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

338Appying the chain rule, we may compute:(3.76)h i1 = f 1 [g i1 ] ,h i1 ,i 2= f 2 [g i1 g i2 ] + f 1 [g i1 ,i 2],h i1 ,i 2 ,i 3= f 3 [g i1 g i2 g i3 ] + f 2 [g i1 g i2 ,i 3+ g i2 g i1 ,i 3+ g i3 g i1 ,i 2] + f 1 [g i1 ,i 2 ,i 3],h i1 ,i 2 ,i 3 ,i 4= f 4 [g i1 g i2 g i3 g i4 ] + f 3 [g i2 g i3 g i1 ,i 4+ g i3 g i1 g i2 ,i 4+ g i1 g i2 g i3 ,i 4++g i1 g i4 g i2 ,i 3+ g i2 g i4 g i1 ,i 3+ g i3 g i4 g i1 ,i 2]++ f 2 [g i1 ,i 2g i3 ,i 4+ g i1 ,i 3g i2 ,i 4+ g i1 ,i 4g i2 ,i 3] ++ f 2 [g i1 g i2 ,i 3 ,i 4+ g i2 g i1 ,i 3 ,i 4+ g i3 g i1 ,i 2 ,i 4+ g i4 g i1 ,i 2 ,i 3] ++ f 1 [g i1 ,i 2 ,i 3 ,i 4] .Introducing the derivations(3.77)n∑Fi 2 := g k1 ,iF 3i :=k 1 =1n∑k 1 =1k 1 =1g k1 ,i(∂+ g i f 2∂g k1∂∂g k1+n∑k 1 ,k 2 =1)∂,∂f 1g k1 ,k 2 ,i(∂+ g i f 2∂g k1 ,k 2∂∂f 1+ f 3)∂,∂f 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .n∑Fi λ ∂n∑ ∂:= g k1 ,i + g k1 ,k∂g 2 ,i + · · · +k1 ∂g k1 ,k 2+n∑k 1 ,...,k λ−1 =1+ g i(f 2k 1 ,k 2 =1∂∂f 1+ f 3g k1 ,...,k λ−1 ,i∂∂g k1 ,...,k λ−1+∂∂f 2+ · · · + f λ)∂,∂f λ−1we observe that the following induction relations hold:(3.78)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 one variable y, it suffices to extract from the expressionof Y i1 ,...,i κprovided by Theorem 3.73 only the terms correspondingto µ 1 λ 1 + · · ·+µ d λ d = κ, dropping all the X terms. After some simplificationsand after a translation by means of an elementary dictionary, we obtaina statement.