I have got a problem when trying to calculate the second term of the expansion of the vector potential $\boldsymbol A$. My textbook gives me a formula yet without proof.

The problem is to prove $$ \iiint (x^ij^k(\boldsymbol x)+ x^kj^i(\boldsymbol x))\,\mathrm d V =0, $$ where $x^i,x^j$ are coordinates and $j^k,j^i$ are components of a current density $\boldsymbol j$. The current density is stable and localized, i.e. vanishes when $\boldsymbol x$ is large enough.

My trial:

$$ x^ij^k(\boldsymbol x) = x^h\delta_h^ij^k(\boldsymbol x) = x^h\partial_h x^i j^k(\boldsymbol x) =\partial_h(x^h x^i j^k(\boldsymbol x)) - \partial_hx^hx^ij^k(\boldsymbol x) - x^hx^i\partial_h j^k(\boldsymbol x) $$

Take the integral over all space, the term $ \partial_h(x^h x^i j^k(\boldsymbol x))$ vanishes (Gauss' theorem), and $\partial_h x^h = \nabla\cdot \boldsymbol x = 3$: $$ 2\iiint x^ij^k(\boldsymbol x)\,\mathrm d V = \iiint x^hx^i\partial_h j^k(\boldsymbol x)\,\mathrm d V. $$

But I don't know what to do next or whether I find the wrong way or not.


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