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

228are the involved partial derivatives of the fundamental function Q that appearinside each column. In summary, □ µ [0 1+l ]comes from □ by changingjust its µ-th column, as Cramer’s rule classically says.Next, the two-ways transfer between local functions G defined in the(x, y, y x )-space and local functions T defined in the (x, a, b)-space, namelythe one-to-one correspondence:G ( ) (x 1 , . . ., x n , y, y x 1, . . .,y x n ←→ T x 1 , . . .,x n , a 1 , . . ., a n , b )through the diffeomorphism (7.28), may be viewed concretely, in the directionwe are interested in, as the following identity:G ( x 1 , . . .,x n , y, y x 1, . . .,y x n)≡≡ T ( x 1 , . . .,x n ,A 1( x 1 , . . ., x n , y, y x 1, . . .,y x n) ( , . . .,Anx 1 , . . .,x n , y, y x 1, . . .,y x n),A n+1( ))x 1 , . . .,x n , y, y x 1, . . .,y x nholding of course in C { x 1 , . . .,x n , y, y x 1, . . ., y x n}. We therefore readilydeduce how the derivation is transferred to the (x, a, b)-space:∂G∂y x l∂∂y x l= ∂A1 · ∂T∂y x l∂a + · · · + ∂An · ∂T1 ∂y x l∂a + ∂T∂An+1 ·n ∂y x l∂a n+1.By applying twice any two such derivations ∂ / ∂y x l 1 and ∂ / ∂y x l 2 to an arbitraryfunction G, we may see, after a few computations, what such a composeddifferentiation corresponds to, in terms of the function T defined inthe (x, a, b)-space:∂ 2 G∂y x l 1 ∂y x l 2==( n+1 ∑µ=1∑n+1µ=1∑n+1ν=1∂A µ∂y xl 1∂A µ∂y xl 1)[∂ ∑n+1∂a µ ν=1∂A ν∂y xl 2]∂A∂T ν∂y xl 2 ∂a ν∂ 2 T∂a µ ∂a + ∑n+1νµ=1∑n+1ν=1∂A µ∂y xl 1[∂ ∂A ν ] ∂T∂a µ ∂y xl 2 ∂a . νHere, by a helpful formal convention, the three Greek letters µ, ν and τ willbe used as summation indices in the total set {1, . . ., n, n+1}, while the fourLatin letters i, j, k, l will always run in the restricted set {1, . . ., n}. Replacingthen the partial derivatives of the A µ by their values (7.28) obtainedpreviously, we thus get:∂ 2 n+1G ∑=∂y x l 1 ∂y x l 2µ=1n+1∑ν=1n+1∑+n+1∑µ=1 ν=1□ µ [0 1+l1 ]□ ν [0 1+l2 ]∂ 2 T∂a µ ∂a ν +□ □{ □µ[0 1+l1 ]· □ · (∂∂a □ν µ [01+l2 ]□)− □ν[01+l2 ] · ( )∂}∂a □ µ□ · □∂T∂a ν