Total differentiation operators as linear vector fields, their flows, invariants and symmetries form the geometry of jet space. In the jet space the dragging of tensor fields obeys the exponential law. The composition of smooth maps induces a composition of jets in corresponding jet spaces. The prolonged total differentiation operators generalize the differentiation of composite function. The relations between Cartan forms under the jet composition are described. MSC (2000): 58A20, 58A40, 28A15.