Interior differential operator
with respect to a surface
A differential operator such that for any function for which it is defined its value at a point
can be calculated from only the values of this function on the smooth surface
defined in the space
,
. An interior differential operator can be computed using derivatives in directions
which lie in the tangent space to
. If one introduces coordinates such that on
,
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then the operator , provided it is interior with respect to
, will not contain, after suitable transformations, derivatives with respect to the variables
(the so-called exterior or extrinsic derivatives). For instance, the operator
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is an interior differential operator with respect to any smooth surface containing a straight line , and with respect to any one of these lines. If the operator
is an interior differential operator with respect to a surface
, then
is said to be a characteristic of the differential equation
.
An operator is sometimes called interior with respect to a surface if, at the points of this surface, the leading order of the extrinsic derivatives is lower than the order of the operator.
Interior differential operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interior_differential_operator&oldid=33241