§5. Differentiable functions of several variables
291
obvious solutions, namely J(x) = ±(1 - X2)~, are not defined on any nei9hbourhood of the point a = 1 - only for -1 ::; x ::; 1 - and, even worse, are
not differentiable at x = 1: their graphs, i.e. the upper and lower semicircles,
here again, have vertical tangents at this point. In this example, the situation
is on the contrary excellent at all points (a, b) where D2F(a, b) = 2b -=I- 0, i.e.
for -1 < a < 1. If b > 0, the formula
y = (1 - X2)t
defines a C l , and even Coo, function on ]-1,1[, which satisfies (1); if b < 0,
one changes the sign of y. It is thus prudent to work at a point (a, b) where
(24.3)
F(a,b) = 0,
simultaneously.
In the first case (existence of a local inverse of a map J from an open
subset G of C. into C), if one puts J = (11,12) with 11 and 12 real, one has
to show - modulo some hypotheses ... - that the map
(24.4)
(x,y) f---+ (l1(x,y),h(x,y))
of G into C. = JR.2 admits an inverse map
(u,v) f---+ 9(U,V) = (9l(U,V),92(U,V))
on an open neighbourhood U of a given point c = (a, b), and, precisely, that
(i) J maps U bijectivelyonto an open neighbourhood V of the point J(c),
(ii) the inverse map 9 : V ----+ U is C l , exactly as in the case of functions
of one real variable. If one assumes the problem solved, then the relation
9[J(Z)] = z shows, by the chain rule (21.13'), that
(24.5)
D9[J(Z)] 0 D J(z) = 1,
on U, where 1 = id is the identity map h 1--+ h, the derivative of the identity
map z 1--+ z. It is therefore necessary for D J (z) to be invertible at all z E U,
and in particular at the point c = (a, b) in question.
This condition can be made more explicit by replacing the linear maps
figuring in (5) with their Jacobian matrices:
(24.6)
or
(24.6')
D19l.Ddl + D29l.Dlh = 1, D192.Ddl + D292.Dd2 = 0,
(24.6") D19l.D211 + D29l.D2/2 = 0, D192.D2/l + D292.D2h = 1,
the derivatives of Ji being calculated at the point c and those of 9i at the
point J(c).