Operatore differenziale: differenze tra le versioni

Versione delle 09:40, 31 mag 2008

Versione delle 21:30, 9 lug 2008

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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]].

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==Aggiunto di un operatore==

Dato un operatore lineare differenziale:

: <math>Tu = \sum_{k=0}^n a_k(x) D^k u</math>

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==Proprietà degli operatori differenziali==

Molte proprietà degli operatori differenziali sono conseguenza delle proprietà delle [[derivata|derivate]], che sono lineari

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==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;_''x''(''E'') &rarr; ''F''_''x'' .

An operator ''P'' is said to be a '''''k''th order differential operator''' if it factors through the [[jet (mathematics)|jet bundle]] ''J''^k(''E''). In other words, there exists a linear mapping of vector bundles

:''i''_''P'' : ''J''^''k''(''E'') &rarr; ''F''

such that ''P'' = ''i''_''P'' o ''j''^''k'' as in the following composition:

:''P'' : &Gamma;_''x''(''E'') &rarr; ''J''^''k''(''E'')_''x'' &rarr; ''F''_''x'' .

A foundational result and characterization is the [[Peetre theorem]].

==Esempi==

* 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]]

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==Voci correlate==

* [[Operatore differenziale lineare]]

{{Portale|matematica}}

{{Portale|matematica}}

[[Categoria:Calcolo a più variabili]]

[[Category:Operatori differenziali]]