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By Choquet-Bruhat Y., DeWitt-Morette C., Dillard-Bleick M.

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X„+1)=0. Looking up the definition of we see this means that 1 = 0 and ^(xi, . . , 3c„)=0, as desired. =4| We have represented a diffeomorphism of 2B" onto the southern hemisphere of 25", taking the center of 2B" to the south pole of 25" and the boundary to the equator. The vector field ^ on 2 F ‘ is transformed into a tangent vector field on the northern hemisphere by the formula i(yiy . • , yn, yn+ l)= - l(yi 9• • . , y m -yn+l) for 1 ^ I ^ n 46 CH. 2, SEC. 3 SMOOTH ANALYSIS Of course, if a continuous vector field on 5" points inward on the boundary, (x, (J(x))<0 for ||x|| = 1, then it also has a zero, because —i* will point outward and (J has the same zeroes as —<^.

The map (13) 9'(л:)= л:+ /'( а:о) ' ‘( ;; - /( л:)) is a contraction of the closed ball Xo + sB into itself A Proof We have /(д :)= /—/'(дго)“ '/'(д:). It follows from lemma 1 that ||/( a:)||^A: on ло + еД so that g is к Lipschitz on that ball, with A:<1. All we have to prove is that g sends xo + гВ into itself. CH. 2, 1 ITERATIVE SEC. PROCEDURES FOR INVERTING A MAP Take any x such that ||x—xo|| 7, the sequence x„ generated by algorithm (7) converges geometrically to the one and only solution off{x) =y in the ball Xq+ sB (16) ||a:„—x|| < constant-k" Proof.

We shall investigate these properties for convex functions in Chapter 4 and for more general functions in Chapter 7. Now for some examples. Example 1 Let Q be an open subset of R", with the usual Lebesgue measure da>. Let there be given a function u: fixR ^-^R , borelian with respect to both variables (co, x). Assume that CH. 1, SEC. 4 DIFFERENTIABLE x) is FUNCTIONALS (5) for any fixed cü e Q, the function (6) there is some a e L^(£î, R) and some constant b such that 19 C* over R*“ I u'y{(o, y)| < a(co)+¿>1y I for ail y e R (7) I \u(a>,0)\dco))da> is well defined and Gateaux differentiable.

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Analysis, Manifolds and Physics. Basics by Choquet-Bruhat Y., DeWitt-Morette C., Dillard-Bleick M.

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