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$$-\iiint\mathbf{x}(\hat{\mathbf{n}}\cdot\mathbf{x})(\nabla\cdot\mathbf{J})\mathrm{d}^3x$$ $$-\iiint x^in_jx^j\partial_kJ^k\mathrm{d}^3x\hat{\mathbf{e}}_i$$ $$\iiint\partial_k(n_jx^ix^j)J^k\mathrm{d}^3x\hat{\mathbf{e}}_i$$ $$\iiint n_j(\delta_i^kx^j + \delta_j^kx^i)J^k\mathrm{d}^3x\hat{\mathbf{e}}_i$$ $$\iiint n_j\delta_i^kx^jJ^k\mathrm{d}^3x\hat{\mathbf{e}}_i + \iiint n_j\delta_j^kx^iJ^k\mathrm{d}^3x\hat{\mathbf{e}}_i$$ $$\iiint n_jx^jJ^i\mathrm{d}^3x\hat{\mathbf{e}}_i + \iiint n_jx^iJ^j\mathrm{d}^3x\hat{\mathbf{e}}_i$$ $$\iiint(n_jx^jJ^i\hat{\mathbf{e}}_i + n_jJ^jx^i\hat{\mathbf{e}}_i)\mathrm{d}^3x$$ $$\iiint\left[(\hat{\mathbf{n}}\cdot\mathbf{x})\mathbf{J} + (\hat{\mathbf{n}}\cdot\mathbf{J})\mathbf{x}\right]\mathrm{d}^3x$$
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Tue, 29 Dec 2020 18:52 GMT