Invariant differential operator

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a> and <a href="/facts/Theoretical_physics/Cs2Y12rQ">theoretical physics</a>, an invariant differential operator is a kind of <a href="/facts/Map_(mathematics)/FABF1l2f">mathematical map</a> from some objects to an object of similar type. These objects are typically <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> on 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
, functions on a <a href="/facts/Manifold/rXPERJtu">manifold</a>, <a href="/facts/Vector_(geometric)/bCLrdksy">vector</a> valued functions, <a href="/facts/Vector_field/ccFNuPPp">vector fields</a>, or, more generally, <a href="/facts/Section_(category_theory)/Ee1wexPw">sections</a> of a <a href="/facts/Vector_bundle/cXACAa1I">vector bundle</a>.
In an invariant differential operator 
 
 
 
 D
 
 
 {\displaystyle D}
 
, the term <a href="/facts/Differential_operator/Nr15J0IE">differential operator</a> indicates that the value 
 
 
 
 D
 f
 
 
 {\displaystyle Df}
 
 of the map depends only on 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 and the <a href="/facts/Derivative/7Ito70Dh">derivatives</a> of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 in 
 
 
 
 x
 
 
 {\displaystyle x}
 
. The word <a href="/facts/Invariant_(mathematics)/9TeNN2ed">invariant</a> indicates that the operator contains some <a href="/facts/Symmetry_in_mathematics/KtLgoqyX">symmetry</a>. This means that there is a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> 
 
 
 
 G
 
 
 {\displaystyle G}
 
 with a <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">group action</a> on the functions (or other objects in question) and this action is preserved by the operator:

D
        (
        g
        ⋅
        f
        )
        =
        g
        ⋅
        (
        D
        f
        )
        .
      
    
    {\displaystyle D(g\cdot f)=g\cdot (Df).}

Usually, the action of the group has the meaning of a <a href="/facts/Change_of_coordinates/Mtd0q4SP">change of coordinates</a> (change of observer) and the invariance means that the operator has the same expression in all admissible coordinates.

Invariant differential operator open-in-new

Invariant differential operator