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pherytic

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I have been reading through Zangwill's Modern Electrodynamics all on my own, and I've just joined here hoping I can post some questions that come up for me. To start, I am confused about something in section 6.7.1, concerning the variation of total energy U of a dielectric in the presence of a charged conductor. This is given by (6.87)

$$\delta U = \int d^3 r \, \vec E \cdot \delta \vec D$$

where E is the total electric field, D is the auxiliary/displacement field.

Then, the books says (6.93)

$$ \vec E = 1/V(∂U/∂ \vec D)$$

I understand (ignoring any center of mass dependence) that using the logic of total differentials I can write

$$\delta U = (∂U/∂ \vec D) \cdot \delta \vec D$$

So it follows that

$$\int d^3 r \, \vec E \cdot \delta \vec D = (∂U/∂ \vec D) \cdot \delta \vec D$$

But the given equation for E only seems valid if E and D are constant over the volume, which isn't generally true. What am I misunderstanding? How does the equation for E follow?

Thanks for any guidance.