ホッジ双対
d η = ( ∂ C ∂ y − ∂ B ∂ z ) d y ∧ d z + ( ∂ A ∂ z − ∂ C ∂ x ) d z ∧ d x + ( ∂ B ∂ x − ∂ A ∂ y ) d x ∧ d y {\displaystyle \mathrm {d} \eta =\left({\frac {\partial C}{\partial y}}-{\frac {\partial B}{\partial z}}\right)\mathrm {d} y\wedge \mathrm {d} z+\left({\frac {\partial A}{\partial z}}-{\frac {\partial C}{\partial x}}\right)\mathrm {d} z\wedge \mathrm {d} x+\left({\frac {\partial B}{\partial x}}-{\frac {\partial A}{\partial y}}\right)\mathrm {d} x\wedge \mathrm {d} y}
ホッジスター作用素を適用することは、次を意味する。
⋆ d η = ( ∂ C ∂ y − ∂ B ∂ z ) d x + ( ∂ A ∂ z − ∂ C ∂ x ) d y + ( ∂ B ∂ x − ∂ A ∂ y ) d z {\displaystyle \star \mathrm {d} \eta =\left({\frac {\partial C}{\partial y}}-{\frac {\partial B}{\partial z}}\right)\mathrm {d} x+\left({\frac {\partial A}{\partial z}}-{\frac {\partial C}{\partial x}}\right)\mathrm {d} y+\left({\frac {\partial B}{\partial x}}-{\frac {\partial A}{\partial y}}\right)\mathrm {d} z}