Let L be a forward operator and M be its adjoint defined using the inner product (f,Lg)=(Mf,g). Typically integration by parts is utilized when the inner product covers all the dependencies of M,L,f and g but what occurs when this is not the case. Here is my specific problem. (f,Lg)=∫∫dE dμ f(x,E,μ) μ dg(x,E,μ)/dx where L = μ d/dx Typically one integrates over dx as well and uses integration by parts to get M=-L. The question really becomes does integration by parts work if my derivative (the one I pick uv' in standard notation) is of a variable that is not integrated over.