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$$\overrightarrow{\nabla}\times\overrightarrow{E} = -\frac{\partial \overrightarrow B}{\partial t}$$

When electromagnetic duality is concerned this rank 2 tensor kicks in:$$G^{\mu\nu}$$

And most of books and sites define $$D_i = G_{i0}$$ and $$H_i = 1/2 \epsilon_{ijk}G_{jk}$$

My question is why is this $$G^{\mu\nu}$$ related to our previously known electric induction D and magnetic intensity H?

In other words,how comes that $$G^{\mu\nu}$$ this new notion that rised after duality came in related to our old notion of electric induction D and magnetic intensity H?