Poynting theorem in a linear anisotropic medium

For symmetric time-independent constitutive tensors,

u = \frac{1}{2} (ε_{ij} E_{i} E_{j} + μ_{ij} H_{i} H_{j}), S = E \times H,

and the macroscopic Maxwell equations give

\partial_{t} u + \nabla \cdot S = - E \cdot J

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