Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

354F λi :=n∑m∑k 1 =1 l 1 =1++g l 1k1 ,in∑∂∂g l 1k1+k 1 ,k 2 ,...,k λ−1 =1 l 1 =1m∑l 1 =1⎛g l 1 ⎝ im∑l 2 =1m∑f l1 ,l 2n∑m∑k 1 ,k 2 =1 l 1 =1g k1 ,k 2 ,...,k λ−1 ,i∂∂f l2++ · · · + ∑m∑l 2 ,l 3 =1g l 1k1 ,k 2 ,i∂∂g l + · · · +1k1 ,k 2∂∂g l +1k1 ,...,k λ−1f l1 ,l 2 ,l 3l 2 ,l 3 ,...,l λf l1 ,l 2 ,l 3 ,...,l λ∂∂f l2 ,l 3+⎞∂⎠,∂f l2 ,l 3 ,...,l λwe observe that the following induction relations hold:(5.19)h i1 ,i 2= F 2i 2(h i1 ),h i1 ,i 2 ,i 3= F 3i 3(h i1 ,i 2) ,. . . . . . . . . . . . . . . . . . . . .h i1 ,i 2 ,...,i λ= F λi λ(hi1 ,i 2 ,...,i λ−1).To obtain the explicit version of the Faà di Bruno in the case of severalvariables (x 1 , . . .,x n ) and several variables (y 1 , . . .,y m ), it suffices to extractfrom the expression of Y j i 1 ,...,i κprovided by Theorem 5.12 only theterms corresponding to µ 1 λ 1 + · · · + µ d λ d = κ, dropping all the X terms.After some simplifications and after a translation by means of an elementarydictionary, we obtain the fourth and the most general multivariate Faàdi Bruno formula.Theorem 5.20. For every integer κ 1 and for every choice of indicesi 1 , . . .,i κ in the set {1, 2, . . ., n}, the κ-th partial derivative of the compositefunction(5.21) h = h(x 1 , . . ., x n ) = f ( g 1 (x 1 , . . .,x n ), . . .,g m (x 1 , . . .,x n ) )