Download PDF by Philip Franklin: Differential and Integral Calculus

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O (3) F(e,N)~ The following commutative diagram establishes the connection between complexes (2) and (3): o~-F(&o )--V~F(~I)-V~F (a 2)~ o - - r(~o~). ~ F(&N)~ ... ~> ~ o -o Recall that complex (I) is called elliptical, if the corresponding symbolic complex is acyolio for any non-zero oovector ~ ~ ~ (eC~. The ellipticity of complex (I) is equivalent to the ellipticity of the dual complex (2) or the conjugate complex (3). This follows from the fact that the symboli~c complex of complex (2) is obtained from the symbolic complex (4) by applying the functor Horn( The proof of the first duality theorem, given below, is based on the following two facts: firstly, the basic result of Hodge-Spencer theory, which states that in each class of cohomologies of an elliptic complex given on a compact oriented Riemannian manifold there exists exactly one harmonic representative.

C. Overdetermined systems of linear partial differential equations - Bull. of the Amer. Math. , 1969, 75 " 2, p. 199-239. 2. G. Formal properties of overdetermined systems of linear partial differential equations, Thesis, Harvard University, Cambridge, Massachusetts, 1964. 3. V. Spencer cohomologies of differential equations. In: Leot. , 1990, vol. 1453, p. 121-136. 4. Goldschmidt H. Duality theorems in deformation theory, Transactions of the Amer. ~[ath. l-50. 5. L. Complex-foliated structures.

A natural pairing between the denote by St@ H o m ( ~ t , ~ t ) a duFor every morphism S~ Hom(~ ,J~) al morphism defined, as usually, by the relation: , ~ sa, bt 2 = ~ a, s t b t > where a ~ ~ (~), b t ~ [ (~t). Proposition I. (J~ t, ~ t ) . Consider a mapping ~ ( ~ , V] ) associating the pair (a, bt), a ~ F ( ~ ), b t ~ [ (j~t) with n-form < V ( a ) , b t ~ -. 45 This mapping is a first-order differential operator, acting from the bundle o Q @ ~ t into the bundle A n ~ " iff the equality + ) = o is satisfied for all covectors ~ 6 C (~')" If the conditions of the proposition are fulfilled, the symbol of the operator ~ ( V , F I) defines the homomorphism w v : ~ @ ~ t _ _ _ ~ ~n-1 ~ ' , acting in the following way: ~(~(~,~))(a{~b t) = ~ A ( V ) a , b t ~ = ~ A w v ( a , bt).

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