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where is the duality pairing between and the vector Explicitly, if is a -form and is a -form, then
The above relation says that the interior product obeys a graded Leibniz rule. An operation satisfying linearity and a Leibniz rule is called a derivation.
Properties
If in local coordinates the vector field is given by
then the interior product is given by
where is the form obtained by omitting from .
By antisymmetry of forms,
and so This may be compared to the exterior derivative which has the property
The interior product relates the exterior derivative and Lie derivative of differential forms by the Cartan formula (also known as the Cartan identity, Cartan homotopy formula[2] or Cartan magic formula):