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Fields as algebraic objects. For vector and tensor fields, use eg. [dg.differential-geometry]. For physical fields, use eg. [mp.mathematical-physics] or [quantum-field-theory].
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How to prove this integral inequality in a 2-D region?
Let $\Omega$ be a 2D region. Now we have a partial differential equation system describing the characteristics of the region:
\begin{align*} \nabla \cdot (h_0^3 P_0 \nabla P_0) &= 0 \\ \nabla \cdot [ …