sections of E to the bundle F. See also intrinsic differential operators. Example: Laplacian Consider the following operator: where and is the sphere centered at with radius . This is in fact the Laplacian. We show will show is a differential operator by Peetre's theorem. The main idea is that since is defined only in terms of 's behavior near , it is local in nature; in particular, if is locally zero, so is , and hence the support cannot grow. The technical proof goes as follows. Let and and be the rank trivial bundles. Then and are simply the space of smooth functions on . As a sheaf, is the set of smooth functions on the open set and restriction is function restriction. To see is indeed a morphism, we need to check for open sets and such that and . This is clear because for , both and are simply , as the eventually sits inside both and anyway. It is easy to check that is linear: and Finally, we check that is local in the sense that . If , then such that in the ball of radius centered at . Thus, for , for , and hence . Therefore, . So by Peetre's theorem, is a differential operator. References Peetre, J., Une caractérisation abstraite des opérateurs différentiels, Math. Scand. 7 (1959), 211-218. Peetre, J., Rectification à l'article Une caractérisation abstraite des opérateurs différentiels, Math. Scand. 8 (1960), 116-120. Terng, C.L., Natural vector bundles and natural differential operators, Am. J. Math. 100 (1978), 775-828. Articles containing proofs Differential operators Theorems in functional analysis