We can express the formula for integration by parts in Cartesian index notation as ∫ ∫ Ω Ω F i j k . . . . G l m n . . . , p d Ω Ω = ∫ ∫ Γ Γ n p F i j k . . . G l m n . . . d Γ Γ − − ∫ ∫ Ω Ω G l m n . . . F i j k . . . , p d Ω Ω .