 In the theory of differential algebra, a differential ideal "I" in a differential ring "R" is an ideal which is mapped to itself by each differential operator.An exterior differential system consists of a smooth manifold formula_1 and a differential ideal

An integral manifold of an exterior differential system formula_3 consists of a submanifold formula_4 having the property that the pullback to formula_5 of all differential forms contained in formula_6 vanishes identically.One can express any partial differential equation system as an exterior differential system with independence condition.