Differenze tra le versioni di "Operatore differenziale"

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{{TS|lingua=inglese|argomento=matematica|data=giugno 2006}}
In [[matematica]] un '''operatore differenziale''' è un [[operatore]] [[trasformazione lineare|lineare]] definito come una funzione dell'operatore [[derivata|differenziazione]].
==Aggiunto di un operatore==
Dato un operatore lineare differenziale:
: <math>Tu = \sum_{k=0}^n a_k(x) D^k u</math>
==Proprietà degli operatori differenziali==
Molte proprietà degli operatori differenziali sono conseguenza delle proprietà delle [[derivata|derivate]], che sono lineari
==Più variabili==
La stessa costruzione può essere usata con le [[derivata parziale|derivate parziali]].
==Descrizione indipendente dalle coordinate==
In [[geometria differenziale]] e in [[geometria algebrica]] è spesso conveniente avere una descrizione degli operatori indipendente dalle [[coordinata|coordinate]].
<!--In [[differential geometry]] and [[algebraic geometry]] it is often convenient to have a [[coordinate]]-independent description of differential operators between two [[vector bundle|vector bundles]]. Let ''E'' and ''F'' be two vector bundles over a [[manifold]] ''M''. An operator is a mapping of [[vector bundle|sections]], ''P'': &Gamma;(''E'') &rarr; &Gamma;(''F'') which maps the [[sheaf (mathematics)|stalk]] of the [[sheaf (mathematics)|sheaf]] of [[sheaf (mathematics)|germs]] of &Gamma;(''E'') at a point ''x'' &isin; ''M'' to the [[fibre bundle|fibre]] of ''F'' at ''x'':
:&Gamma;<sub>''x''</sub>(''E'') &rarr; ''F''<sub>''x''</sub> .
An operator ''P'' is said to be a '''''k''th order differential operator''' if it factors through the [[jet (mathematics)|jet bundle]] ''J''<sup>k</sup>(''E''). In other words, there exists a linear mapping of vector bundles
:''i''<sub>''P''</sub> : ''J''<sup>''k''</sup>(''E'') &rarr; ''F''
such that ''P'' = ''i''<sub>''P''</sub> o ''j''<sup>''k''</sup> as in the following composition:
:''P'' : &Gamma;<sub>''x''</sub>(''E'') &rarr; ''J''<sup>''k''</sup>(''E'')<sub>''x''</sub> &rarr; ''F''<sub>''x''</sub> .
A foundational result and characterization is the [[Peetre theorem]].
* In applications to the physical sciences, operators such as the [[Laplace operator]] play a major role in setting up and solving [[partial differential equation]]s.
* In [[differential topology]] the [[exterior derivative]] and [[Lie derivative]] operators have intrinsic meaning.
* In [[abstract algebra]], the concept of a [[derivation (abstract algebra)|derivation]] allows for generalizations of differential operators which do not require the use of calculus. Frequently such generalizations are employed in [[algebraic geometry]] and [[commutative algebra]]. See also [[jet (algebraic geometry)]].
==See also==
* [[Difference operator]]
* [[Delta operator]]
* [[Elliptic operator]]
* [[Fractional calculus]]
==Voci correlate==
* [[Operatore differenziale lineare]]
[[Categoria:Calcolo a più variabili]]
[[Category:Operatori differenziali]]
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