ExplicitPartialD[exp] inserts the definitions for
LeftRightPartialD, LeftRightPartialD2,
LeftRightNablaD, LeftRightNablaD2,
LeftNablaD and RightNablaD
Overview, ExpandPartialD, LeftRightPartialD, LeftRightPartialD2, LeftRightNablaD, LeftRightNablaD2, LeftNablaD, RightNablaD.
LeftRightPartialD[\[Mu]]
ExplicitPartialD[%]\overleftrightarrow{\partial }_{\mu }
\frac{1}{2} \left(\vec{\partial }_{\mu }-\overleftarrow{\partial }_{\mu }\right)
LeftRightPartialD2[\[Mu]]
ExplicitPartialD[%]\overleftrightarrow{\partial }_{\mu }
\overleftarrow{\partial }_{\mu }+\vec{\partial }_{\mu }
LeftRightPartialD[OPEDelta]
ExplicitPartialD[%]\overleftrightarrow{\partial }_{\Delta }
\frac{1}{2} \left(\vec{\partial }_{\Delta }-\overleftarrow{\partial }_{\Delta }\right)
16 LeftRightPartialD[OPEDelta]^4
ExplicitPartialD[%]16 \overleftrightarrow{\partial }_{\Delta }^4
\left(\vec{\partial }_{\Delta }-\overleftarrow{\partial }_{\Delta }\right){}^4
Notice that by definition \nabla^i = \partial_i = - \partial^i, where the last equality depends on the metric signature.
LeftNablaD[i]
ExplicitPartialD[%]\overleftarrow{\nabla }^i
-\overleftarrow{\partial }_i
RightNablaD[i]
ExplicitPartialD[%]\vec{\nabla }^i
-\vec{\partial }_i
LeftRightNablaD[i]
ExplicitPartialD[%]\overleftrightarrow{\nabla }_i
\frac{1}{2} \overleftarrow{\partial }_i-\vec{\partial }_i
LeftRightNablaD2[\[Mu]]
ExplicitPartialD[%]\overleftrightarrow{\nabla }_{\mu }
-\overleftarrow{\partial }_{\mu }-\vec{\partial }_{\mu }