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(Invariant local form theorem for n.d.o.). Let D(M): C?(V(M))--C?(W(M)) be a n.d.o. of order k. Then D looks the same in every coordinate system, i.e., there exists a map P: E V- W lal (k a n-tuple such that locally D(f)(x)>P(Djf(x)) under every local coordinate system.

ding T9:R'--R'. Then there exists a unique n.d.o. D(M):C'(Vi(M)) -*C??(V2(M)) such that D(R n) = p. Proof. Given an n-manifold M, we construct D(M): C??(V1(M)) C ??( V2(M )) as follows: Suppose s E C ( V1(M )), and Tp: R n ->UCM is a chart. Define D(M)(s)rU=(9(-l)*P(T9*s). The assumption on P implies that this gives a well-defined n.d.o. with the required property

Suppose V1(M) and V2(M) are two n.v.b. and P:C??(V,(Rn)) ->C (V2(Rn)) is a differential operator such that Tp*P= Pap* for every embed-

consider the linear space C?(F(M)) of all smooth sections of F(M), and for an embedding p:M-*N there is an induced map p*:C?(F(N))*C??(F(M)) given by qg*s= F()-'osoT. Thus M-C??(F(M)) is a functor from 9ln to the category of real vector spaces. Definition 0.3. A natural differential operator (n.d.o.) from one n.v.b. F1 to another n.v.b. F2 is a family of differential operators {D(M):C??(Fi(M)) -> C?(F2(M)), M an n-manifold} such that (P*D(N)= D(M)q* for every embedding 9p: M-*N.