Magnetostatics: Explaining Jackson's Derivation from 5.20 to 5.21

[shehry1]

Can anyone explain to me how Jackson goes from equation 5.20 to 5.21 (Magnetostatics - Derivation of the curl of B in terms of the current density).

He says that he's used integration by parts but I can't see how he got rid of the first term (the one that involves integrals only) when integrating by parts.

Regards

 

[gabbagabbahey]

It looks like he uses the vector identity $$\mathbf{A} \cdot (\mathbf{\nabla}f)=\mathbf{\nabla} \cdot (f \mathbf{A})-f(\mathbf{\nabla}\cdot \mathbf{A})$$ with $$f=\frac{1}{|\mathbf{x}-\mathbf{x}'|}$$ and $$\mathbf{A}=\mathbf{J}(\mathbf{x}')$$

The $$\int \mathbf{\nabla}' \cdot \frac{\mathbf{J}(\mathbf{x}')}{|\mathbf{x}-\mathbf{x}'|} d^3x'$$ term can be transformed to a surface integral using the divergence theorem, and since the boundary of all space is at $$x' \to \infty$$ the integrand of that term is zero leaving you with equation 5.21