In mathematics and theoretical physics, an invariant differential operator is a mathematical map from some objects to an object of similar type. These objects are typically functions on, functions on a manifold, vector valued functions, vector fields, or, more generally, sections of a vector bundle.
In an invariant differential operator, the word differential indicates that the value of the image depends only on and the derivatives of in . The word invariant indicates that the operator contains some symmetry. This means that there is a group that has an action on the functions (or other objects in question) and this action commutes with the action of the operator:
Usually, the action of the group has the meaning of a change of coordinates (change of observer) and the invariance means that the operator has the same expression in all admissible coordinates.
Read more about Invariant Differential Operator: Invariance On Homogeneous Spaces, Invariance in Terms of Abstract Indices, Examples, Conformal Invariance
Famous quotes containing the word differential:
“But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
—Antonin Artaud (1896–1948)