1550251515-Classical_Complex_Analysis__Gonzalez_

400 Chapter^6

Solving for aii, we get

.. p2

a'^1 = -yAi; (6.24-19)

where Aij is the cofactor of aii in J. Since

0

.~] = p2n

p2

we have p^2 = IJl^2 fn, and (6.24-19) becomes

(sgn J)IJl(n-2)/naij = Aij (6.24-20)

which are then-dimensional Cauchy-Riemann equations.
Another form of these equations can be obtained in the following
manner. Using (6.24-19), we find that

2

(ai1)2+···+(ain)2 = L(aiIAiI +···+ainA;n)

J

= p2

and for i f::. j,

il ·1 in "n P

2

a a^3 +···+a a^3 = -y(a il A;1+···+a in A;n)

=0

Since '\lfi = aiie;, equation (6.24-21) can be written as

(i=l, ... ,n)

and (6.24-22) as

(i f::. j)

(6.24-21)

(6.24-22)

Hence the mapping defined by a function f = (!1, ... r) of class c<^1 )(U)
at a point p where J f::. 0 is conformal iff all the components functions have
gradients with the same magnitude at that point and the gradients of any
two different components are orthogonal. The common magnitude of the
gradients is precisely the magnification ratio at p, and the second condition
means that the hypersurfaces Ji = ci meet orthogonally at p. Assuming

that the components Ji are of class C(^2 )(U), one obtains from (6.24-20)

~ ~ (IJl(n-2)/n BJ~)= O

L... 8xJ 8xJ

j=l '