Integration by parts and vector differential forms in higher order   variational calculus on fibred manifolds
  Infinitesimal variation of Action functional in classical (non-quantum) field theory with higher derivatives is presented in terms of well-defined intrinsic geometric objects independent of the particular field which varies. 'Integration by parts' procedure for this variation is then described in purely formal language and is shown to consist in application of nonlinear Green formula to the vertical differential of the Lagrangian. Euler-Lagrange expressions and the Green operator are calculated by simple pull-backs of certain vector bundle valued differential forms associated with the given variational problem. 
Roman Ya. Matsyuk
math.DG math.CA