Poyntings theorem and energy inequality for an anomalous field

The energy flow F of the anomalous field outside the domain V2 with the anomalous domain D, is also always non-negative, 

The proof of this formula is similar to the proof of formula (8.111). Based on formulae (9.6) and (9.7), we can obtain, after some algebraic transformations, the following important energy inequality, derived originally by Singer (1995) and Pankratov et al. (1995)  

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