx Wxl' Vz =A [ Wx Ux] , Wz =A [ Ux Vx l

Vy Wy Wy Uy Uy Vy

where .A = 1/VE. The last nine equations are the generalized Cauchy-

Riemann equations in ~^3 •

By using the equations in the first column above, one obtains

(VEux)x + (VEuy)y + (VEuz)z = 0
which is the generalization of the Laplace equation. The same equation is
also satisfied by v and by w.
The present author [51] has used the same approach to extend these
results to ~n (n 2:: 2). We consider in ~n a regular arc 1: [a, b] ~ ~n
defined by

a::::; u::::; b, i = 1, ... ,n

where the summation convention is used and { ei, ... , en} is the sta.ndard

orthonormal basis in ~n.

Let f = (!^1 , ... , Jn) be a vector function defined in some region n c ~n

containing 1*, with components Ji of class C(^1 )(U), and let

f('y) = r: f(x) = yi(u)ei,

be the image of I under f, so that

y i( u ) = Ji( x ,^1 ... ,x n)

for each i. Since

=

du oxi du

it follows that

I

()Ji I ..

J = Jr(x) = 8xi = la'J I =f. 0

at x = p is a sufficient condition for r to be also regular at the correspond-

ing point f(p ). In what follows it is assumed that the condition Jr(P) =f. 0
is satisfied.