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alt="StartLayout 1st Row 1st Column negative bold-script upper E dot StartFraction partial-differential bold-script upper D Over partial-differential t EndFraction 2nd Column equals minus one half bold-script upper E dot StartFraction partial-differential Over partial-differential t EndFraction left-parenthesis epsilon 0 bold-script upper E right-parenthesis minus one half bold-script upper E dot StartFraction partial-differential Over partial-differential t EndFraction bold-script upper P minus one half StartFraction partial-differential Over partial-differential t EndFraction bold-script upper E dot left-parenthesis epsilon 0 bold-script upper E right-parenthesis minus one half StartFraction partial-differential Over partial-differential t EndFraction left-parenthesis bold-script upper E dot bold-script upper P right-parenthesis 2nd Row 1st Column Blank 2nd Column minus one half bold-script upper E dot StartFraction partial-differential Over partial-differential t EndFraction bold-script upper P plus one half bold-script upper P dot StartFraction partial-differential Over partial-differential t EndFraction bold-script upper E comma EndLayout"/>
Grouping the first two, middle two, and last two terms of the right-hand side reformulates this relation as
(2.61)
which, using again , becomes
(2.62)
Finally, grouping the first two terms of the right-hand side of this relation yields
(2.63)
Similarly, the term in (2.57) becomes
(2.64)
Substituting now (2.63) and (2.64) into (2.57) finally yields the bianisotropic Poynting theorem:
(2.65)
where is the energy density, is the Poynting vector, and and are the impressed source power densities, and and are the induced polarization power densities, respectively, which are defined by
(2.66a)
(2.66b)
(2.66c)
(2.66d)
(2.66e)
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