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

43remaining terms afterwards:(3.71) ⎧⎪⎨0 = A x + B y +⎪⎩+ A 1− B a+ 2C b− 1 3 D 2− 2 3 A 1+ 1 6 D 2+ E 3+ B a− 2C b.Here, each remaining term is also underlined, with a number or with a letterappended. For reasons of typographical readability, we never underline thesign, + or − of each term; however, it should be understood that every termalways includes its (not underlined) sign. Until the end of Section 4, we shalluse the roman alphabetic letters a, b, c, etc. inside an octagon to exhibitthe vanishing terms. As readily checked by the eyes, we indeed have −B a+B a= 0 and 2 C b−2 C b= 0. Also, until the end of Section 4, we shall usethe numbers 1, 2, 3, etc. inside a square □ to exhibit the remaining terms,collected in a certain order. The numbers have the following signification:after the simplifications, the equation (3.71) may be written⎧⎨ 0 = A x + B y +(3.72)⎩ + 1 3 A − 1 6 D + E.Here, the plainly underlined terms A x + B y do not count in the numbering(their number is zero, for instance) and the first term of the second line 1 A 3correspond to the addition of all terms 1 in (3.69). Analogously, the secondterm − 1 D correspond to the addition of all terms 2 in (3.69). Again, this6guiding facilitates the checking of the correctness of the computation, usingsimply the eyes. No hidden delicate computational step is “left to the reader”for the convenience of the writer.This principle will be constantly used until the end of Section 4; it hasbeen systematically used in [M2004] and it could be applied in various othercontexts. Again, the advantage is that it enables to check the correctness ofall the formal computations just by reading, without having to write anythingmore. This is also useful for the author.3.73. Choice of an ordering. Until the end of Section 4, we shall have todeal with terms G, H, L, M, Θ together with indices and partial derivativesup to order two. In order to organize the formal expressions in a waywhich provides an easier deciphering, it is convenient to introduce an orderbetween these differential monomials. In a symbolic index-free notation, wechoose:(3.74) G < H < L < M < Θ.It follows for instance that G < GH < GL < HHL < HLMΘ. Also, if asum appears, we choose: GM < ∑ GM.