$$-\iiint\mathbf{x}(\hat{\mathbf{n}}\cdot\mathbf{x})(\nabla\cdot\mathbf{J})\mathrm{d}^3x$$

$$-\iiint x^injx^j\partialkJ^k\mathrm{d}^3x\hat{\mathbf{e}}_i$$

$$\iiint\partialk(njx^ix^j)J^k\mathrm{d}^3x\hat{\mathbf{e}}_i$$

$$\iiint nj(\deltai^kx^j + \deltaj^kx^i)J^k\mathrm{d}^3x\hat{\mathbf{e}}i$$

$$\iiint nj\deltai^kx^jJ^k\mathrm{d}^3x\hat{\mathbf{e}}i + \iiint nj\deltaj^kx^iJ^k\mathrm{d}^3x\hat{\mathbf{e}}i$$

$$\iiint njx^jJ^i\mathrm{d}^3x\hat{\mathbf{e}}i + \iiint njx^iJ^j\mathrm{d}^3x\hat{\mathbf{e}}i$$

$$\iiint(njx^jJ^i\hat{\mathbf{e}}i + njJ^jx^i\hat{\mathbf{e}}i)\mathrm{d}^3x$$

$$\iiint\left[(\hat{\mathbf{n}}\cdot\mathbf{x})\mathbf{J} + (\hat{\mathbf{n}}\cdot\mathbf{J})\mathbf{x}\right]\mathrm{d}^3x$$