Constructing p, n-forms from p-forms via the Hodge star operator and the exterior derivative

Jun-Jin Peng

Table A2. Definitions for n-forms.

n-form Expression ${{\bf{U}}}^{{ik}}$ ${\hat{O}}^{i}{\bf{F}}\wedge \star {\tilde{O}}^{k}{\bf{H}}$ ${\check{{\bf{U}}}}_{{\rm{\Delta }}}^{i}$ ${\bf{F}}\wedge \star {{\rm{\Delta }}}^{i}{\bf{H}}$ ${\check{{\bf{U}}}}_{{\mathbb{P}}}^{k}$ ${\bf{F}}\wedge \star {{\mathbb{P}}}^{k}{\bf{H}}$ ${{\bf{U}}}_{{\rm{\Delta }}}^{{ik}}$ ${{\rm{\Delta }}}^{i}{\bf{F}}\wedge \star {{\rm{\Delta }}}^{k}{\bf{H}}$ ${{\bf{U}}}_{{\mathbb{P}}}^{{ik}}$ ${{\mathbb{P}}}^{i}{\bf{F}}\wedge \star {{\mathbb{P}}}^{k}{\bf{H}}$ ${{\bf{U}}}_{{\rm{\Delta }},{\mathbb{P}}}^{{ik}}$ ${{\rm{\Delta }}}^{i}{\bf{F}}\wedge \star {{\mathbb{P}}}^{k}{\bf{H}}$ ${{\bf{U}}}_{{\mathbb{P}},{\rm{\Delta }}}^{{ik}}$ ${{\mathbb{P}}}^{i}{\bf{F}}\wedge \star {{\rm{\Delta }}}^{k}{\bf{H}}$ ${{\bf{L}}}_{\hat{m}\tilde{n}}$ ${\sum }_{k=0}^{\tilde{n}}{\gamma }_{i}{\lambda }_{k}{{\bf{U}}}^{{ik}}$ ${\check{{\bf{L}}}}_{\check{m}\check{n}}$ ${\sum }_{i=0}^{\check{m}}{\rho }_{i}{\check{{\bf{U}}}}_{{\rm{\Delta }}}^{i}+{\sum }_{k=0}^{\check{n}}{\sigma }_{k}{\check{{\bf{U}}}}_{{\mathbb{P}}}^{k}$