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

Then thanks to a straightforward application of the rule of Cramer for 2 × 2linear systems, we derive six useful formulas.Lemma. ([19], p. 9) All the six first order derivatives A x , A y , A yx , B x ,B y , B yx of the two functions A and B with respect to their three arguments(x, y, y x ) may be expressed as follows in terms of the second jet J 2 (Q) ofthe defining function Q:A x = Q b Q xx − Q x Q xbQ a Q xb − Q b Q xa,Q xbA y =,Q a Q xb − Q b Q xaB y =−Q bA yx =,Q a Q xb − Q b Q xaB yx =B x = Q x Q xa − Q a Q xx,Q a Q xb − Q b Q xa−Q xa,Q a Q xb − Q b Q xaQ aQ a Q xb − Q b Q xa.For future abbreviation, we shall denote the single appearing denominator— which evidently is the common determinant of all the three 2 × 2linear systems involved above — simply by a square symbol:∆ := Q a Q xb − Q b Q xa .The two-ways transfer between functions G defined in the (x, y, y x )-spaceand functions T defined in the (x, a, b)-space, namely the one-to-one correspondence:G(x, y, y x ) ←→ T(x, a, b)may be read very concretely as the following two equivalent identities:G(x, y, y x ) ≡ T ( x, A(x, y, y x ), B(x, y, y x ) )G ( x, Q(x, a, b), Q x (x, a, b) ) ≡ T(x, a, b),holding in K{x, y, y x } and in K{x, a, b} respectively. By differentiatingthe first identity, the chain rule shows how the three first-order derivationoperators (basic vector fields) ∂ x , ∂ y and ∂ yx living in the (x, y, y x )-spaceare transformed into the (x, a, b)-space:∂∂x = ∂ ( ) ( )∂x + Qb Q xx − Q x Q xb ∂∆ ∂a + Qx Q xa − Q a Q xx ∂∆ ∂b( ) (Qxb ∂∂a + −Qxa∂∂y =∂∂y x=) ∂∆∆ ∂b( ) ( ) −Qb ∂∆ ∂a + Qa ∂∆ ∂b .Lemma. The total differentiation operator D = ∂ x +y x ∂ y +F ∂ yx associatedto y xx = F(x, y, y x ) simply transfers to the basic derivation operator alongthe x-direction:D ←→ ∂ x .207