Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 251second prolongation u (2) = pr (2) f(x, y) is given by [Olver (1986)]((u; u x , u y ; u xx , u xy , u yy ) = f; ∂f∂x , ∂f∂y ; ∂2 f∂x 2 , ∂ 2 )f∂x∂y , ∂2 f∂y 2 , (3.66)all evaluated at (x, y).The nth prolongation pr (n) f(x) is also known as the n−jet of f. In otherwords, the nth prolongation pr (n) f(x) represents the Taylor polynomial ofdegree n for f at the point x, since the derivatives of order ≤ n determinethe Taylor polynomial and vice versa.3.9.2.2 Prolongations of Differential EquationsA system S of nth order DEs in p independent and q dependent variablesis given as a system of equations [Olver (1986)]∆ r (x, u (n) ) = 0, (r = 1, ..., l), (3.67)involving x = (x 1 , ..., x p ), u = (u 1 , ..., u q ) and the derivatives ofu with respect to x up to order n. The functions ∆(x, u (n) ) =(∆ 1 (x, u (n) ), ..., ∆ l (x, u (n) )) are assumed to be smooth in their arguments,so ∆ : X ×U (n) → R l represents a smooth map from the jet space X ×U (n)to some lD Euclidean space (see section 4.14.12.5 below). The DEs themselvestell where the given map ∆ vanishes on the jet space X × U (n) , andthus determine a submanifold{}S ∆ = (x, u (n) ) : ∆(x, u (n) ) = 0 ⊂ X × U (n) (3.68)of the total the jet space X × U (n) .We can identify the system of DEs (3.67) with its corresponding submanifoldS ∆ (3.68). From this point of view, a smooth solution of the givensystem of DEs is a smooth function u = f(x) such that [Olver (1986)]∆ r (x, pr (n) f(x)) = 0,(r = 1, ..., l),whenever x lies in the domain of f. This is just a restatement of the factthat the derivatives ∂ J f α (x) of f must satisfy the algebraic constraintsimposed by the system of DEs. This condition is equivalent to the statementthat the graph of the prolongation pr (n) f(x) must lie entirely within thesubmanifold S ∆ determined by the system:Γ (n)f≡{(x, pr (n) f(x))}⊂ S ∆ ={}∆(x, u (n) ) = 0 .